Solution to Exercise

Question 1. You should switch to either the purple box or the magenta box.

There was a 1 in 4 chance at the outset that your original choice, the red box, contained the prize. This does not change when I open the box which I know to be empty. There was a 3 in 4 chance that it was either the orange box, the purple box or the magenta box before I opened the box and by opening the orange box, which I know to be empty, that can be eliminated. So the chance it is either the purple box or the magenta box is now 3 in 4 in total (or 3/8 each), compared to 1 in 4 for your original choice, the red box.

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

There was a 1 in 4 chance at the outset that your original choice, the black box, contained the prize. There was a 3 in 4 chance that it was either the white box, the grey box or the brown 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 white box, which might have contained the prize, this means there are now two boxes left (grey and brown). Each of these started with a 1 in 4 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 each of the remaining boxes rises to 1/3 in each case. So it makes no difference whether you switch or not.

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