Stick Triangle
Suppose you break a stick into three pieces and try to assemble them into a triangle. If you choose the two breaking points at random points in the interval, the theoretical probability that the pieces are able to form a triangle is or 25%. To see how well this theoretical probability works in practice, select two points on the line below at random by clicking on the image then click 'Assemble Triangle' to see if a triangle can be made out of those pieces. To try to make another triangle yourself, click 'Start a new triangle' and select new breaking points. You can also let Maple generate random sets of breaking points.
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Explanation of probability
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Assume the endpoints of the stick are at the points 0 and 1.
Let and be the two breaking points, chosen at random from the values in the interval .
Let us assume at first that . The three sides of the triangle have lengths , , and . The three conditions that must be satisfied for a triangle to be formed are thus:
,
,
.
Simplifying these inequalities gives us:
, , and .
We can rearrange this to give:
, .
The total probability for this case can thus be given by the integral:
.
Considering that the other case (that ) is equally likely, the total probability that a triangle can be formed is .
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