The intuitive explanation I like is this: Suppose there are 10000 boxes. You pick one. Monty Hall knows where the prize is. He opens 9998 boxes, leaving just two closed: your original choice, and one other one. Should you switch? Before Monty Hall opened any boxes, the chances of the prize being in one of the boxes other than the one you chose was 9999 in 10000. This probability does not change when he opens 9998 of these. So you should choose the remaining box, not your original choice.

The same argument applies with 3 boxes, but intuition is much less reliable in this case.

This problem is confusing partly because the premises are often incorrectly stated... as in TFA.

If it is given that Monty always picks a door that does not have a prize behind it, then indeed the correct answer is to switch doors. But otherwise, there is no advantage in switching: 1/3 of the time he would expose the prize, and in the other 2/3 of cases you are just as likely to win without switching.

Many previous threads: https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...

(Sharing for curiosity purposes.)