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Solution: Monty Hall Problem – in a nutshell.

April 24, 2019

Solution to Exercise

Question 1. You should switch to the red box.

There was a 1 in 3 chance at the outset that your original choice, the blue box, contained the prize. This does not change when I open the box which I know to be empty. There was a 2 in 3 chance that it was either the red box or the yellow box before I opened the box and by opening the yellow box, which I know to be empty, that can be eliminated. So the chance it is the red box is now 2 in 3, compared to 1 in 3 for your original choice, the blue box.

Question 2. It makes no difference whether you switch or not.

There was a 1 in 3 chance at the outset that your original choice, the green box, contained the prize. There was a 2 in 3 chance that it was either the pink box or the violet box before I opened the box. By randomly opening a box (I don’t know which box contains the prize), I am giving you no new information. It is the same as asking you to choose a box to open. If you randomly opened the pink box, which might have contained the prize, this means there are now two boxes left (green and violet). Each of these started with a 1 in 3 chance of containing the prize. I have not deliberately eliminated a box potentially containing the prize, so I have given you no new information to indicate which box contains the prize. So the chance of both remaining boxes rises to ½ in each case. So it makes no difference whether you switch or not.

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