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<Startup-Code startupcode="with(plots): with(plottools): with(Statistics):\012LMan := PLOT(POLYGONS([[-4.3, 11.6], [-5.0, 6.6], [-6.0, 1.6], [-6.8, 0.], [-2\134\012.6, 0.], [-2.6, 1.1], [-4.6, 1.6], [-2.4, 6.6], [-.7, 11.6]],COLOUR(RGB,0.,1.0\134\0120000000,1.00000000)),POLYGONS([[1.90673726, 14.45778812], [2.08536837, 15.2529\134\0128071], [2.21534137, 16.04733776], [2.29460653, 16.82833178], [2.32191376, 17.5\134\0128364605], [2.29683243, 18.30136879], [2.21975809, 18.97018109], [2.09190624, \01219.57953538], [1.91529318, 20.11982178], [1.69270420, 20.58251966], [1.4276496\134\0127, 20.96033199], [1.124309657, 21.24730044], [.787468009, 21.43889934], [.4224\134\01236925, 21.53210708], [.34973159e-1, 21.52545371], [-.368812751, 21.41904415],\012[-.782552865, 21.21455656], [-1.199722247, 20.91521581], [-1.613741889, 20.525\134\01274269], [-2.018082456, 20.05227944], [-2.406367255, 19.50229283], [-2.77247280\134\0120, 18.88445650], [-3.110625393, 18.20851409], [-3.415492160, 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[2.38, 6.6], [.58, 11.6]],COLOUR(\012RGB,0.,1.00000000,1.00000000)),POLYGONS([[1.4, 21.], [1.4, 22.], [1.4, 23.], [\0122.4, 22.9], [3.0, 23.], [3.1, 23.1], [3.0, 23.6], [3.15, 23.9], [3.1, 24.], [3\134\012.15, 24.1], [3.1, 24.5], [3.5, 24.45], [3.6, 24.5], [3.5, 24.8], [3.1, 25.2],\012[3.07, 25.5], [3.2, 25.7], [3.24, 26.1], [3.0, 27.], [2.4, 27.6], [1.4, 27.9],\012[.4, 27.9], [-.6, 27.5], [-1.0, 27.], [-1.3, 26.], [-1.4, 25.], [-1.1, 24.], [\012-.8, 23.], [-.8, 22.], [-1.1, 21.]],COLOUR(RGB,1.00000000,.62745098,.47843137)\012),CURVES([[1.4, 21.], [1.4, 22.], [1.4, 23.], [2.4, 22.90000000], [3.00000000,\01223.], [3.10000000, 23.10000000], [3.00000000, 23.60000000], [3.15000000, 23.90\134\012000000], [3.10000000, 24.], [3.15000000, 24.10000000], [3.10000000, 24.5000000\134\0120], [3.50000000, 24.45000000], [3.60000000, 24.50000000], [3.50000000, 24.8000\134\0120000], [3.10000000, 25.20000000], [3.07000000, 25.50000000], [3.20000000, 25.7\134\0120000000], [3.24000000, 26.10000000], [3.00000000, 27.], [2.4, 27.60000000], [1\134\012.4, 27.90000000], [.4, 27.90000000], [-.6, 27.50000000], [-1.000000000, 27.],\012[-1.300000000, 26.], [-1.400000000, 25.], [-1.100000000, 24.], [-.800000000, \01223.], [-.800000000, 22.], [-1.100000000, 21.]],COLOUR(RGB,0.,0.,0.)),CURVES([[\0122.90000000, 25.50000000], [2.60000000, 25.30000000], [2.90000000, 25.20000000]\012],COLOUR(RGB,0.,0.,0.)),CURVES([[.700000000, 25.20000000], [.600000000, 25.500\134\01200000], [.4, 25.60000000], [.200000000, 25.50000000], [0., 25.], [.200000000,\01224.30000000], [.4, 24.10000000]],COLOUR(RGB,0.,0.,0.)));\012RMan := PLOT(POLYGONS([[4.3, 11.60000000], [5.0, 6.600000000], [6.0, 1.6000000\134\01200], [6.8, 0.], [2.6, 0.], [2.6, 1.100000000], [4.6, 1.600000000], [2.4, 6.600\134\012000000], [.7, 11.60000000]],COLOUR(RGB,0.,1.00000000,1.00000000)),POLYGONS([[-\134\0121.90673726, 14.45778812], [-2.08536837, 15.25298071], [-2.21534137, 16.0473377\134\0126], [-2.29460653, 16.82833178], [-2.32191376, 17.58364605], [-2.29683243, 18.3\134\0120136879], [-2.21975809, 18.97018109], [-2.09190624, 19.57953538], [-1.91529318\012, 20.11982178], [-1.69270420, 20.58251966], [-1.42764967, 20.96033199], [-1.12\134\0124309657, 21.24730044], [-.787468009, 21.43889934], [-.422436925, 21.53210708],\012[-.34973159e-1, 21.52545371], [.368812751, 21.41904415], [.782552865, 21.21455\134\012656], [1.199722247, 20.91521581], [1.613741889, 20.52574269], [2.018082456, 20\134\012.05227944], [2.406367255, 19.50229283], [2.772472800, 18.88445650], [3.1106253\134\01293, 18.20851409], [3.415492160, 17.48512561], [3.682265176, 16.72569934], [3.9\134\01206737265, 15.94221188], [4.085368372, 15.14701929], [4.215341376, 14.35266224]\012, [4.294606527, 13.57166822], [4.321913764, 12.81635395], [4.296832437, 12.098\134\01263121], [4.219758092, 11.42981891], [4.091906239, 10.82046462], [3.915293179,\01210.28017822], [3.692704203, 9.817480340], [3.427649671, 9.439668012], [3.12430\134\0129657, 9.152699564], [2.787468009, 8.961100655], [2.422436925, 8.867892917], [2\134\012.034973159, 8.874546290], [1.631187249, 8.980955846], [1.217447135, 9.18544344\134\0124], [.800277753, 9.484784191], [.386258111, 9.874257306], [-.18082456e-1, 10.3\134\0124772056], [-.406367252, 10.89770717], [-.772472800, 11.51554350], [-1.11062539\134\0125, 12.19148591], [-1.41549216, 12.91487439], [-1.68226517, 13.67430066], [-1.9\134\0120673726, 14.45778812]],COLOUR(RGB,0.,1.00000000,1.00000000)),POLYGONS([[3.02,\01211.60000000], [.22, 6.600000000], [-2.28, 1.600000000], [-2.6, 0.], [-6.8, 0.]\012, [-6.03, 1.100000000], [-3.68, 1.600000000], [-2.38, 6.600000000], [-.58, 11.\134\01260000000]],COLOUR(RGB,0.,1.00000000,1.00000000)),POLYGONS([[-1.4, 21.00000000]\012, [-1.4, 22.00000000], [-1.4, 23.00000000], [-2.4, 22.90000000], [-3.0, 23.000\134\01200000], [-3.1, 23.10000000], [-3.0, 23.60000000], [-3.15, 23.90000000], [-3.1,\01224.00000000], [-3.15, 24.10000000], [-3.1, 24.50000000], [-3.5, 24.45000000],\012[-3.6, 24.50000000], [-3.5, 24.80000000], [-3.1, 25.20000000], [-3.07, 25.5000\134\0120000], [-3.2, 25.70000000], [-3.24, 26.10000000], [-3.0, 27.00000000], [-2.4,\01227.60000000], [-1.4, 27.90000000], [-.4, 27.90000000], [.6, 27.50000000], [1.0\012, 27.00000000], [1.3, 26.00000000], [1.4, 25.00000000], [1.1, 24.00000000], [.\134\0128, 23.00000000], [.8, 22.00000000], [1.1, 21.00000000]],COLOUR(RGB,1.00000000,\012.62745098,.47843137)),CURVES([[-1.4, 21.00000000], [-1.4, 22.00000000], [-1.4,\01223.00000000], [-2.4, 22.90000000], [-3.00000000, 23.00000000], [-3.10000000, \01223.10000000], [-3.00000000, 23.60000000], [-3.15000000, 23.90000000], [-3.1000\134\0120000, 24.00000000], [-3.15000000, 24.10000000], [-3.10000000, 24.50000000], [-\134\0123.50000000, 24.45000000], [-3.60000000, 24.50000000], [-3.50000000, 24.8000000\134\0120], [-3.10000000, 25.20000000], [-3.07000000, 25.50000000], [-3.20000000, 25.7\134\0120000000], [-3.24000000, 26.10000000], [-3.00000000, 27.00000000], [-2.4, 27.60\134\012000000], [-1.4, 27.90000000], [-.4, 27.90000000], [.6, 27.50000000], [1.000000\134\012000, 27.00000000], [1.300000000, 26.00000000], [1.400000000, 25.00000000], [1.\134\012100000000, 24.00000000], [.800000000, 23.00000000], [.800000000, 22.00000000],\012[1.100000000, 21.00000000]],COLOUR(RGB,0.,0.,0.)),CURVES([[-2.90000000, 25.500\134\01200000], [-2.60000000, 25.30000000], [-2.90000000, 25.20000000]],COLOUR(RGB,0.,\0120.,0.)),CURVES([[-.700000000, 25.20000000], [-.600000000, 25.50000000], [-.4,\01225.60000000], [-.200000000, 25.50000000], [-0., 25.00000000], [-.200000000, 24\134\012.30000000], [-.4, 24.10000000]],COLOUR(RGB,0.,0.,0.)));\012LArm := PLOT(POLYGONS([[1.3, 19.], [1.289749112, 19.16293320], [1.259158109, \01219.32329685], [1.208709432, 19.47856192], [1.139198684, 19.62627978], [1.05172\134\0122093, 19.76412083], [.947659216, 19.88991124], [.828651186, 20.00166722], [.69\134\0126574834, 20.09762630], [.553513079, 20.17627517], [.401722092, 20.23637347], [\012.243595708, 20.27697343], [.81627675e-1, 20.29743475], [-.81627676e-1, 20.2974\134\0123475], [-.243595709, 20.27697343], [-.401722092, 20.23637347], [-.553513079, \01220.17627517], [-.696574833, 20.09762630], [-.828651186, 20.00166722], [-.94765\134\0129217, 19.88991124], [-1.051722094, 19.76412083], [-1.139198685, 19.62627978],\012[-1.208709432, 19.47856192], [-1.259158110, 19.32329685], [-1.289749112, 19.16\134\012293320], [-1.3, 19.00000000], [-1.289749112, 18.83706680], [-1.259158109, 18.6\134\0127670315], [-1.208709432, 18.52143808], [-1.139198684, 18.37372022], [-1.051722\134\012093, 18.23587917], [-.947659216, 18.11008876], [-.828651185, 17.99833278], [-.\134\012696574832, 17.90237370], [-.553513078, 17.82372483], [-.401722091, 17.76362653\012], [-.243595708, 17.72302657], [-.81627674e-1, 17.70256525], [.81627676e-1, 17\134\012.70256525], [.243595710, 17.72302657], [.401722093, 17.76362653], [.553513079,\01217.82372483], [.696574834, 17.90237370], [.828651186, 17.99833278], [.94765921\134\0127, 18.11008876], [1.051722094, 18.23587917], [1.139198685, 18.37372023], [1.20\134\0128709432, 18.52143808], [1.259158110, 18.67670315], [1.289749112, 18.83706680],\012[1.3, 19.00000000]],COLOUR(RGB,0.,1.00000000,1.00000000)),POLYGONS([[.4, 20.1]\012, [-1.3, 18.5], [1.4, 13.2], [6.4, 15.2], [5.9, 16.5], [2.4, 15.8]],COLOUR(RGB\012,0.,1.00000000,1.00000000)),POLYGONS([[8.7, 16.1], [8.68817205, 16.28799985],\012[8.65287474, 16.47303483], [8.59466473, 16.65218683], [8.51446002, 16.82263051\012], [8.41352549, 16.98167788], [8.29345294, 17.12682066], [8.15613598, 17.25576\134\012986], [8.00374019, 17.36649189], [7.83866894, 17.45724058], [7.66352549, 17.52\134\012658478], [7.48107197, 17.57343088], [7.29418578, 17.59704009], [7.10581422, 17\134\012.59704009], [6.91892803, 17.57343088], [6.73647451, 17.52658477], [6.56133106,\01217.45724058], [6.39625981, 17.36649189], [6.24386402, 17.25576986], [6.1065470\134\0126, 17.12682066], [5.98647451, 16.98167788], [5.88553998, 16.82263051], [5.8053\134\0123527, 16.65218683], [5.74712526, 16.47303483], [5.71182795, 16.28799985], [5.7\012, 16.10000000], [5.71182795, 15.91200015], [5.74712526, 15.72696517], [5.80533\134\012527, 15.54781317], [5.88553998, 15.37736949], [5.98647451, 15.21832212], [6.10\134\012654706, 15.07317934], [6.24386402, 14.94423013], [6.39625981, 14.83350811], [6\134\012.56133106, 14.74275942], [6.73647451, 14.67341522], [6.91892803, 14.62656912],\012[7.10581422, 14.60295991], [7.29418578, 14.60295991], [7.48107197, 14.62656912\012], [7.66352549, 14.67341523], [7.83866894, 14.74275942], [8.00374019, 14.83350\134\012811], [8.15613598, 14.94423014], [8.29345294, 15.07317934], [8.41352549, 15.21\134\012832212], [8.51446002, 15.37736949], [8.59466473, 15.54781317], [8.65287474, 15\134\012.72696517], [8.68817205, 15.91200015], [8.7, 16.10000000]],COLOUR(RGB,0.,0.,0.\012)));\012RArm := PLOT(POLYGONS([[-1.3, 19.00000000], [-1.289749112, 19.16293320], [-1.2\134\01259158109, 19.32329685], [-1.208709432, 19.47856192], [-1.139198684, 19.6262797\134\0128], [-1.051722093, 19.76412083], [-.947659216, 19.88991124], [-.828651186, 20.\134\01200166722], [-.696574834, 20.09762630], [-.553513079, 20.17627517], [-.40172209\134\0122, 20.23637347], [-.243595708, 20.27697343], [-.81627675e-1, 20.29743475], [.\134\01281627676e-1, 20.29743475], [.243595709, 20.27697343], [.401722092, 20.23637347\012], [.553513079, 20.17627517], [.696574833, 20.09762630], [.828651186, 20.00166\134\012722], [.947659217, 19.88991124], [1.051722094, 19.76412083], [1.139198685, 19.\134\01262627978], [1.208709432, 19.47856192], [1.259158110, 19.32329685], [1.28974911\134\0122, 19.16293320], [1.3, 19.00000000], [1.289749112, 18.83706680], [1.259158109,\01218.67670315], [1.208709432, 18.52143808], [1.139198684, 18.37372022], [1.05172\134\0122093, 18.23587917], [.947659216, 18.11008876], [.828651185, 17.99833278], [.69\134\0126574832, 17.90237370], [.553513078, 17.82372483], [.401722091, 17.76362653], [\012.243595708, 17.72302657], [.81627674e-1, 17.70256525], [-.81627676e-1, 17.7025\134\0126525], [-.243595710, 17.72302657], [-.401722093, 17.76362653], [-.553513079, \01217.82372483], [-.696574834, 17.90237370], [-.828651186, 17.99833278], [-.94765\134\0129217, 18.11008876], [-1.051722094, 18.23587917], [-1.139198685, 18.37372023],\012[-1.208709432, 18.52143808], [-1.259158110, 18.67670315], [-1.289749112, 18.83\134\012706680], [-1.3, 19.00000000]],COLOUR(RGB,0.,1.00000000,1.00000000)),POLYGONS([\012[-.4, 20.10000000], [1.3, 18.50000000], [-1.4, 13.20000000], [-6.4, 15.2000000\134\0120], [-5.9, 16.50000000], [-2.4, 15.80000000]],COLOUR(RGB,0.,1.00000000,1.00000\134\012000)),POLYGONS([[-8.7, 16.10000000], [-8.68817205, 16.28799985], [-8.65287474,\01216.47303483], [-8.59466473, 16.65218683], [-8.51446002, 16.82263051], [-8.4135\134\0122549, 16.98167788], [-8.29345294, 17.12682066], [-8.15613598, 17.25576986], [-\134\0128.00374019, 17.36649189], [-7.83866894, 17.45724058], [-7.66352549, 17.5265847\134\0128], [-7.48107197, 17.57343088], [-7.29418578, 17.59704009], [-7.10581422, 17.5\134\0129704009], [-6.91892803, 17.57343088], [-6.73647451, 17.52658477], [-6.56133106\012, 17.45724058], [-6.39625981, 17.36649189], [-6.24386402, 17.25576986], [-6.10\134\012654706, 17.12682066], [-5.98647451, 16.98167788], [-5.88553998, 16.82263051],\012[-5.80533527, 16.65218683], [-5.74712526, 16.47303483], [-5.71182795, 16.28799\134\012985], [-5.7, 16.10000000], [-5.71182795, 15.91200015], [-5.74712526, 15.726965\134\01217], [-5.80533527, 15.54781317], [-5.88553998, 15.37736949], [-5.98647451, 15.\134\01221832212], [-6.10654706, 15.07317934], [-6.24386402, 14.94423013], [-6.3962598\134\0121, 14.83350811], [-6.56133106, 14.74275942], [-6.73647451, 14.67341522], [-6.9\134\0121892803, 14.62656912], [-7.10581422, 14.60295991], [-7.29418578, 14.60295991],\012[-7.48107197, 14.62656912], [-7.66352549, 14.67341523], [-7.83866894, 14.74275\134\012942], [-8.00374019, 14.83350811], [-8.15613598, 14.94423014], [-8.29345294, 15\134\012.07317934], [-8.41352549, 15.21832212], [-8.51446002, 15.37736949], [-8.594664\134\01273, 15.54781317], [-8.65287474, 15.72696517], [-8.68817205, 15.91200015], [-8.\134\0127, 16.10000000]],COLOUR(RGB,0.,0.,0.)));\012LChest := PLOT(POLYGONS([[1.3, 18.5], [1.289749112, 18.66293320], [1.259158109\012, 18.82329685], [1.208709432, 18.97856192], [1.139198684, 19.12627978], [1.051\134\012722093, 19.26412083], [.9476592155, 19.38991124], [.8286511865, 19.50166722],\012[.6965748336, 19.59762630], [.5535130793, 19.67627517], [.4017220919, 19.73637\134\012347], [.2435957085, 19.77697343], [.8162767502e-1, 19.79743475], [-.8162767556\134\012e-1, 19.79743475], [-.2435957090, 19.77697343], [-.4017220925, 19.73637347], [\012-.5535130787, 19.67627517], [-.6965748330, 19.59762630], [-.8286511861, 19.501\134\01266722], [-.9476592168, 19.38991124], [-1.051722094, 19.26412083], [-1.13919868\134\0125, 19.12627978], [-1.208709432, 18.97856192], [-1.259158110, 18.82329685], [-1\134\012.289749112, 18.66293320], [-1.3, 18.50000000], [-1.289749112, 18.33706680], [-\134\0121.259158109, 18.17670315], [-1.208709432, 18.02143808], [-1.139198684, 17.8737\134\0122022], [-1.051722093, 17.73587917], [-.9476592160, 17.61008876], [-.8286511853\012, 17.49833278], [-.6965748321, 17.40237370], [-.5535130778, 17.32372483], [-.4\134\012017220914, 17.26362653], [-.2435957079, 17.22302657], [-.8162767449e-1, 17.202\134\01256525], [.8162767609e-1, 17.20256525], [.2435957095, 17.22302657], [.401722093\134\0120, 17.26362653], [.5535130792, 17.32372483], [.6965748335, 17.40237370], [.828\134\0126511865, 17.49833278], [.9476592172, 17.61008876], [1.051722094, 17.73587917],\012[1.139198685, 17.87372023], [1.208709432, 18.02143808], [1.259158110, 18.17670\134\012315], [1.289749112, 18.33706680], [1.3, 18.50000000]],COLOUR(RGB,0.,1.00000000\012,1.00000000)),POLYGONS([[.4, 17.1], [9.2, 18.7], [8.7, 20.], [0., 19.7]],\012COLOUR(RGB,0.,1.00000000,1.00000000)),POLYGONS([[11.5, 19.6], [11.48817205, 19\134\012.78799985], [11.45287474, 19.97303483], [11.39466473, 20.15218683], [11.314460\134\01202, 20.32263051], [11.21352549, 20.48167788], [11.09345294, 20.62682066], [10.\134\01295613598, 20.75576986], [10.80374019, 20.86649189], [10.63866894, 20.95724058]\012, [10.46352549, 21.02658478], [10.28107197, 21.07343088], [10.09418578, 21.097\134\01204009], [9.905814221, 21.09704009], [9.718928028, 21.07343088], [9.536474509,\01221.02658477], [9.361331063, 20.95724058], [9.196259808, 20.86649189], [9.04386\134\0124016, 20.75576986], [8.906547058, 20.62682066], [8.786474508, 20.48167788], [8\134\012.685539979, 20.32263051], [8.605335271, 20.15218683], [8.547125258, 19.9730348\134\0123], [8.511827948, 19.78799985], [8.5, 19.60000000], [8.511827948, 19.41200015]\012, [8.547125258, 19.22696517], [8.605335271, 19.04781317], [8.685539980, 18.877\134\01236949], [8.786474508, 18.71832212], [8.906547058, 18.57317934], [9.043864017,\01218.44423013], [9.196259809, 18.33350811], [9.361331064, 18.24275942], [9.53647\134\0124510, 18.17341522], [9.718928029, 18.12656912], [9.905814222, 18.10295991], [\01210.09418578, 18.10295991], [10.28107197, 18.12656912], [10.46352549, 18.173415\134\01223], [10.63866894, 18.24275942], [10.80374019, 18.33350811], [10.95613598, 18.\134\01244423014], [11.09345294, 18.57317934], [11.21352549, 18.71832212], [11.3144600\134\0122, 18.87736949], [11.39466473, 19.04781317], [11.45287474, 19.22696517], [11.4\134\0128817205, 19.41200015], [11.5, 19.60000000]],COLOUR(RGB,0.,0.,0.)));\012RChest := PLOT(POLYGONS([[-1.3, 18.50000000], [-1.289749112, 18.66293320], [-1\134\012.259158109, 18.82329685], [-1.208709432, 18.97856192], [-1.139198684, 19.12627\134\012978], [-1.051722093, 19.26412083], [-.9476592155, 19.38991124], [-.8286511865,\01219.50166722], [-.6965748336, 19.59762630], [-.5535130793, 19.67627517], [-.401\134\0127220919, 19.73637347], [-.2435957085, 19.77697343], [-.8162767502e-1, 19.79743\134\012475], [.8162767556e-1, 19.79743475], [.2435957090, 19.77697343], [.4017220925,\01219.73637347], [.5535130787, 19.67627517], [.6965748330, 19.59762630], [.828651\134\0121861, 19.50166722], [.9476592168, 19.38991124], [1.051722094, 19.26412083], [1\134\012.139198685, 19.12627978], [1.208709432, 18.97856192], [1.259158110, 18.8232968\134\0125], [1.289749112, 18.66293320], [1.3, 18.50000000], [1.289749112, 18.33706680]\012, [1.259158109, 18.17670315], [1.208709432, 18.02143808], [1.139198684, 17.873\134\01272022], [1.051722093, 17.73587917], [.9476592160, 17.61008876], [.8286511853,\01217.49833278], [.6965748321, 17.40237370], [.5535130778, 17.32372483], [.401722\134\0120914, 17.26362653], [.2435957079, 17.22302657], [.8162767449e-1, 17.20256525],\012[-.8162767609e-1, 17.20256525], [-.2435957095, 17.22302657], [-.4017220930, 17\134\012.26362653], [-.5535130792, 17.32372483], [-.6965748335, 17.40237370], [-.82865\134\01211865, 17.49833278], [-.9476592172, 17.61008876], [-1.051722094, 17.73587917],\012[-1.139198685, 17.87372023], [-1.208709432, 18.02143808], [-1.259158110, 18.17\134\012670315], [-1.289749112, 18.33706680], [-1.3, 18.50000000]],COLOUR(RGB,0.,1.000\134\01200000,1.00000000)),POLYGONS([[-.4, 17.10000000], [-9.2, 18.70000000], [-8.7, \01220.00000000], [-0., 19.70000000]],COLOUR(RGB,0.,1.00000000,1.00000000)),\012POLYGONS([[-11.5, 19.60000000], [-11.48817205, 19.78799985], [-11.45287474, 19\134\012.97303483], [-11.39466473, 20.15218683], [-11.31446002, 20.32263051], [-11.213\134\01252549, 20.48167788], [-11.09345294, 20.62682066], [-10.95613598, 20.75576986],\012[-10.80374019, 20.86649189], [-10.63866894, 20.95724058], [-10.46352549, 21.02\134\012658478], [-10.28107197, 21.07343088], [-10.09418578, 21.09704009], [-9.9058142\134\01221, 21.09704009], [-9.718928028, 21.07343088], [-9.536474509, 21.02658477], [-\134\0129.361331063, 20.95724058], [-9.196259808, 20.86649189], [-9.043864016, 20.7557\134\0126986], [-8.906547058, 20.62682066], [-8.786474508, 20.48167788], [-8.685539979\012, 20.32263051], [-8.605335271, 20.15218683], [-8.547125258, 19.97303483], [-8.\134\012511827948, 19.78799985], [-8.5, 19.60000000], [-8.511827948, 19.41200015], [-8\134\012.547125258, 19.22696517], [-8.605335271, 19.04781317], [-8.685539980, 18.87736\134\012949], [-8.786474508, 18.71832212], [-8.906547058, 18.57317934], [-9.043864017,\01218.44423013], [-9.196259809, 18.33350811], [-9.361331064, 18.24275942], [-9.53\134\0126474510, 18.17341522], [-9.718928029, 18.12656912], [-9.905814222, 18.10295991\012], [-10.09418578, 18.10295991], [-10.28107197, 18.12656912], [-10.46352549, 18\134\012.17341523], [-10.63866894, 18.24275942], [-10.80374019, 18.33350811], [-10.956\134\01213598, 18.44423014], [-11.09345294, 18.57317934], [-11.21352549, 18.71832212],\012[-11.31446002, 18.87736949], [-11.39466473, 19.04781317], [-11.45287474, 19.22\134\012696517], [-11.48817205, 19.41200015], [-11.5, 19.60000000]],COLOUR(RGB,0.,0.,0\134\012.)));\012LChin := PLOT(POLYGONS([[1.3, 18.5], [1.289749112, 18.66293320], [1.259158109,\01218.82329685], [1.208709432, 18.97856192], [1.139198684, 19.12627978], [1.05172\134\0122093, 19.26412083], [.9476592155, 19.38991124], [.8286511865, 19.50166722], [.\134\0126965748336, 19.59762630], [.5535130793, 19.67627517], [.4017220919, 19.7363734\134\0127], [.2435957085, 19.77697343], [.8162767502e-1, 19.79743475], [-.8162767556e-\134\0121, 19.79743475], [-.2435957090, 19.77697343], [-.4017220925, 19.73637347], [-.\134\0125535130787, 19.67627517], [-.6965748330, 19.59762630], [-.8286511861, 19.50166\134\012722], [-.9476592168, 19.38991124], [-1.051722094, 19.26412083], [-1.139198685,\01219.12627978], [-1.208709432, 18.97856192], [-1.259158110, 18.82329685], [-1.28\134\0129749112, 18.66293320], [-1.3, 18.50000000], [-1.289749112, 18.33706680], [-1.2\134\01259158109, 18.17670315], [-1.208709432, 18.02143808], [-1.139198684, 17.8737202\134\0122], [-1.051722093, 17.73587917], [-.9476592160, 17.61008876], [-.8286511853, \01217.49833278], [-.6965748321, 17.40237370], [-.5535130778, 17.32372483], [-.401\134\0127220914, 17.26362653], [-.2435957079, 17.22302657], [-.8162767449e-1, 17.20256\134\012525], [.8162767609e-1, 17.20256525], [.2435957095, 17.22302657], [.4017220930,\01217.26362653], [.5535130792, 17.32372483], [.6965748335, 17.40237370], [.828651\134\0121865, 17.49833278], [.9476592172, 17.61008876], [1.051722094, 17.73587917], [1\134\012.139198685, 17.87372023], [1.208709432, 18.02143808], [1.259158110, 18.1767031\134\0125], [1.289749112, 18.33706680], [1.3, 18.50000000]],COLOUR(RGB,0.,1.00000000,1\134\012.00000000)),POLYGONS([[.4, 17.1], [9.3, 20.7], [8.8, 22.], [0., 19.7]],COLOUR(\012RGB,0.,1.00000000,1.00000000)),POLYGONS([[11.6, 21.6], [11.58817205, 21.787999\134\01285], [11.55287474, 21.97303483], [11.49466473, 22.15218683], [11.41446002, 22.\134\01232263051], [11.31352549, 22.48167788], [11.19345294, 22.62682066], [11.0561359\134\0128, 22.75576986], [10.90374019, 22.86649189], [10.73866894, 22.95724058], [10.5\134\0126352549, 23.02658478], [10.38107197, 23.07343088], [10.19418578, 23.09704009],\012[10.00581422, 23.09704009], [9.818928028, 23.07343088], [9.636474509, 23.02658\134\012477], [9.461331063, 22.95724058], [9.296259808, 22.86649189], [9.143864016, 22\134\012.75576986], [9.006547058, 22.62682066], [8.886474508, 22.48167788], [8.7855399\134\01279, 22.32263051], [8.705335271, 22.15218683], [8.647125258, 21.97303483], [8.6\134\01211827948, 21.78799985], [8.6, 21.60000000], [8.611827948, 21.41200015], [8.647\134\012125258, 21.22696517], [8.705335271, 21.04781317], [8.785539980, 20.87736949],\012[8.886474508, 20.71832212], [9.006547058, 20.57317934], [9.143864017, 20.44423\134\012013], [9.296259809, 20.33350811], [9.461331064, 20.24275942], [9.636474510, 20\134\012.17341522], [9.818928029, 20.12656912], [10.00581422, 20.10295991], [10.194185\134\01278, 20.10295991], [10.38107197, 20.12656912], [10.56352549, 20.17341523], [10.\134\01273866894, 20.24275942], [10.90374019, 20.33350811], [11.05613598, 20.44423014]\012, [11.19345294, 20.57317934], [11.31352549, 20.71832212], [11.41446002, 20.877\134\01236949], [11.49466473, 21.04781317], [11.55287474, 21.22696517], [11.58817205,\01221.41200015], [11.6, 21.60000000]],COLOUR(RGB,0.,0.,0.)));\012RChin := PLOT(POLYGONS([[-1.3, 18.50000000], [-1.289749112, 18.66293320], [-1.\134\012259158109, 18.82329685], [-1.208709432, 18.97856192], [-1.139198684, 19.126279\134\01278], [-1.051722093, 19.26412083], [-.9476592155, 19.38991124], [-.8286511865,\01219.50166722], [-.6965748336, 19.59762630], [-.5535130793, 19.67627517], [-.401\134\0127220919, 19.73637347], [-.2435957085, 19.77697343], [-.8162767502e-1, 19.79743\134\012475], [.8162767556e-1, 19.79743475], [.2435957090, 19.77697343], [.4017220925,\01219.73637347], [.5535130787, 19.67627517], [.6965748330, 19.59762630], [.828651\134\0121861, 19.50166722], [.9476592168, 19.38991124], [1.051722094, 19.26412083], [1\134\012.139198685, 19.12627978], [1.208709432, 18.97856192], [1.259158110, 18.8232968\134\0125], [1.289749112, 18.66293320], [1.3, 18.50000000], [1.289749112, 18.33706680]\012, [1.259158109, 18.17670315], [1.208709432, 18.02143808], [1.139198684, 17.873\134\01272022], [1.051722093, 17.73587917], [.9476592160, 17.61008876], [.8286511853,\01217.49833278], [.6965748321, 17.40237370], [.5535130778, 17.32372483], [.401722\134\0120914, 17.26362653], [.2435957079, 17.22302657], [.8162767449e-1, 17.20256525],\012[-.8162767609e-1, 17.20256525], [-.2435957095, 17.22302657], [-.4017220930, 17\134\012.26362653], [-.5535130792, 17.32372483], [-.6965748335, 17.40237370], [-.82865\134\01211865, 17.49833278], [-.9476592172, 17.61008876], [-1.051722094, 17.73587917],\012[-1.139198685, 17.87372023], [-1.208709432, 18.02143808], [-1.259158110, 18.17\134\012670315], [-1.289749112, 18.33706680], [-1.3, 18.50000000]],COLOUR(RGB,0.,1.000\134\01200000,1.00000000)),POLYGONS([[-.4, 17.10000000], [-9.3, 20.70000000], [-8.8, \01222.00000000], [-0., 19.70000000]],COLOUR(RGB,0.,1.00000000,1.00000000)),\012POLYGONS([[-11.6, 21.60000000], [-11.58817205, 21.78799985], [-11.55287474, 21\134\012.97303483], [-11.49466473, 22.15218683], [-11.41446002, 22.32263051], [-11.313\134\01252549, 22.48167788], [-11.19345294, 22.62682066], [-11.05613598, 22.75576986],\012[-10.90374019, 22.86649189], [-10.73866894, 22.95724058], [-10.56352549, 23.02\134\012658478], [-10.38107197, 23.07343088], [-10.19418578, 23.09704009], [-10.005814\134\01222, 23.09704009], [-9.818928028, 23.07343088], [-9.636474509, 23.02658477], [-\134\0129.461331063, 22.95724058], [-9.296259808, 22.86649189], [-9.143864016, 22.7557\134\0126986], [-9.006547058, 22.62682066], [-8.886474508, 22.48167788], [-8.785539979\012, 22.32263051], [-8.705335271, 22.15218683], [-8.647125258, 21.97303483], [-8.\134\012611827948, 21.78799985], [-8.6, 21.60000000], [-8.611827948, 21.41200015], [-8\134\012.647125258, 21.22696517], [-8.705335271, 21.04781317], [-8.785539980, 20.87736\134\012949], [-8.886474508, 20.71832212], [-9.006547058, 20.57317934], [-9.143864017,\01220.44423013], [-9.296259809, 20.33350811], [-9.461331064, 20.24275942], [-9.63\134\0126474510, 20.17341522], [-9.818928029, 20.12656912], [-10.00581422, 20.10295991\012], [-10.19418578, 20.10295991], [-10.38107197, 20.12656912], [-10.56352549, 20\134\012.17341523], [-10.73866894, 20.24275942], [-10.90374019, 20.33350811], [-11.056\134\01213598, 20.44423014], [-11.19345294, 20.57317934], [-11.31352549, 20.71832212],\012[-11.41446002, 20.87736949], [-11.49466473, 21.04781317], [-11.55287474, 21.22\134\012696517], [-11.58817205, 21.41200015], [-11.6, 21.60000000]],COLOUR(RGB,0.,0.,0\134\012.)));\012LBlock := PLOT(POLYGONS([[-.1390779871, 19.63421459], [-.2817201259, 19.713621\134\01294], [-.4331898742, 19.77452535], [-.5910984617, 19.81596433], [-.7529555722,\01219.83728535], [-.9162086200, 19.83815218], [-1.078283012, 19.81855114], [-1.23\134\0126622740, 19.77879134], [-1.388730679, 19.71949984], [-1.532208017, 19.64161169\012], [-1.664792013, 19.54635523], [-1.784391752, 19.43523272], [-1.889121061, 19\134\012.30999661], [-1.977328304, 19.17262197], [-2.047622398, 19.02527527], [-2.0988\134\01294759, 18.87028026], [-2.130336803, 18.71008130], [-2.141452656, 18.54720482],\012[-2.132067034, 18.38421949], [-2.102327932, 18.22369567], [-2.052704365, 18.06\134\012816493], [-1.983978924, 17.92008008], [-1.897235448, 17.78177650], [-1.7938419\134\01234, 17.65543532], [-1.675428964, 17.54304903], [-1.543863982, 17.44639003], [-\134\0121.401221844, 17.36698267], [-1.249752096, 17.30607926], [-1.091843508, 17.2646\134\0124028], [-.9299863974, 17.24331926], [-.7667333495, 17.24245243], [-.6046589569\012, 17.26205348], [-.4463192287, 17.30181327], [-.2942112892, 17.36110478], [-.1\134\012507339515, 17.43899292], [-.181499555e-1, 17.53424939], [.1014497822, 17.64537\134\012190], [.2061790915, 17.77060800], [.2943863336, 17.90798264], [.3646804287, 18\134\012.05532934], [.4159527898, 18.21032435], [.4473948338, 18.37052331], [.45851068\134\01278, 18.53339980], [.4491250639, 18.69638513], [.4193859629, 18.85690895], [.36\134\01297623951, 19.01243969], [.3010369464, 19.16052454], [.2142934786, 19.29882811]\012, [.1108999643, 19.42516929], [-.75130053e-2, 19.53755558], [-.1390779871, 19.\134\01263421459]],COLOUR(RGB,0.,1.00000000,1.00000000)),POLYGONS([[-1.550968146, 19.4\134\0127122324], [-1.123128490, 17.17623887], [4.795483955, 16.58460831], [5.81405351\134\0125, 21.87256785], [4.449990082, 22.15422535], [3.147961701, 18.83086529]],\012COLOUR(RGB,0.,1.00000000,1.00000000)),POLYGONS([[6.299424932, 24.29422319], [6\134\012.134837845, 24.38584706], [5.960065060, 24.45612023], [5.777862847, 24.5039344\134\0123], [5.591104646, 24.52853561], [5.402735740, 24.52953579], [5.215726826, 24.5\134\0120691920], [5.033027146, 24.46104251], [4.857517973, 24.39262925], [4.691967207\012, 24.30275831], [4.538985665, 24.19284701], [4.400985974, 24.06462872], [4.280\134\012144469, 23.92012552], [4.178366880, 23.76161632], [4.097258306, 23.59160090],\012[4.038097891, 23.41276050], [4.001818605, 23.22791554], [3.988992617, 23.03998\134\012115], [3.999822197, 22.85192114], [4.034136537, 22.66670136], [4.091394499, 22\134\012.48724281], [4.170693087, 22.31637567], [4.270781708, 22.15679462], [4.3900819\134\01215, 22.01101635], [4.526712264, 21.88133986], [4.678518015, 21.76981023], [4.8\134\01243105102, 21.67818636], [5.017877887, 21.60791319], [5.200080100, 21.56009899]\012, [5.386838301, 21.53549781], [5.575207207, 21.53449763], [5.762216121, 21.557\134\01211422], [5.944915810, 21.60299090], [6.120424973, 21.67140417], [6.285975740,\01221.76127511], [6.438957282, 21.87118641], [6.576956974, 21.99940470], [6.69779\134\0128478, 22.14390790], [6.799576067, 22.30241710], [6.880684641, 22.47243252], [6\134\012.939845056, 22.65127292], [6.976124342, 22.83611788], [6.988950329, 23.0240522\134\0127], [6.978120750, 23.21211227], [6.943806410, 23.39733206], [6.886548448, 23.5\134\0127679061], [6.807249860, 23.74765775], [6.707161239, 23.90723880], [6.587861032\012, 24.05301707], [6.451230683, 24.18269356], [6.299424932, 24.29422319]],COLOUR\012(RGB,0.,0.,0.)));\012RBlock := PLOT(POLYGONS([[.1390779871, 19.63421459], [.2817201259, 19.71362194\012], [.4331898742, 19.77452535], [.5910984617, 19.81596433], [.7529555722, 19.83\134\012728535], [.9162086200, 19.83815218], [1.078283012, 19.81855114], [1.236622740,\01219.77879134], [1.388730679, 19.71949984], [1.532208017, 19.64161169], [1.66479\134\0122013, 19.54635523], [1.784391752, 19.43523272], [1.889121061, 19.30999661], [1\134\012.977328304, 19.17262197], [2.047622398, 19.02527527], [2.098894759, 18.8702802\134\0126], [2.130336803, 18.71008130], [2.141452656, 18.54720482], [2.132067034, 18.3\134\0128421949], [2.102327932, 18.22369567], [2.052704365, 18.06816493], [1.983978924\012, 17.92008008], [1.897235448, 17.78177650], [1.793841934, 17.65543532], [1.675\134\012428964, 17.54304903], [1.543863982, 17.44639003], [1.401221844, 17.36698267],\012[1.249752096, 17.30607926], [1.091843508, 17.26464028], [.9299863974, 17.24331\134\012926], [.7667333495, 17.24245243], [.6046589569, 17.26205348], [.4463192287, 17\134\012.30181327], [.2942112892, 17.36110478], [.1507339515, 17.43899292], [.\134\012181499555e-1, 17.53424939], [-.1014497822, 17.64537190], [-.2061790915, 17.770\134\01260800], [-.2943863336, 17.90798264], [-.3646804287, 18.05532934], [-.415952789\134\0128, 18.21032435], [-.4473948338, 18.37052331], [-.4585106878, 18.53339980], [-.\134\0124491250639, 18.69638513], [-.4193859629, 18.85690895], [-.3697623951, 19.01243\134\012969], [-.3010369464, 19.16052454], [-.2142934786, 19.29882811], [-.1108999643,\01219.42516929], [.75130053e-2, 19.53755558], [.1390779871, 19.63421459]],COLOUR(\012RGB,0.,1.00000000,1.00000000)),POLYGONS([[1.550968146, 19.47122324], [1.123128\134\012490, 17.17623887], [-4.795483955, 16.58460831], [-5.814053515, 21.87256785], [\012-4.449990082, 22.15422535], [-3.147961701, 18.83086529]],COLOUR(RGB,0.,1.00000\134\012000,1.00000000)),POLYGONS([[-6.299424932, 24.29422319], [-6.134837845, 24.3858\134\0124706], [-5.960065060, 24.45612023], [-5.777862847, 24.50393443], [-5.591104646\012, 24.52853561], [-5.402735740, 24.52953579], [-5.215726826, 24.50691920], [-5.\134\012033027146, 24.46104251], [-4.857517973, 24.39262925], [-4.691967207, 24.302758\134\01231], [-4.538985665, 24.19284701], [-4.400985974, 24.06462872], [-4.280144469,\01223.92012552], [-4.178366880, 23.76161632], [-4.097258306, 23.59160090], [-4.03\134\0128097891, 23.41276050], [-4.001818605, 23.22791554], [-3.988992617, 23.03998115\012], [-3.999822197, 22.85192114], [-4.034136537, 22.66670136], [-4.091394499, 22\134\012.48724281], [-4.170693087, 22.31637567], [-4.270781708, 22.15679462], [-4.3900\134\01281915, 22.01101635], [-4.526712264, 21.88133986], [-4.678518015, 21.76981023],\012[-4.843105102, 21.67818636], [-5.017877887, 21.60791319], [-5.200080100, 21.56\134\012009899], [-5.386838301, 21.53549781], [-5.575207207, 21.53449763], [-5.7622161\134\01221, 21.55711422], [-5.944915810, 21.60299090], [-6.120424973, 21.67140417], [-\134\0126.285975740, 21.76127511], [-6.438957282, 21.87118641], [-6.576956974, 21.9994\134\0120470], [-6.697798478, 22.14390790], [-6.799576067, 22.30241710], [-6.880684641\012, 22.47243252], [-6.939845056, 22.65127292], [-6.976124342, 22.83611788], [-6.\134\012988950329, 23.02405227], [-6.978120750, 23.21211227], [-6.943806410, 23.397332\134\01206], [-6.886548448, 23.57679061], [-6.807249860, 23.74765775], [-6.707161239,\01223.90723880], [-6.587861032, 24.05301707], [-6.451230683, 24.18269356], [-6.29\134\0129424932, 24.29422319]],COLOUR(RGB,0.,0.,0.)));\012\012showboxers:= proc(rx,lcon,rcon,ldown,rdown,lbar1,lbar2)  \012local arms, x;\012   x:= 13.5;\012   if rx &gt; 13.5 then arms:= [LArm, RArm]; x:= rx\012   else arms:= [[LArm,LBlock,LChin,LChest][lcon],[RArm,RBlock,RChin,RChest][rcon]];\012   end if;\012   if ldown &gt;0 then\012     display(rotate(arms[1],ldown*Pi/2,[-7,0]),rotate(LMan,ldown*Pi/2,[-7,0]),\012         translate(arms[2],x,0),translate(RMan,x,0),op(lbar1),op(lbar2),scaling=constrained,axes=none, view=[-35..55,DEFAULT])\012   elif rdown &gt; 0 then\012     display(arms[1],LMan,translate(rotate(arms[2],-rdown*Pi/2,[7,0]),x,0),\012                          translate(rotate(RMan,-rdown*Pi/2,[7,0]),x,0), \012                          op(lbar1),op(lbar2),scaling=constrained,axes=none, view=[-35..55,DEFAULT])\012   else\012     display(arms[1],LMan,translate(arms[2],x,0),translate(RMan,x,0),\012          op(lbar1),op(lbar2),scaling=constrained,axes=none, view=[-35..55,DEFAULT])\012   end if\012end proc;\012\012moveboxers:= proc(lcon,rcon,lhit,loldhit,rhit,roldhit)\012local ldown, rdown, lbars0, lbars1; global frame0, Res;\012ldown:= charfcn[0](lhit);\012rdown:= charfcn[0](rhit);\012lbars0:= lifebar(true,loldhit,loldhit), lifebar(false,roldhit,roldhit);\012lbars1:= lifebar(true,lhit,loldhit),lifebar(false,rhit,roldhit); \012frame0:= showboxers(18.5,1,1,0,0,lbars0);\012Res:= [seq(showboxers(13.5+0.5*i,1,1,0,0,lbars0),i=10..1,-3),\012showboxers(13.5, lcon,rcon,0,0,lbars0)$3,\012seq(showboxers(13.5+0.5*i,1,1,ldown*i/10,rdown*i/10,lbars1),i=1..10,3)];\012if lhit*rhit=0 then \012  Res:= [op(Res),display(Res[-1],textplot([8,30,&quot;Round Over: Press Reset&quot;],font=[TIMES,ROMAN,15]))] \012end if;\012display(Res, insequence=true);\012end proc;\012\012outcome1:= proc(pleft)\012  round(Sample(Bernoulli(pleft),1)[1])\012end proc;\012\012outcome2:= proc(phead,pchest)\012local x;\012  x:= Sample(Uniform(0,1),1)[1];\012  if x &lt; phead then 1\012  elif x &lt; phead+pchest then 2\012  else 3\012  end if\012end proc; \012\012lifebar:= proc(qleft, level, oldlevel)\012   uses plots, plottools; \012   local bx0,db,x1,x2, bmax;\012   if qleft then bx0:= -20; db:= 10; bmax:= LHP0 else bx0:= 40; db:= -10; bmax:= RHP0 end if;\012   x1:= bx0+level*db/bmax;\012   if level = oldlevel then \012      [rectangle([bx0,30],[x1, 35],colour=blue),\012      rectangle([x1,30],[bx0+db,35],colour=orange),\012      textplot([bx0+1.1*db,32,level])]\012   else\012      x2:= bx0+oldlevel*db/bmax;\012      [rectangle([bx0,30],[x1,35],colour=blue),\012      rectangle([x1,30],[x2,35],colour=red),\012      rectangle([x2,30],[bx0+db,35],colour=orange),\012      textplot([bx0+1.1*db,32,level])]\012   end if\012end proc;\012\012button1:= proc()\012global lhit,rhit, lrounds1, rrounds1; \012local loldhit, roldhit, anim, arms;\012use DocumentTools in \012Do(res = outcome1(%LProbArea1));\012loldhit:= lhit; roldhit:= rhit;\012if res = 1 then arms:= 3,1; rhit:= rhit-1 \012  else arms:= 1,3; lhit:= lhit-1 \012end if;\012ldown:= piecewise(lhit=0,1,0);\012rdown:= piecewise(rhit=0,1,0);\012anim:=  \012moveboxers(arms,\012     lhit,loldhit,rhit,roldhit);\012SetProperty(Plot1,value,frame0,refresh=true);\012SetProperty(Plot1,value, anim,refresh=true);\012SetProperty(Plot1,play,true);\012if lhit * rhit = 0 then \012   SetProperty(GoButton1,enabled,false);\012   if lhit = 0 then \012     rrounds1:= rrounds1+1;\012     Do(%RRounds1=rrounds1);\012   else\012     lrounds1:= lrounds1+1;\012     Do(%LRounds1 = lrounds1); \012   end if\012end if;\012end use; \012end proc;\012\012lrounds1:= 0; rrounds1:= 0;\012DocumentTools[Do](%LRounds1 = 0);\012DocumentTools[Do](%RRounds1 = 0);\012lrounds2:= 0; rrounds2:= 0;\012DocumentTools[Do](%LRounds2 = 0);\012DocumentTools[Do](%RRounds2 = 0);\012\012move2:= proc(lmove)\012global lhit,rhit, loldhit, roldhit, lrounds2, rrounds2;\012local  anim, rmove, res, p, arms;\012use DocumentTools in\012Do(rmove = outcome2(%pHead,%pChest));\012loldhit:= lhit; roldhit:= rhit;\012if lmove = 1 then\012  if rmove = 1 then  # Head-Head\012    Do(p = %hhp1);\012    res:= outcome1(p);\012    if res = 0 then \012      Do(lhit = lhit - %hh1a);\012      Do(rhit = rhit - %hh1b);\012      arms:= 1,3; \012    else\012      Do(lhit = lhit - %hh2a);\012      Do(rhit = rhit - %hh2b);\012      arms:= 3, 1;\012    end if\012  elif rmove = 2 then # Head-Chest\012    Do(p = %hcp1);\012    res:= outcome1(p);\012    if res = 0 then\012      Do(lhit = lhit - %hc1a);\012      Do(rhit = rhit - %hc1b);\012      arms:= 1,4;\012    else\012      Do(lhit = lhit - %hc2a);\012      Do(rhit = rhit - %hc2b);\012      arms:= 3,1;\012    end if     \012  else # Head-Block\012    Do(lhit = lhit - %hb1a);\012    Do(rhit = rhit - %hb1b);\012    arms:= 3,2;\012  end if\012elif lmove = 2 then # Chest\012  if rmove = 1 then # Chest-Head\012    Do(p = %chp1);\012    res:= outcome1(p);\012    if res = 0 then\012      Do(lhit = lhit - %ch1a);\012      Do(rhit = rhit - %ch1b);\012      arms := 1,3;\012    else\012      Do(lhit = lhit - %ch2a);\012      Do(rhit = rhit - %ch2b);\012      arms := 4,1;\012    end if\012  elif rmove = 2 then # Chest-Chest\012    Do(p = %ccp1);\012    res:= outcome1(p);\012    if res = 0 then\012      Do(lhit = lhit - %cc1a);\012      Do(rhit = rhit - %cc1b);\012      arms := 1,4;\012    else\012      Do(lhit = lhit - %cc2a);\012      Do(rhit = rhit - %cc2b);\012      arms := 4,1;\012    end if\012  else # Chest-Block\012    Do(lhit = lhit - %cb1a);\012    Do(rhit = rhit - %cb1b);\012    arms:= 4,1;\012  end if\012 else  # Block\012  if rmove = 1 then # Block-Head\012    Do(lhit = lhit - %bh1a);\012    Do(rhit = rhit - %bh1b);\012    arms:= 2,3;\012  elif rmove=2 then # Block-Chest\012    Do(lhit = lhit - %bc1a);\012    Do(rhit = rhit - %bc1b);\012    arms:= 1,4;\012  else # Block-Block\012    Do(p = %bbp1);\012    res:= outcome1(p);\012    if res = 0 then\012      Do(lhit = lhit - %bb1a);\012      Do(rhit = rhit - %bb1b);\012      arms := 2,1;\012    else\012      Do(lhit = lhit - %bb2a);\012      Do(rhit = rhit - %bb2b);\012      arms := 1,2;\012    end if\012  end if\012end if;\012lhit:= max(0,lhit);\012rhit:= max(0,rhit);\012ldown:= piecewise(lhit=0,1,0);\012rdown:= piecewise(rhit=0,1,0);\012anim:=  \012moveboxers(arms,\012     lhit,loldhit,rhit,roldhit);\012SetProperty(Plot2,value,frame0,refresh=true);\012SetProperty(Plot2,value, anim,refresh=true);\012SetProperty(Plot2,play,true);\012if lhit*rhit=0 then \012  SetProperty(HButton,enabled,false);   \012  SetProperty(CButton,enabled,false);   \012  SetProperty(BButton,enabled,false);   \012  if lhit = 0 then \012     rrounds2:= rrounds2+1;\012     Do(%RRounds2=rrounds2);\012   else\012     lrounds2:= lrounds2+1;\012     Do(%LRounds2 = lrounds2); \012   end if \012end if;\012if Do(%OptCheck) and (lhit &gt; 0) and (rhit &gt; 0) then \012  ShowOppStrats();\012end if;            \012end use\012end proc;\012\012ShowOppStrats:= proc()\012uses DocumentTools;\012Do(%pHead = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lhit,rhit,2,1])));\012Do(%pChest = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lhit,rhit,2,2]))); \012Do(%pBlock = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lhit,rhit,2,3])));\012end proc;\012\012\012WinProbs1:= Array(1..10,1..10);\012DocumentTools:-SetProperty(PTextArea1,value,0.5);\012DocumentTools:-SetProperty(Plot1,delay,300);\012DocumentTools:-SetProperty(Plot2,delay,300);\012DocumentTools:-SetProperty(Plot1,value,plots[textplot]([0,0,&quot;Press Reset to start&quot;],axes=none));\012DocumentTools:-SetProperty(Plot2,value,plots[textplot]([0,0,&quot;Press Reset to start&quot;],axes=none));\012\012UpdateWinProbs1:= proc()\012  local p, h1, h2, r;\012  global WinProbs1;\012  uses DocumentTools;\012  Do(p = %PTextArea1);\012  if not(p::numeric and p &gt;= 0 and p &lt;= 1) then\012    error &quot;Expected a number from 0 to 1, got &quot;,p\012  end if;\012  for h1 from 1 to 6 do\012    for h2 from 1 to 6 do    \012      WinProbs1[h2,h1]:= convert(sprintf(&quot;%4.4f&quot;,add(binomial(h2-r+h1-1,h1-1)*p^h1*(1-p)^(h2-r),\012         r=1..h2)),name)\012  end do end do;\012  SetProperty(WinDT1,update);\012end proc;\012\012UpdateWinProbs1():\012\012OutcomeData:= Array(1..14,1..5);\012OutcomeData[1..14,1]:= &lt;1,1,1,1,1,2,2,2,2,2,3,3,3,3&gt;;\012OutcomeData[1..14,2]:= &lt;1,1,2,2,3,1,1,2,2,3,1,2,3,3&gt;;\012# gather data\012for r from 1 to 14 do\012  s1:= [&quot;h&quot;,&quot;c&quot;,&quot;b&quot;][OutcomeData[r,1]];\012  s2:= [&quot;h&quot;,&quot;c&quot;,&quot;b&quot;][OutcomeData[r,2]];\012  if r &gt; 1 and OutcomeData[r,2]=OutcomeData[r-1,2] then s3:= &quot;2&quot; else s3:= &quot;1&quot; end if;\012  parse(sprintf(&quot;DocumentTools[Do](qq = %c%c%cp%c)&quot;,&quot;%&quot;,s1,s2,s3),statement);\012  OutcomeData[r,3]:= qq;\012  parse(sprintf(&quot;DocumentTools[Do](qq = %c%c%c%ca)&quot;,&quot;%&quot;,s1,s2,s3),statement);\012  OutcomeData(r,4):= qq;\012  parse(sprintf(&quot;DocumentTools[Do](qq = %c%c%c%cb)&quot;,&quot;%&quot;,s1,s2,s3),statement);\012  OutcomeData(r,5):= qq;\012end do; \012\012MixedStrats:= Array(1..2,1..3);\012WinProbs2:= Array(1..10,1..10);\012Win2Out:= Array(1..10,1..10);\012Win3Out:= Array(1..10,1..10);\012\012UpdateWinProbs2:= proc() \012  local h1,h2,r,t, oldf;\012  global WinProbs2, Win2Out;\012  for h1 from 1 to 10 do\012    for h2 from 1 to 10 do\012      t:= 0;\012      for r from 1 to 14 do\012        if h1 &lt;= OutcomeData(r,4) then next end if;\012        if h2 &lt;= OutcomeData(r,5) then oldf:= 1 \012        else oldf:= WinProbs2[h1-OutcomeData[r,4],h2-OutcomeData[r,5]]\012        end if;\012        t:= t + MixedStrats[1,OutcomeData[r,1]] * MixedStrats[2,OutcomeData[r,2]]\012         * OutcomeData[r,3] * oldf;\012      end do;\012      WinProbs2[h1,h2]:= t;\012      Win2Out[h1,h2]:= convert(sprintf(&quot;%4.4f&quot;, t),name); \012    end do\012  end do;\012  DocumentTools[SetProperty](Win2DT,update);\012end proc:\012\012use DocumentTools in\012  Do(MixedStrats[1,1]=%YHArea):\012  Do(MixedStrats[1,2]=%YCArea):\012  Do(MixedStrats[1,3]=%YBArea):\012  Do(MixedStrats[2,1]=%OHArea):\012  Do(MixedStrats[2,2]=%OCArea):\012  Do(MixedStrats[2,3]=%OBArea):\012end use:\012\012UpdateWinProbs2():\012\012WinProbs3:= Array(1..10,1..10);\012OMixedStrats:= Array(1..10,1..10,1..2,1..3);\012\012OpponentOpt:= proc()\012  local h1,h2,c, i, r, oldf,x, res;\012  global WinProbs3, Win3Out, OMixedStrats;\012  for h1 from 1 to 10 do\012    for h2 from 1 to 10 do\012      for i from 1 to 3 do c[i]:= 0 end do;\012      for r from 1 to 14 do\012        if h1 &lt;= OutcomeData[r,4] then next end if;\012        if h2 &lt;= OutcomeData[r,5] then oldf:= 1 \012        else oldf:= WinProbs3[h1-OutcomeData[r,4],h2-OutcomeData[r,5]]\012        end if;\012        c[OutcomeData[r,1]]:= c[OutcomeData[r,1]] + OutcomeData[r,3]*x[OutcomeData[r,2]]*oldf;\012      end do;\012    res:= Optimization[LPSolve](v,{seq(v &gt;= c[i],i=1..3),seq(x[i]&gt;=0,i=1..3),x[1]+x[2]+x[3]=1},  \012             maximize=false);\012    WinProbs3[h1,h2]:= res[1];\012    Win3Out[h1,h2]:= convert(sprintf(&quot;%4.4f&quot;, res[1]),name); \012    for i from 1 to 3 do OMixedStrats[h1,h2,2,i]:= subs(res[2],x[i]) end do;\012    end do\012  end do;\012end proc;\012\012MyOpt:= proc()\012  local h1,h2,c, i, r, oldf,x, res;\012  global WinProbs3, OMixedStrats;\012  for h1 from 1 to 10 do\012    for h2 from 1 to 10 do\012      for i from 1 to 3 do c[i]:= 0 end do;\012      for r from 1 to 14 do\012        if h1 &lt;= OutcomeData[r,4] then next end if;\012        if h2 &lt;= OutcomeData[r,5] then oldf:= 1 \012        else oldf:= WinProbs3[h1-OutcomeData[r,4],h2-OutcomeData[r,5]]\012        end if;\012        c[OutcomeData[r,2]]:= c[OutcomeData[r,2]] + OutcomeData[r,3]*x[OutcomeData[r,1]]*oldf;\012      end do;\012    res:= Optimization[LPSolve](v,{seq(v &lt;= c[i],i=1..3),seq(x[i]&gt;=0,i=1..3),x[1]+x[2]+x[3]=1},  \012             maximize=true);\012    for i from 1 to 3 do OMixedStrats[h1,h2,1,i]:= subs(res[2],x[i]) end do;\012    end do\012  end do;\012end proc;\012\012OpponentOpt(); MyOpt();"/>
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<Text-field style="Title" layout="Title"><Font style="Text" size="24">Street-Fighting Maple</Font></Text-field>
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<Text-field style="Normal" layout="Normal">Fighting games such as the <Annotation tag_id="2"><Font style="Normal" encoding="UTF-8">Street Fighter\302\256</Font></Annotation> series are very popular.  If you haven't played them yourself, you've seen your friends (or perhaps your children) playing them.  But you may not be aware of the mathematical side of such games.</Text-field>
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<Text-field style="Text" layout="Normal"></Text-field>
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<Text-field style="Text" layout="Normal"><Font style="Normal">In the simplest form of such a game, your character and an opponent are fighting against each other.  Each starts out with a certain number of  <Font bold="true">hit points</Font>, which may be displayed in a <Font bold="true">life bar</Font>.As the game progresses, whenever one player succeeds in hitting the other the player who was hit loses a certain number of hit points.  Whenever a player's hit points are reduced to 0, that player loses this round of the game and the other player wins.  Thus the objective of the game is to</Font></Text-field>
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<Text-field style="Text" layout="Normal"><Font style="Normal">reduce your opponent's hit points to 0 before your own reaches 0.  There may be several rounds in the game, but we'll consider just a single round.</Font></Text-field>
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<Text-field style="Heading 1" layout="Heading 1">A simple fight</Text-field></Title><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L46" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block>
<Text-field style="Heading 1" layout="Heading 1"></Text-field><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L11" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Font style="Normal">Here is a </Font><Annotation tag_id="3"><Font style="Normal">rather crude</Font></Annotation><Font style="Normal"> Maple implementation of a simple fighting game.  Your character is the one on the left.  There's no strategy involved here: just press the &quot;Go&quot; button to bring the fighters together.  Each time you press &quot;Go&quot;, one fighter will hit the other, and the one hit will lose 1 hit point.  The &quot;Reset&quot; button will restore both fighters to their original hit points.</Font></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L12" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L13" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">                     <EC-Button id="GoButton1" caption="Go" pixel-width="300" pixel-height="200" enabled="true" visible="true" show-border="true" user-size="false"><MapleCode action="click-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
button1()
end use; 
</MapleCode></EC-Button>                              <EC-Button id="Reset1" caption="Reset" pixel-width="300" pixel-height="200" enabled="true" visible="true" show-border="true" user-size="false"><MapleCode action="click-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(LHP0 = %LHP0Box);
Do(RHP0 = %RHP0Box);
lhit:= LHP0;
rhit:= RHP0;
Do(%Plot1 = showboxers(18.5,1,1,0,0,
    lifebar(true,lhit,lhit),
    lifebar(false,rhit,rhit)));
SetProperty(GoButton1,enabled,true);
end use; 
</MapleCode></EC-Button>                          </Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L19" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><EC-Plot id="Plot1" visible="true" pixel-width="800" pixel-height="400" continuous="false" delay="300" clickx="1.7976931348623157E308" clicky="1.7976931348623157E308" startx="-2.292592729081761" starty="20.542592238868608" endx="0.0" endy="0.0" clickdefault="false" labelreference="L20" drawcaption="false" captionposition="1" captionalignment="0"><MapleCode action="drag-end-event-handler"></MapleCode><MapleCode action="click-event-handler"></MapleCode><MapleCode action="drag-event-handler"></MapleCode><Plot height="-1" type="two-dimensional" width="-1" plot-scale="1.0" plot-xtrans="0.0" plot-ytrans="0.0" gridlinevisibility="1" legendvisibility="false">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</Plot></EC-Plot></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L5227" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Rounds  won:    You  <EC-TextArea id="LRounds1" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="0" editable="true" wrapping="true"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(lrounds1=%LRounds1);
end use; 
</MapleCode></EC-TextArea>  Opponent  <EC-TextArea id="RRounds1" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="0" editable="true" wrapping="true"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(rrounds1=%RRounds1);
end use; 
</MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L15" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L23" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The initial numbers of hit points, and the probabilities of a hit for each fighter, can be</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L24" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">changed below (<Font bold="true">do not</Font> press Enter: just click somewhere out of the text area when done).  Note that the two probabilities will always add up to 1.  You can also give names to the two fighters.</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L27" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L28" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Table interior="group" showinput="true" alignment="left" exterior="all" width="100%" labelreference="L29" showlabel="true" captionalignment="0" title="" order="row" drawcaption="false" randomized="false" captionposition="1" plotalignlists="" pagebreak="cell" postexecute="insert" showgroup="true"><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L30" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L31" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><EC-TextArea id="LName" enabled="true" visible="true" visiblecharacterwidth="15" visiblerows="1" contents="You" editable="true" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L32" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><EC-TextArea id="RName" enabled="true" visible="true" visiblecharacterwidth="15" visiblerows="1" contents="Opponent" editable="true" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L33" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Initial hit points</Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L34" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-ComboBox id="LHP0Box" enabled="true" visible="true" selecteditem="7">1,2,3,4,5,6,7,8,9,10<MapleCode action="selection-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(LHP0 = %LHP0Box);
end use; 
</MapleCode></EC-ComboBox></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L35" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><EC-ComboBox id="RHP0Box" enabled="true" visible="true" selecteditem="6">1,2,3,4,5,6,7,8,9,10<MapleCode action="selection-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(RHP0 = %RHP0Box);
end use; 
</MapleCode></EC-ComboBox><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L36" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Probability of scoring a hit</Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L37" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="LProbArea1" tooltip="left fighter hits right fighter with this probability" enabled="true" visible="true" visiblecharacterwidth="15" visiblerows="1" contents="2/5" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(t = %LProbArea1);
if not(t::numeric and t &gt;= 0 and t &lt;= 1) then
  error &quot;Expected a numeric value between 0 and 1, got &quot;,t
end if;
Do(%RProbArea1 = 1 - t)
end use; 
</MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L38" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="RProbArea1" enabled="true" visible="true" visiblecharacterwidth="15" visiblerows="1" contents="3/5" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(t = %RProbArea1);
if not(t::numeric and 0&lt;=t and t&lt;=1) then
  error &quot;Expected a number between 0 and 1, got &quot;,t
end if;
Do(%LProbArea1 = 1 - t)
end use; 
</MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell></Table-Row></Table><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L25" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block>
</Section><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L22" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Winning probabilities</Text-field></Title><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L816" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L857" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The mathematical question here is: given the initial hit points of the two fighters and their probabilities of scoring a hit, what is the probability that your fighter wins the round?  </Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L859" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L818" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The answer can be based on the <Hyperlink linktarget="Help:Statistics,Distributions,Binomial" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">binomial distribution</Font></Hyperlink>, which says that in <Font italic="true">n</Font> independent trials with probability <Font italic="true">p</Font> of success in each trial, the probability of having exactly <Font italic="true">x</Font> successes is <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkobWZlbmNlZEdGJDYmLUkmbWZyYWNHRiQ2KC1GIzYoLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0Y3LyUrZm9yZWdyb3VuZEdRK1swLDE2MCw4MF1GJy8lLHBsYWNlaG9sZGVyR0Y5LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y5RjotRiM2Jy1GNDYlUSJ4RidGN0Y6RjcvRj5RLFsyMDAsMCwyMDBdRidGQEY6LyUubGluZXRoaWNrbmVzc0dRIjBGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZQLyUpYmV2ZWxsZWRHUSZmYWxzZUYnL0Y7USdub3JtYWxGJy9JK21zZW1hbnRpY3NHRiRRKWJpbm9taWFsRidGWC1JI21vR0YkNi1RIn5GJ0ZWLyUmZmVuY2VHRlUvJSpzZXBhcmF0b3JHRlUvJSlzdHJldGNoeUdGVS8lKnN5bW1ldHJpY0dGVS8lKGxhcmdlb3BHRlUvJS5tb3ZhYmxlbGltaXRzR0ZVLyUnYWNjZW50R0ZVLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGaW8tSSVtc3VwR0YkNiUtRjQ2JVEicEYnRjdGOi1GIzYlRkZGN0Y6LyUxc3VwZXJzY3JpcHRzaGlmdEdGTUZlbi1GXXA2JS1GLDYkLUYjNiYtSSNtbkdGJDYkUSIxRidGVi1GZm42LVEoJm1pbnVzO0YnRlZGaW5GW29GXW9GX29GYW9GY29GZW8vRmhvUSwwLjIyMjIyMjJlbUYnL0ZbcEZkcUZfcEZWRlYtRiM2J0YzRmBxRkZGN0Y6RmRwRlY=">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</Equation> for any integer <Font italic="true">x</Font> from 0 to <Font italic="true">n</Font>.</Text-field>
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<Text-field style="Text" layout="Normal">Suppose you have <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+</Equation> hit points and your opponent has <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+</Equation> hit points.  Imagine that instead of stopping when one fighter's hit points reach 0, the round continued for a total of <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> hits.  At the end, the sum of the hit points of the two fighters will be 1: the loser will have 0 or negative hit points, the winner will still have at least one hit point.  So we can tell from this extended version of the round which fighter wins.  The probability that you are the winner, i.e. that you have scored at least <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y+</Equation> hits, is thus obtained by adding up the probability of scoring <Font italic="true">x</Font> hits in the <Font italic="true">n</Font> rounds for <Font italic="true">x </Font>from <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+</Equation> to <Font italic="true">n</Font>.</Text-field>
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<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
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<Text-field style="Text" layout="Normal">Here's another way of calculating the result, without imagining continuing the round.  In order for you to win with exactly <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=</Equation> hit points left, there must be a total of <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> hits; out of the first <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> hits you score <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RKCZtaW51cztGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlgtRjs2JFEiMUYnRj5GPg==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RKCZtaW51cztGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlgtRjs2JFEiMUYnRj5GPg==</Equation>, and then you score the last hit to finish off your opponent.  Since the first <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> hits and the <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>th hit are independent, the probability that you do this  is

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Thus the total probability that you win the round can be obtained by adding these up for <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=</Equation> from 1 to </Text-field>
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<Text-field style="Text" layout="Normal">You could consider this as <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> where <Font italic="true">F</Font> is the cumulative distribution function for a <Hyperlink linktarget="Help:Statistics,Distributions,NegativeBinomial" hyperlink="true"><Font style="Text" underline="true" foreground="[0,128,128]">Negative Binomial</Font></Hyperlink> random variable with parameters <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+</Equation> and <Font italic="true">p</Font>.  This is the distribution for the number of trials until the <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y+</Equation>'th success with probability <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=</Equation> of success in each one. </Text-field>
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<Input>
<Text-field style="Text" layout="Normal">These probabilities are not only valid at the beginning of the round: at any time during the round, given that you now have <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+</Equation> hit points left and your opponent has <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+</Equation>, your probability of winning the round is given by the formula above.

The following table gives your probability <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>of winning, based on your probability <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=</Equation> of scoring a hit, for each pair <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> up to 6.</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L851" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">   </Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L845" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Table interior="group" showinput="true" alignment="left" exterior="all" width="60.04343105320304%" labelreference="L913" showlabel="true" captionalignment="0" title="" drawtitle="false" order="row" drawcaption="false" randomized="false" captionposition="1" plotalignlists="" pagebreak="cell" postexecute="insert" showgroup="true"><Table-Column weight="309" separator="true"></Table-Column><Table-Column weight="244" separator="true"></Table-Column><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L914" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Your probability of scoring a hit</Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L915" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="PTextArea1" enabled="true" visible="true" visiblecharacterwidth="15" visiblerows="2" contents=".5" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
UpdateWinProbs1();

end use; 
</MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell></Table-Row></Table><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L969" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L973" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">                                            <EC-Label id="Label1" caption="Opponent's  Hit Points" enabled="true" visible="true" visiblecharacterwidth="20" pixel-width="300" pixel-height="200" user-size="false" user-width="true"></EC-Label></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L852" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><EC-Label id="Label4" caption="Your Hit Points" enabled="true" visible="true" visiblecharacterwidth="5" pixel-width="300" pixel-height="200" user-size="false" user-width="false"></EC-Label><EC-RTableBrowser id="WinDT1" enabled="true" visiblerows="6" visiblecolumns="6" visible="true" editable="false" variable="WinProbs1" columnnames="&quot;1&quot;,&quot;2&quot;,&quot;3&quot;,&quot;4&quot;,&quot;5&quot;,&quot;6&quot;,&quot;7&quot;,&quot;8&quot;,&quot;9&quot;,&quot;10&quot;" rownames="&quot;1&quot;,&quot;2&quot;,&quot;3&quot;,&quot;4&quot;,&quot;5&quot;,&quot;6&quot;,&quot;7&quot;,&quot;8&quot;,&quot;9&quot;,&quot;10&quot;" rowheader="true" columnheader="true" handle="140626276"><MapleCode action="content-event-handler"></MapleCode></EC-RTableBrowser></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L954" drawlabel="true">
<Input>
<Text-field superscript="false" placeholder="false" executable="false" selection-placeholder="false" italic="false" size="12" bold="false" subscript="false" family="Times New Roman" opaque="false" underline="false" background="[255,255,255]" readonly="false" foreground="[0,0,0]" alignment="centred" firstindent="0" spacebelow="0" leftmargin="0" linespacing="0.0" initial="0" linebreak="space" rightmargin="0" bulletsuffix="" spaceabove="0" bullet="none" pagebreak-before="false"></Text-field>
</Input>
</Group></Presentation-Block>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">More moves</Text-field></Title><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L974" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1266" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">To make the game more interesting, we might consider allowing the fighters to choose among several different types of move.  For each possible pair of moves for the two fighters, there may be several possible outcomes with various probabilities; these may involve a fighter losing more than one hit point.  I'll suppose there are three possible moves: a punch to the head, a punch to the chest, and a block.  Here are the nine possible pairs, the outcomes and their probabilities.  The outcomes and probabilities can be changed, but each possible outcome must involve one fighter losing at least one hit point.  Of course the probabilities of the outcomes for each pair of moves must add up to 1.  In some of the cases there is just one possible outcome, which of course has probability 1.  </Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1268" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L975" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Notice that if you knew which move your opponent would choose, you would be able to do better:  &quot;Block&quot; beats &quot;Head&quot;, &quot;Chest&quot; beats &quot;Block&quot; and (on the average) &quot;Head&quot; beats &quot;Chest&quot;.  Thus this is a game where you are trying to outguess your opponent.</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L976" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L977" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Table interior="group" showinput="true" alignment="left" exterior="all" width="100%" labelreference="L1015" showlabel="true" captionalignment="0" title="" order="row" drawcaption="false" randomized="false" captionposition="1" plotalignlists="" pagebreak="cell" postexecute="insert" showgroup="true"><Table-Column weight="100" separator="true"></Table-Column><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1016" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Head vs Head</Text-field>
</Input>
</Group></Presentation-Block><Table interior="group" showinput="true" alignment="left" exterior="all" width="100%" labelreference="L1036" showlabel="true" captionalignment="0" title="" order="row" drawcaption="false" randomized="false" captionposition="1" plotalignlists="" pagebreak="cell" postexecute="insert" showgroup="true"><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1037" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Probability <EC-TextArea id="hhp1" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".5" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pp = %hhp1);
if not(pp::numeric and pp &gt;= 0 and pp &lt;= 1) then
  error &quot;Expected a number from 0 to 1, got %1&quot;,pp
end if;
Do(%hhp2 = 1-pp);
end use; 
</MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1038" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">You lose <EC-TextArea id="hh1a" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="3" editable="true" wrapping="true"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1039" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"> Opponent loses<EC-TextArea id="hh1b" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="0" editable="false" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1040" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="hhp2" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".5" editable="true" wrapping="true"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pp = %hhp2);
if not(pp::numeric and pp &gt;= 0 and pp &lt;= 1) then
  error &quot;Expected a number from 0 to 1, got %1&quot;,pp
end if;
Do(%hhp1 = 1-pp);
end use; 
</MapleCode></EC-TextArea></Text-field>
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<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="hh2a" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="0" editable="false" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
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<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="hh2b" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="3" editable="true" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
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<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1044" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
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</Group></Presentation-Block></Table-Cell></Table-Row></Table></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1017" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Head vs Chest</Text-field>
</Input>
</Group></Presentation-Block><Table interior="group" showinput="true" alignment="left" exterior="all" width="100%" labelreference="L1081" showlabel="true" captionalignment="0" title="" order="row" drawcaption="false" randomized="false" captionposition="1" plotalignlists="" pagebreak="cell" postexecute="insert" showgroup="true"><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1082" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="hcp1" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".54" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pp = %hcp1);
if not(pp::numeric and pp &gt;= 0 and pp &lt;= 1) then
  error &quot;Expected a number from 0 to 1, got %1&quot;,pp
end if;
Do(%hcp2 = 1-pp);
end use; 
</MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1083" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="hc1a" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="2" editable="true" wrapping="true"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1084" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="hc1b" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="0" editable="false" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1085" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><Image height="215" width="337" zoomable="false" labelreference="L1104" drawcaption="false" captionalignment="0" 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<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1086" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="hcp2" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".46" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pp = %hcp2);
if not(pp::numeric and pp &gt;= 0 and pp &lt;= 1) then
  error &quot;Expected a number from 0 to 1, got %1&quot;,pp
end if;
Do(%hcp1 = 1-pp);
end use; 
</MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1087" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="hc2a" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="0" editable="false" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1088" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="hc2b" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="3" editable="true" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1089" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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</Group></Presentation-Block></Table-Cell></Table-Row></Table></Table-Cell></Table-Row><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1019" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Chest vs Head</Text-field>
</Input>
</Group></Presentation-Block><Table interior="group" showinput="true" alignment="left" exterior="all" width="100%" labelreference="L1134" showlabel="true" captionalignment="0" title="" order="row" drawcaption="false" randomized="false" captionposition="1" plotalignlists="" pagebreak="cell" postexecute="insert" showgroup="true"><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1135" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="chp1" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".46" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pp = %chp1);
if not(pp::numeric and pp &gt;= 0 and pp &lt;= 1) then
  error &quot;Expected a number from 0 to 1, got %1&quot;,pp
end if;
Do(%chp2 = 1-pp);
end use; 
</MapleCode></EC-TextArea></Text-field>
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<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="chp2" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".54" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pp = %chp2);
if not(pp::numeric and pp &gt;= 0 and pp &lt;= 1) then
  error &quot;Expected a number from 0 to 1, got %1&quot;,pp
end if;
Do(%chp1 = 1-pp);
end use; 
</MapleCode></EC-TextArea></Text-field>
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<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="ch2b" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="2" editable="true" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
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<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
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</Group></Presentation-Block><Table interior="group" showinput="true" alignment="left" exterior="all" width="100%" labelreference="L1160" showlabel="true" captionalignment="0" title="" order="row" drawcaption="false" randomized="false" captionposition="1" plotalignlists="" pagebreak="cell" postexecute="insert" showgroup="true"><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Column weight="100" separator="true"></Table-Column><Table-Row align="top" separator="true"><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1161" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="ccp1" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".5" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pp = %ccp1);
if not(pp::numeric and pp &gt;= 0 and pp &lt;= 1) then
  error &quot;Expected a number from 0 to 1, got %1&quot;,pp
end if;
Do(%ccp2 = 1-pp);
end use; 
</MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1162" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="cc1a" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="2" editable="true" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1163" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="cc1b" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="0" editable="false" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell><Table-Cell columnspan="1" backgroundstyle="0" rowspan="1" fillcolor="[255,255,255]"><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1164" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="ccp2" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".5" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pp = %ccp2);
if not(pp::numeric and pp &gt;= 0 and pp &lt;= 1) then
  error &quot;Expected a number from 0 to 1, got %1&quot;,pp
end if;
Do(%ccp1 = 1-pp);
end use; 
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<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="cc2b" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="2" editable="true" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
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<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
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<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1024" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Block vs Block</Text-field>
</Input>
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<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1225" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="bbp1" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".5" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pp = %bbp1);
if not(pp::numeric and pp &gt;= 0 and pp &lt;= 1) then
  error &quot;Expected a number from 0 to 1, got %1&quot;,pp
end if;
Do(%bbp2 = 1-pp);
end use; 
</MapleCode></EC-TextArea></Text-field>
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<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="bb1b" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="0" editable="false" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
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<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1228" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
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<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="bbp2" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".5" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pp = %bbp2);
if not(pp::numeric and pp &gt;= 0 and pp &lt;= 1) then
  error &quot;Expected a number from 0 to 1, got %1&quot;,pp
end if;
Do(%bbp1 = 1-pp);
end use; 
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<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><EC-TextArea id="bb2b" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="1" editable="true" wrapping="false"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
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<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
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<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block></Table-Cell></Table-Row></Table><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1259" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1261" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Here is a simulation of the game.  The fighter on the right will use a &quot;mixed strategy&quot;: at each turn he will randomly choose one of the three possible moves, using probabilities as set below.  The probabilities must add up to 1 (and will be adjusted accordingly).  You, controlling the fighter on the left, get to choose which move to make by clicking the appropriate button.  If you check the &quot;Optimize&quot; box, the opponent's mixed strategy will be optimized: see the section Optimal Play below. </Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1260" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1262" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Opponent's strategy: probabilities for</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1263" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Head: <EC-TextArea id="pHead" enabled="true" visible="true" visiblecharacterwidth="12" visiblerows="1" contents="0.6270005400" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pH = %pHead);
if not(pH::numeric and pH &gt;=0 and pH &lt;= 1) then
  error &quot;Expected a number between 0 and 1, got %1&quot;,pH
end if;
Do(pC = %pChest);
if pH + pC &gt;= 1 then
   pC:= 1 - pH;
   Do(%pChest = pC)
end if;
pB:= 1 - pH - pC;
Do(%pBlock = pB);
end use; 
</MapleCode></EC-TextArea>  Chest:  <EC-TextArea id="pChest" enabled="true" visible="true" visiblecharacterwidth="12" visiblerows="1" contents="0.2962508374" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pC = %pChest);
if not(pC::numeric and pC &gt;=0 and pC &lt;= 1) then
  error &quot;Expected a number between 0 and 1, got %1&quot;,pC
end if;
Do(pH = %pHead);
if pH + pC &gt;= 1 then
   pH:= 1 - pC;
   Do(%pHead = pH)
end if;
pB:= 1 - pH - pC;
Do(%pBlock = pB);
end use; 
</MapleCode></EC-TextArea>  Block:  <EC-TextArea id="pBlock" enabled="true" visible="true" visiblecharacterwidth="12" visiblerows="1" contents="0.0767486226" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pB = %pBlock);
if not(pB::numeric and pB &gt;=0 and pB &lt;= 1) then
  error &quot;Expected a number between 0 and 1, got %1&quot;,pB
end if;
Do(pC = %pChest);
if pB + pC &gt;= 1 then
   pC:= 1 - pB;
   Do(%pChest = pC)
end if;
pH:= 1 - pB - pC;
Do(%pHead = pH);
end use; 
</MapleCode></EC-TextArea>    <EC-CheckBox id="OptCheck" groupname="" caption="Optimize" pixel-width="300" pixel-height="200" enabled="true" visible="true" selected="true" user-size="false"><MapleCode action="click-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(q = %OptCheck);
if q then
  OpponentOpt();
  Do(%pHead = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lhit,rhit,2,1])));
  Do(%pChest = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lhit,rhit,2,2])));
  Do(%pBlock = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lhit,rhit,2,3])));
  SetProperty(pHead,editable,false);
  SetProperty(pChest,editable,false);
  SetProperty(pBlock,editable,false);
else
  SetProperty(pHead,editable,true);
  SetProperty(pChest,editable,true);
  SetProperty(pBlock,editable,true);
end if;
end use; 
</MapleCode></EC-CheckBox></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1264" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1272" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Initial hit points: You <EC-ComboBox id="LHPBox2" enabled="true" visible="true" selecteditem="10">1,2,3,4,5,6,7,8,9,10<MapleCode action="selection-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(LHP0 = %LHPBox2)
end use; 
</MapleCode></EC-ComboBox>  Opponent  <EC-ComboBox id="RHPBox2" enabled="true" visible="true" selecteditem="10">1,2,3,4,5,6,7,8,9,10<MapleCode action="selection-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(RHP0 = %RHPBox2)
end use; 
</MapleCode></EC-ComboBox></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1275" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1276" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><EC-Button id="HButton" caption="Head" pixel-width="300" pixel-height="200" enabled="true" visible="true" show-border="true" user-size="false"><MapleCode action="click-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.

move2(1);

end use; 
</MapleCode></EC-Button><EC-Button id="CButton" caption="Chest" pixel-width="300" pixel-height="200" enabled="true" visible="true" show-border="true" user-size="false"><MapleCode action="click-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
move2(2)
end use; 
</MapleCode></EC-Button><EC-Button id="BButton" caption="Block" pixel-width="300" pixel-height="200" enabled="true" visible="true" show-border="true" user-size="false"><MapleCode action="click-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
move2(3)
end use; 
</MapleCode></EC-Button>     <EC-Button id="Button3" caption="Reset" pixel-width="300" pixel-height="200" enabled="true" visible="true" show-border="true" user-size="false"><MapleCode action="click-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(LHP0 = %LHPBox2);
Do(RHP0 = %RHPBox2);
lhit:= LHP0; rhit:= RHP0;
if Do(%OptCheck) then
  OpponentOpt();
  ShowOppStrats(); 
end if;
Do(%Plot2 = showboxers(18.5,1,1,0,0,
    lifebar(true,lhit,lhit),
    lifebar(false,rhit,rhit)));
SetProperty(HButton,enabled,true);
SetProperty(CButton,enabled,true);
SetProperty(BButton,enabled,true);
end use; 
 
</MapleCode></EC-Button></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L1292" drawlabel="true">
<Input>
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<Text-field style="Text" layout="Normal">Rounds  won:    You  <EC-TextArea id="LRounds2" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="0" editable="true" wrapping="true"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(lrounds2=%LRounds2);
end use; 
</MapleCode></EC-TextArea>  Opponent  <EC-TextArea id="RRounds2" enabled="true" visible="true" visiblecharacterwidth="5" visiblerows="1" contents="0" editable="true" wrapping="true"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(rrounds2=%RRounds2);
end use; 
</MapleCode></EC-TextArea></Text-field>
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<Input>
<Text-field style="Text" layout="Normal"></Text-field>
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</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Winning probabilities</Text-field></Title><Presentation-Block>
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<Input>
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<Input>
<Text-field style="Text" layout="Normal">Let's suppose we know the mixed strategies the two fighters will use at each turn.  Calculating the winning probabilities for this game is more complicated than for the first game.   We won't get a single formula to do it.  Instead, we'll calculate each <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUklbXN1YkdGJDYlLUYsNiVRImhGJ0YvRjItRiM2JS1JI21uR0YkNiRRIjFGJy9GM1Enbm9ybWFsRidGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGRi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZRLyUqc3ltbWV0cmljR0ZRLyUobGFyZ2VvcEdGUS8lLm1vdmFibGVsaW1pdHNHRlEvJSdhY2NlbnRHRlEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GTDYtUSJ+RidGRkZPL0ZTRlFGVEZWRlhGWkZmbkZobi9GXG9Gam4tRjs2JUY9LUYjNiUtRkM2JFEiMkYnRkZGL0YyRkhGRkZGRkY=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUklbXN1YkdGJDYlLUYsNiVRImhGJ0YvRjItRiM2JS1JI21uR0YkNiRRIjFGJy9GM1Enbm9ybWFsRidGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGRi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZRLyUqc3ltbWV0cmljR0ZRLyUobGFyZ2VvcEdGUS8lLm1vdmFibGVsaW1pdHNHRlEvJSdhY2NlbnRHRlEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GTDYtUSJ+RidGRkZPL0ZTRlFGVEZWRlhGWkZmbkZobi9GXG9Gam4tRjs2JUY9LUYjNiUtRkM2JFEiMkYnRkZGL0YyRkhGRkZGRkY=</Equation> in terms of the other <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImlGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRImpGJ0YvRjJGQUZBRkE=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImlGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRImpGJ0YvRjJGQUZBRkE=</Equation> where <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSYmbGVxO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yNzc3Nzc4ZW1GJy9GTkZTRjUtSSVtc3ViR0YkNiUtRiw2JVEiaEYnRi9GMi1GIzYlLUkjbW5HRiQ2JFEiMUYnRjlGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGOQ==">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</Equation> and <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> (and <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.  We know of course that <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> if  <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5Rjk=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5Rjk=</Equation> (you've already lost if you have 0 hit points) and <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> (you've already won if your opponent has 0).  We can calculate <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnRjpGPkY+Rj4=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnRjpGPkY+Rj4=</Equation>, and then <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY7NiRRIjJGJ0Y+Rj5GPkY+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY7NiRRIjJGJ0Y+Rj5GPkY+</Equation> using that information, then <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY7NiRRIjNGJ0Y+Rj5GPkY+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY7NiRRIjNGJ0Y+Rj5GPkY+</Equation>, and all other <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRImpGJ0YvRjJGPkY+Rj4=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRImpGJ0YvRjJGPkY+Rj4=</Equation> that we're interested in; then <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>, etc.</Text-field>
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<Text-field style="Text" layout="Normal"></Text-field>
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<Input>
<Text-field style="Text" layout="Normal">To calculate <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUklbXN1YkdGJDYlLUYsNiVRImhGJ0YvRjItRiM2JS1JI21uR0YkNiRRIjFGJy9GM1Enbm9ybWFsRidGL0YyLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIsRidGRi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZRLyUqc3ltbWV0cmljR0ZRLyUobGFyZ2VvcEdGUS8lLm1vdmFibGVsaW1pdHNHRlEvJSdhY2NlbnRHRlEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GOzYlRj0tRiM2JS1GQzYkUSIyRidGRkYvRjJGSEZGRkZGRg==">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</Equation>, we look at the possibilities for what happens on the first turn.   The first fighter chooses his move (&quot;head&quot;, &quot;chest&quot; or &quot;block&quot;) with probabilities <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> respectively, according to his mixed strategy.  The second player independently chooses his move with probabilities <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> according to his mixed strategy.  Thus the probability that they choose moves <Font italic="true">i</Font> and <Font italic="true">j</Font> respectively is <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.  The outcome may be determined (e.g. for &quot;head&quot; vs &quot;block&quot; fighter 1 loses 1 hit point), or there may be further probabilities to consider: let's say with probability <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> the outcome is that fighter 1 loses <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+</Equation> hit points and fighter 2 loses <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+</Equation> (where one of these is 0 and the other positive).   This leaves fighter 1 with <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> and fighter 2 with <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> (with negative values being replaced by 0).  We would then be in a situation where your probability of winning is <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.  To get <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> we add up all the  contributions <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> for all <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.  For example, with <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRjs2JFEiM0YnRj5GPg==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRjs2JFEiM0YnRj5GPg==</Equation> and <Equation executable="false" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRjs2JFEiM0YnRj4tRkQ2LVEifkYnRj5GR0ZKRkxGTkZQRlJGVC9GV1EmMC4wZW1GJy9GWkZcb0Y+</Equation> and moves &quot;head&quot; vs &quot;chest&quot;, the probabilities are <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> 0.54, <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>, corresponding to terms <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.  <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation>It would be rather tedious to do all this by hand, but of course for the computer it's an easy calculation.</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4073" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4078" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Here, then, are your probabilities of winning.  For now we'll suppose both players use the same mixed strategy at every turn, as specified below.</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4080" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4082" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Your mixed strategy: probabilities for </Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4083" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4084" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Head: <EC-TextArea id="YHArea" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents="1/3" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pH = %YHArea);
if not(pH::numeric and pH &gt;=0 and pH &lt;= 1) then
  error &quot;Expected a number between 0 and 1, got %1&quot;,pH
end if;
Do(pC = %YCArea);
if pH + pC &gt;= 1 then
   pC:= 1 - pH;
   Do(%YCArea = pC)
end if;
pB:= 1 - pH - pC;
Do(%YBArea = pB);
MixedStrats[1,1..3]:= &lt;pH|pC|pB&gt;;
UpdateWinProbs2();
end use; 
</MapleCode></EC-TextArea>  Chest:  <EC-TextArea id="YCArea" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents="1/3" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pC = %YCArea);
if not(pC::numeric and pC &gt;=0 and pC &lt;= 1) then
  error &quot;Expected a number between 0 and 1, got %1&quot;,pC
end if;
Do(pH = %YHArea);
if pH + pC &gt;= 1 then
   pH:= 1 - pC;
   Do(%YHArea = pH)
end if;
pB:= 1 - pH - pC;
Do(%YBArea = pB);
MixedStrats[1,1..3]:= &lt;pH|pC|pB&gt;;
UpdateWinProbs2()
end use; 
</MapleCode></EC-TextArea>  Block:  <EC-TextArea id="YBArea" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents="1/3" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pB = %YBArea);
if not(pB::numeric and pB &gt;=0 and pB &lt;= 1) then
  error &quot;Expected a number between 0 and 1, got %1&quot;,pB
end if;
Do(pC = %YCArea);
if pB + pC &gt;= 1 then
   pC:= 1 - pB;
   Do(%YCArea = pC)
end if;
pH:= 1 - pB - pC;
Do(%YHArea = pH);
MixedStrats[1,1..3]:= &lt;pH|pC|pB&gt;;
UpdateWinProbs2()
end use; 
</MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4088" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4089" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Opponent's mixed strategy: probabilities for </Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4090" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4091" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Head: <EC-TextArea id="OHArea" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".45" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pH = %OHArea);
if not(pH::numeric and pH &gt;=0 and pH &lt;= 1) then
  error &quot;Expected a number between 0 and 1, got %1&quot;,pH
end if;
Do(pC = %OCArea);
if pH + pC &gt;= 1 then
   pC:= 1 - pH;
   Do(%OCArea = pC)
end if;
pB:= 1 - pH - pC;
Do(%OBArea = pB);
MixedStrats[2,1..3]:= &lt;pH|pB|pC&gt;;
UpdateWinProbs2();
end use; 
</MapleCode></EC-TextArea>  Chest:  <EC-TextArea id="OCArea" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents="0.45" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pC = %OCArea);
if not(pC::numeric and pC &gt;=0 and pC &lt;= 1) then
  error &quot;Expected a number between 0 and 1, got %1&quot;,pC
end if;
Do(pH = %OHArea);
if pH + pC &gt;= 1 then
   pH:= 1 - pC;
   Do(%OHArea = pH)
end if;
pB:= 1 - pH - pC;
Do(%OBArea = pB);
MixedStrats[2,1..3]:= &lt;pH|pB|pC&gt;;
UpdateWinProbs2();
end use; 
</MapleCode></EC-TextArea>  Block:  <EC-TextArea id="OBArea" enabled="true" visible="true" visiblecharacterwidth="10" visiblerows="1" contents=".10" editable="true" wrapping="false"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(pB = %OBArea);
if not(pB::numeric and pB &gt;=0 and pB &lt;= 1) then
  error &quot;Expected a number between 0 and 1, got %1&quot;,pB
end if;
Do(pC = %OCArea);
if pB + pC &gt;= 1 then
   pC:= 1 - pB;
   Do(%OCArea = pC)
end if;
pH:= 1 - pB - pC;
Do(%OHArea = pH);
end use; 
</MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4095" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4096" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">                                    <EC-Label id="Label2" caption="Opponent's Hit Points" enabled="true" visible="true" visiblecharacterwidth="20" pixel-width="300" pixel-height="200" user-size="false" user-width="true"></EC-Label></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4074" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"> <EC-Label id="Label3" caption="Your hit points" enabled="true" visible="true" visiblecharacterwidth="5" pixel-width="300" pixel-height="200" user-size="false" user-width="false"></EC-Label> <EC-RTableBrowser id="Win2DT" enabled="true" visiblerows="6" visiblecolumns="6" visible="true" editable="true" variable="Win2Out" columnnames="&quot;1&quot;,&quot;2&quot;,&quot;3&quot;,&quot;4&quot;,&quot;5&quot;,&quot;6&quot;,&quot;7&quot;,&quot;8&quot;,&quot;9&quot;,&quot;10&quot;" rownames="&quot;1&quot;,&quot;2&quot;,&quot;3&quot;,&quot;4&quot;,&quot;5&quot;,&quot;6&quot;,&quot;7&quot;,&quot;8&quot;,&quot;9&quot;,&quot;10&quot;" rowheader="true" columnheader="true" handle="140627172"><MapleCode action="content-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.

end use; 
</MapleCode></EC-RTableBrowser></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4061" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block>
</Section>
<Section collapsed="false" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title>
<Text-field style="Heading 1" layout="Heading 1">Optimal play</Text-field></Title><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L4062" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L6056" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">In the last section we assumed that the same mixed strategies were used no matter how many hit points each fighter had.  That's not very realistic.  A good strategy when you're winning might not be a good strategy when you're losing.  But how can you find the best strategy?  What does &quot;best&quot; mean in this context?</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L6058" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L6684" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Game theory can answer this question.  This game is an example of a &quot;two-person zero-sum game&quot;, where your payoff is 1 if you win the round and 0 if you lose.  Game theory says that each player has an optimal strategy, and the game has a value <Font italic="true">v</Font>: if you use your optimal strategy, you ensure yourself probability at least <Font italic="true">v</Font> of winning the round, no matter what strategy your opponent uses.  On the other hand, by using his optimal strategy, your opponent ensures that your probability of winning the round is at most <Font italic="true">v</Font>, no matter what you do.  If you use a strategy that is not optimal, you can't do any better than probability <Font italic="true">v</Font> against an opponent who uses his optimal strategy, and you might do worse.</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L6686" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
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<Text-field style="Text" layout="Normal">The complete round, involving many different moves, would be very complicated to analyze using game theory.  Fortunately, however, each move can be analyzed as a two-person zero-sum game, in which your payoff is your probability of winning the round.  Again we will look at each situation where the hit points are <Equation executable="false" style="2D Math" input-equation="" 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after already analyzing  all other pairs of hit points <Equation executable="false" style="2D Math" input-equation="" 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with  <Equation executable="false" style="2D Math" input-equation="" 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and <Equation executable="false" style="2D Math" input-equation="" 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so we will know your probability of winning the round given any possible result of this move.   Let <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImlGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRImpGJ0YvRjJGQUZBRkE=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEid0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImlGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRImpGJ0YvRjJGQUZBRkE=</Equation> be the expected probability of winning the round if you choose move <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=</Equation> and your opponent chooses move <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiakYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=</Equation>.  To get your optimal mixed strategy, you solve the following linear programming problem:</Text-field>
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<Text-field style="Text" layout="Normal">maximize <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=</Equation> </Text-field>
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<Text-field style="Text" layout="Normal">subject to</Text-field>
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<Text-field style="Text" layout="Normal">   <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>    </Text-field>
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<Text-field style="Text" layout="Normal">The interpretation of this is that <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation> are your mixed strategies, and <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=</Equation> is the value of the game, your probability of winning in this situation with hit points <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.  The right side of the first inequality,<Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2I1EhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn">JSFH</Equation><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>, would be your probability of winning if you used mixed strategy  <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>  and the opponent uses move 1 (&quot;head&quot;), and similarly the other two inequalities come from your opponents's other two moves.</Text-field>
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<Text-field style="Text" layout="Normal">For example, with <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=</Equation><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkobWZlbmNlZEdGJDYkLUYjNiYtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYkUSIxRicvRjtRJ25vcm1hbEYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLEYnRkMvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGTi8lKnN5bW1ldHJpY0dGTi8lKGxhcmdlb3BHRk4vJS5tb3ZhYmxlbGltaXRzR0ZOLyUnYWNjZW50R0ZOLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JUYzLUYjNiUtRkA2JFEiMkYnRkNGN0Y6RkVGQ0ZDLUZJNi1RIn5GJ0ZDRkwvRlBGTkZRRlNGVUZXRllGZW4vRmluRmduLUZJNi1RIj1GJ0ZDRkxGZW9GUUZTRlVGV0ZZL0ZmblEsMC4yNzc3Nzc4ZW1GJy9GaW5GW3BGYm8tRiw2JC1GIzYmRj9GSEY/RkNGQy1GNDYjUSFGJ0ZD">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkobWZlbmNlZEdGJDYkLUYjNiYtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJoRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYkUSIxRicvRjtRJ25vcm1hbEYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiLEYnRkMvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGTi8lKnN5bW1ldHJpY0dGTi8lKGxhcmdlb3BHRk4vJS5tb3ZhYmxlbGltaXRzR0ZOLyUnYWNjZW50R0ZOLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JUYzLUYjNiUtRkA2JFEiMkYnRkNGN0Y6RkVGQ0ZDLUZJNi1RIn5GJ0ZDRkwvRlBGTkZRRlNGVUZXRllGZW4vRmluRmduLUZJNi1RIj1GJ0ZDRkxGZW9GUUZTRlVGV0ZZL0ZmblEsMC4yNzc3Nzc4ZW1GJy9GaW5GW3BGYm8tRiw2JC1GIzYmRj9GSEY/RkNGQy1GNDYjUSFGJ0ZD</Equation>, the first three inequalities are</Text-field>
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<Text-field style="Text" layout="Normal">    <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
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<Text-field style="Text" layout="Normal">where, for example, the <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFElMC41NEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiUtRiw2JFEiMkYnRi9GUUZULyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMkYv">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JFElMC41NEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYjNiUtRiw2JFEiMkYnRi9GUUZULyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMkYv</Equation> comes from the fact that if you use &quot;chest&quot; and your opponent uses &quot;head&quot;, with probability 0.54 the opponent loses 2 hit points (and you win the round) while with probability 0.46 you lose 3 hit points (and you lose the round).  The optimal solution for this problem is <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>,<Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.  Thus linear programming is saying that in this situation you should always use &quot;chest&quot;, obtaining a probability of winning at least 0.5.  The opponent's optimal strategy is also to always use &quot;chest&quot;.   If your opponent chooses anything else when you use &quot;chest&quot;, it would increase your probability of winning.  If you choose anything else when your opponent uses &quot;chest&quot;, it would decrease your probability of winning.</Text-field>
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<Text-field style="Text" layout="Normal">It's easy to see why this is the case.  With both fighters having just 1 hit point, &quot;chest&quot; is a better move than &quot;head&quot;, no matter what your opponent does.  And once you know that your opponent is not going to use &quot;head&quot;, there's no reason to use &quot;block&quot; either.</Text-field>
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<Text-field style="Text" layout="Normal">On the other hand, with <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>, if you use &quot;chest&quot; and your opponent uses &quot;head&quot; and the opponent loses 2 hit points, the hit points will go to <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIzRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMUYnRjRGNEY0RjQ=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIzRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMUYnRjRGNEY0RjQ=</Equation>, which is not a sure win.  From the linear programming problem for hit points <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIzRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMUYnRjRGNEY0RjQ=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIzRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMUYnRjRGNEY0RjQ=</Equation>, your probability of winning in that situation is  <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEtMC42NzYwNTYzMzgwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi8=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEtMC42NzYwNTYzMzgwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRi8=</Equation>.  So instead of .54, the coefficient of <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+</Equation> in the first inequality would be <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>.  The three inequalities for this linear programming problem turn out to be</Text-field>
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<Text-field style="Text" layout="Normal">   <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
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<Text-field style="Text" layout="Normal">Note that in this case there would be an advantage to using &quot;head&quot; if you knew your opponent was using &quot;chest&quot;, because <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEtMC42MzQ5Mjk1Nzc1RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEifkYnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1GMzYtUSI+RidGL0Y2RjlGO0Y9Rj9GQUZDL0ZGUSwwLjI3Nzc3NzhlbUYnL0ZJRk5GMi1GLDYkUSQwLjVGJ0YvRi8=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEtMC42MzQ5Mjk1Nzc1RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEifkYnRi8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjgvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRy1GMzYtUSI+RidGL0Y2RjlGO0Y9Rj9GQUZDL0ZGUSwwLjI3Nzc3NzhlbUYnL0ZJRk5GMi1GLDYkUSQwLjVGJ0YvRi8=</Equation>.   None of the moves is better than another in all situations, and in fact the optimal solution here involves all three moves:<Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>
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<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7016" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Your opponent's optimal mixed strategies can be obtained from a similar linear programming problem, which turns out to be the dual of the one for your optimal mixed strategies.</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7017" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L6994" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">The following table presents your probability of winning with optimal play, for each pair </Text-field>
</Input>
</Group>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L6996" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
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<Text-field style="Text" layout="Normal"> </Text-field>
</Input>
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<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7002" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">                                    <EC-Label id="Label5" caption="Opponent's Hit Points" enabled="true" visible="true" visiblecharacterwidth="20" pixel-width="300" pixel-height="200" user-size="false" user-width="true"></EC-Label></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7003" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal"> <EC-Label id="Label6" caption="Your hit points" enabled="true" visible="true" visiblecharacterwidth="5" pixel-width="300" pixel-height="200" user-size="false" user-width="false"></EC-Label> <EC-RTableBrowser id="Win3OutDT" enabled="true" visiblerows="6" visiblecolumns="6" visible="true" editable="true" variable="Win3Out" columnnames="&quot;1&quot;,&quot;2&quot;,&quot;3&quot;,&quot;4&quot;,&quot;5&quot;,&quot;6&quot;,&quot;7&quot;,&quot;8&quot;,&quot;9&quot;,&quot;10&quot;" rownames="&quot;1&quot;,&quot;2&quot;,&quot;3&quot;,&quot;4&quot;,&quot;5&quot;,&quot;6&quot;,&quot;7&quot;,&quot;8&quot;,&quot;9&quot;,&quot;10&quot;" rowheader="true" columnheader="true" handle="140627236"><MapleCode action="content-event-handler"></MapleCode></EC-RTableBrowser></Text-field>
</Input>
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<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L6997" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">
Here you can see the optimal strategies for you and your opponent for any given </Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7005" drawlabel="true">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="" display="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">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</Equation></Text-field>
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<Text-field style="Text" layout="Normal"></Text-field>
</Input>
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<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7006" drawlabel="true">
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<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7007" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Your hit points: <EC-ComboBox id="YHPBox" enabled="true" visible="true" selecteditem="2">1,2,3,4,5,6,7,8,9,10<MapleCode action="selection-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(lh = %YHPBox);
Do(rh = %OHPBox);
Do(%YHead = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,1,1])));
Do(%YChest = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,1,2]))); 
Do(%YBlock = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,1,3])));
Do(%OHead = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,2,1])));
Do(%OChest = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,2,2]))); 
Do(%OBlock = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,2,3])));
end use; 
</MapleCode></EC-ComboBox>  Opponent's hit points: <EC-ComboBox id="OHPBox" enabled="true" visible="true" selecteditem="6">1,2,3,4,5,6,7,8,9,10<MapleCode action="selection-event-handler">use DocumentTools in 
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use: 
#    Do( %component_name );
# and
#    Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
#    Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(lh = %YHPBox);
Do(rh = %OHPBox);
Do(%YHead = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,1,1])));
Do(%YChest = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,1,2]))); 
Do(%YBlock = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,1,3])));
Do(%OHead = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,2,1])));
Do(%OChest = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,2,2]))); 
Do(%OBlock = sprintf(&quot;%0.10f&quot;,abs(OMixedStrats[lh,rh,2,3])));
end use; 
</MapleCode></EC-ComboBox></Text-field>
</Input>
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<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7024" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Your optimal mixed strategy:  </Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7019" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Head <EC-TextArea id="YHead" enabled="true" visible="true" visiblecharacterwidth="13" visiblerows="1" contents="0.7502961993" editable="false" wrapping="true"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea> Chest  <EC-TextArea id="YChest" enabled="true" visible="true" visiblecharacterwidth="13" visiblerows="1" contents="0.2497038007" editable="false" wrapping="true"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea>  Block <EC-TextArea id="YBlock" enabled="true" visible="true" visiblecharacterwidth="13" visiblerows="1" contents="0.0000000000" editable="false" wrapping="true"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7025" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Opponent's optimal mixed strategy:</Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7026" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">Head <EC-TextArea id="OHead" enabled="true" visible="true" visiblecharacterwidth="13" visiblerows="1" contents="0.0000000000" editable="false" wrapping="true"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea> Chest  <EC-TextArea id="OChest" enabled="true" visible="true" visiblecharacterwidth="13" visiblerows="1" contents="0.6845416422" editable="false" wrapping="true"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea>  Block <EC-TextArea id="OBlock" enabled="true" visible="true" visiblecharacterwidth="13" visiblerows="1" contents="0.3154583578" editable="false" wrapping="true"><MapleCode action="content-event-handler"></MapleCode></EC-TextArea></Text-field>
</Input>
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<Input>
<Text-field style="Text" layout="Normal"></Text-field>
</Input>
</Group></Presentation-Block><Presentation-Block>
<Group view="presentation" hide-input="false" hide-output="true" inline-output="false" labelreference="L7040" drawlabel="true">
<Input>
<Text-field style="Text" layout="Normal">If your opponent is using an optimal mixed strategy, your expected payoff will be the same for all of your moves that have nonzero probabilities in your optimal mixed strategy.  So e.g. if you have 2 hit points and your opponent has 6, your optimal mixed strategy involves &quot;Head&quot; and &quot;Chest&quot; but not &quot;Block&quot;.  If you know the opponent is using his optimal mixed strategy, you could actually use any combination of &quot;Head&quot; and &quot;Chest&quot; and have the same expected payoff: in this case probability 0.1644 of winning.  However, against a clever opponent it would be dangerous to use anything other than your optimal mixed strategy, because if the opponent could figure out which strategy you were using he could adjust his strategy to take advantage of that.</Text-field>
</Input>
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<Input>
<Text-field style="Text" layout="Normal"></Text-field>
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