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
Think of it as rolling a three-sided die (a 2d triangle with the selected “side” being the edge pointing up), since you don’t really have any information at first, so it’s random. When you roll said die, there’s only a 1/3 chance the car will be picked, therefore, it’s more likely that it’s in one of the doors you didn’t choose, or in this case, in one of the sides in the base. The important thing is: The host will NEVER reveal a side (door) with the car in it, always a goat, so, if there was a 2/3 chance the car was on the bottom, that didn’t change, since it is still a three-sided die, no matter how many or which of those are revealed or hidden, therefore the chance you picked wrong is still 2 in 3, so the real question the host is asking, probability-wise, when they ask if you want to switch, is if you want to invert the game and bet the car is in one of the base sides.
Try thinking of the same situation with a 6-sided die, and you’ll see that it starts to become more and more likely that the car will not be on top, so the safest bet is to say it’s not the one you chose.
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