Steve Schiffman’s notes on the infamous Monty Hall problem
Numberphile video narrated by Lisa Goldberg:
https://www.youtube.com/ embed/4Lb-6rxZxx0?start=0&end=315
Brady Haran’s explanation:
https://www.youtube.com/embed/ 7u6kFlWZOWg?start=0&end=253
Steve's full explanation:
The concentrated 2/3 idea is a valid way to look at it, but it does not explain why switching does not help you under the scenario that Monty opens a door at random (rather than always opens a door with a Zonk). In other words, why does the switching strategy not always help whether Monty had opened the door at random or not? Actually, depending on what game rule you choose, your chances of winning if you play under optimal strategy is either 1/3, 1/2 or 2/3 (!).
This is a subtle but key point: we need to know what the game rules are if Monty were to open a door at random and it were to reveal a car. We never had to worry about this if Monty always chose a door to reveal a Zonk.
There are two possible rules we need to consider in this case where Monty opens a door to reveal a car: whether you (the player) get a “do-over” (i.e. neither win nor lose, but rather get to play again) or not.
First possible rule: no “do-over”, i.e. if Monty opens a door and it reveals a car you LOSE and the next player comes up to take your place. Under this rule, 1/3 of the time you will lose without even getting a chance to switch, because 1/3 of the time Monty will choose the door with the car if he is indeed just choosing a door at random. For the other 2/3 of the time (where Monty randomly choses a door and it does NOT reveal the car) then there is indeed a 50-50 chance that the car is behind either remaining door and so switching doesn’t help you; switching or not you would choose the Zonk 1/2 of that 2/3 (i.e. 1/3) of the time. Thus if Monty chooses his door at random, and you will LOSE 1/3 + 1/3 = 2/3 of the time. In other words under the no “do-over” rule your chances of winning are 1/3.
Second possible rule: if Monty opens a door and it reveals a car you get a “do- over”: you don’t lose or win, you get to play again. In this case you are always looking at the “other 2/3 of the time” described in the paragraph above (where Monty randomly choses a door and it does NOT reveal the car). Recall that in the case there is indeed a 50-50 chance that the car is behind either remaining door and so switching doesn’t help you; switching or not you would choose the Zonk 1/2 of of the time. Thus if Monty chooses his door at random, and the rules are that you get a “do-over” if Monty chooses a door with the car behind it, whether or not you switch doors your chances of winning are 1/2.
My analysis of scenarios where Monty chooses door at random
But if Monty knows where the car is, and he must always open a door to reveal a Zonk, then Lisa Goldberg’s argument holds and by always switching your chances of winning are 2/3.
My analysis of (classic) case where Monty never reveals the car
Here is a Monty Hall game simulator you can play with.
