Monty Hall Probability
You find yourself on a game show and are asked to play the following game:
A prize has been hidden behind one of three doors, selected randomly. You must select a door, and the show's host will then open one of the other doors, which is not the door hiding the prize. You are then given the opportunity to switch your door with the one remaining door, before the host opens your door to show you whether you've won the prize.
To maximize your chances of winning, should you switch doors? Try your luck in this interactive simulation, and watch your statistics.
In the beginning, you know that you have a 1/3 chance of choosing the right door. You might think that, after one of the options is eliminated, now your door and the remaining door both have an equal 1/2 chance to be hiding the prize. This would seem reasonable, as probability suggests that distribution of chance should be even across your options. However, this is not the case, because information is in fact given regarding the choices. By switching doors, you actually raise your chances of winning from 1/3 to 2/3. Here is an explanation.
In the beginning, each door has an equal chance of hiding the prize. You choose a door, and it has 1/3 chance of being the right one. Let's say you are going to stay with this door. At this point, we don't really care about the other doors anymore, because our choice remains static regardless of any information given. Because we're staying with this door, and if it's the right door, we will assuredly win, then we have 1/3 chance of winning. Therefore staying with the door you first chose has effectively a 1/3 chance of winning.
Let's say you were in fact wrong in your original door choice. Then what happens is that the show's host eliminates the other wrong choice for you (because he can't reveal the winning door). At this point, you will win the prize if you switch. Now, since there was a 2/3 chance of being wrong on the first choice, this means you have a 2/3 chance of winning if you switch!
With that, you can see how math will help you if you are ever on a game show!
Still not convinced? Try the Monty Hall game out yourself, below. Or, if that's too slow, you can use the automated game system below the statistics for faster results.
Click a button to select that door as your original door. The host will then open a door without the prize. At this point, you may either select your door again or switch to the remaining door in an attempt to choose the door with the prize behind it. The winning door will then be revealed. Click "New Round" afterwards to start the next round.
Alternatively, use the slider to change the number of rounds to simulate, select your option in the radio buttons, and click the button to have the computer play rounds for you.
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