Game One: You are given a choice of three doors. You pick number one. The host opens one of the other two doors, having been given instructions that, if you pick the car, the host is to open one of the other doors, and if you pick a goat, the host opens the other door with a goat. Stalemate. It is a predetermined outcome.
Game Two: The prior game’s outcome stands. The new choice you have is do you keep door number one, or do you switch?
How do you have a 2/3 chance of winning if you switch?
In: 107
It helps me to think in terms of sets, rather than the three individual options.
There are three choices, A,B and C. You choose one, it doesn’t matter which. You can think of the three choices now being divided into two sets: the ones you chose and the ones you didn’t choose. It’s pretty obvious the set of ones you did not choose is twice as likely to contain the prize. It’s twice as big.
Then Monty Hall reveals one of the two in that Did-Not-Choose set. The 2/3 likelihood that the prize is in that second set does not change, it simply collapses down now to the one unrevealed choice in the Did-Not-Choose set. The set still contains 2/3 of the probability…we just have more clarity where the prize must be, if it’s inside that’s set.
Isn’t this simply a second game?, you asked. I get that. It sure feels that way. And if Monty Hall revealed one of the two options in that set at _random_ then yes, that would be the case. But he does NOT choose at random. He always only intentionally reveals “goats.” A game of chance is not a game of chance when one of the players is intentionally crafting the outcome.
This feels more intuitive if we change the numbers. Instead of three choices to pick from, let’s make it 100. You make your choic. Let’s say you pick #1 out of the 100. It’s 99% likely that the prize is in that huge Did-Not-Choose set.
Monty knows which one is the prize, and which of his 98 others are goats.
_So he reveals 98 goats._ There’s only one option left in the Did-Not-Choose set.
That 99% probability still exists. It’s just been concentrated, purified, into that one remaining option in his set. If it was 99% likely before, it’s still 99% likely now.
But your final choice is still between your set and his set. I’d go for his set.
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