Monty Hall

You may have heard a bunch of hype lately about the Monty Hall problem, that keeps raising its head from time to time. The problem, stated unambiguously (and thus verbosely), is this:

Suppose you're on a game show and you're given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?

Here is how I solved the problem, in my head:

Said another way:

Said yet another way:
Each door has a ⅓ chance of being the car. If we split the doors into: the one I picked, and the two I didn't pick, the "one I picked" has ⅓ chance of being the car, and the chance that the car is in the "two I didn't pick" is ⅔. One of the two I didn't pick is revealed as a goat door, so it has a 0 chance. Therefore the other door I didn't pick must have (⅔ - 0 = ⅔) chance of being the car.

There is another path to consider:
If I know beforehand that my first choice doesn't matter, because after I make it Monty will eliminate one of the doors, I can "throw away" my first choice. After Monty gets rid of one of the doors, I know that there is a ½ chance that I will pick the car door. By discarding the past, I now have a 50% chance of getting the car. That's true and correct and all, because if you eliminate past knowledge, you really do have a ½ chance of picking the right door. But why would you do that? There's a perfectly valid option that gives you a ⅔ chance, which is much better than ½. Why not pay attention to past information and use it to your advantage?

... Matty /<

Matthew Kerwin

CC BY-SA 4.0

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