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I find it's easier to explain the answer when you consider 100 doors. You've selected 1 door out of 100 - which is a 1/100 chance of getting it right. The host

I find it's easier to explain the answer when you consider 100 doors. You've selected 1 door out of 100 - which is a 1/100 chance of getting it right. The host then opens 98 doors, leaving you with 2 doors - the one you selected, and one other. Are you going to change your choice? No matter what you selected initially, the host was still going to open 98 doors and give you the chance to swap your choice. Would you? It's obviously not a 1/2 selection at that point, is it? There were still 100 options, but now 98 of them are eliminated. Your original door was (and is) 1/100, making the other option 99/100 for a total of 100/100. If you extend that out to 1,000,000 doors the game is even easier to play. What are the odds that you just happened to select that one special door out of 1,000,000 options?

sk5t 12y ago
Exactly--this case shows at an intuitively obvious level that the original problem looks like even odds, but the elimination of only non-winning doors always provides valuable information.
SiVal 12y ago
I'll toss in another nudge to intuition. You select one of the hundred doors. Then Monty takes the other 99 and opens door after door, displaying goats, but when he reaches one of them, he carefully skips over it, then continues to the last door showing goat after goat. Then he says, "Okay, the car is either behind the door you selected or the one I selected by skipping over it. There's only one car, so if you picked the right one, then I had to pick a random goat door to skip, but if the door you picked was the wrong one, the door I skipped was the one with the car. And, unlike you, I know which door has the car. Would you like to keep the door that you originally picked or switch to the one that I just picked?" You don't know where the car is; he does. Do you want your pick or his?

sk5t 12y ago

Exactly--this case shows at an intuitively obvious level that the original problem looks like even odds, but the elimination of only non-winning doors always provides valuable information.

SiVal 12y ago

I'll toss in another nudge to intuition. You select one of the hundred doors. Then Monty takes the other 99 and opens door after door, displaying goats, but when he reaches one of them, he carefully skips over it, then continues to the last door showing goat after goat. Then he says, "Okay, the car is either behind the door you selected or the one I selected by skipping over it. There's only one car, so if you picked the right one, then I had to pick a random goat door to skip, but if the door you picked was the wrong one, the door I skipped was the one with the car. And, unlike you, I know which door has the car. Would you like to keep the door that you originally picked or switch to the one that I just picked?" You don't know where the car is; he does. Do you want your pick or his?