You’re not really arguing against the whole crowd there, a lot of people (wrongly) hold the same opinion. The problem is thinking of the door swap as an independent event when it’s not; the result is directly related to the original choice of door. If we label the doors A, B, and C and put the prize behind door A, here’s the possible options:
Edit: I see you added a table to your comment, but you’re miscounting pretty badly there. You’re giving double weight to initial choice being correct.
It is technically true that when you pick A the presenter can open either B or C, but then you need to account for that in your odds; it’s 50% either way so the win/loss rate is halved. In other words:
As shown, including which door the presenter opens does not affect the odds. When sticking, you win (16.5% + 16.5% = 33%) and lose (33% + 33% = 66%), when swapping you win (33% + 33% = 66%) and lose (16.5% + 16.5% = 33%).
You’re not really arguing against the whole crowd there, a lot of people (wrongly) hold the same opinion. The problem is thinking of the door swap as an independent event when it’s not; the result is directly related to the original choice of door. If we label the doors A, B, and C and put the prize behind door A, here’s the possible options:
Initial Choice A - Stick: win - Swap: lose Initial Choice B: - Stick: lose - Swap: win Initial Choice C: - Stick: lose - Swap: winTwo out of three times swapping wins.
Edit: I see you added a table to your comment, but you’re miscounting pretty badly there. You’re giving double weight to initial choice being correct.
It is technically true that when you pick A the presenter can open either B or C, but then you need to account for that in your odds; it’s 50% either way so the win/loss rate is halved. In other words:
Initial Choice A - 33% - Presenter opens B - 50% - Stick: win (16.5%) - Swap: lose (16.5%) - Presenter opens C - 50% - Stick: win (16.5%) - Swap: lose (16.5%) Initial Choice B - 33% - Presenter opens C - 100% - Stick: lose (33%) - Swap: win (33%) Initial Choice C - 33% - Presenter opens B - 100% - Stick: lose (33%) - Swap: win (33%)As shown, including which door the presenter opens does not affect the odds. When sticking, you win (16.5% + 16.5% = 33%) and lose (33% + 33% = 66%), when swapping you win (33% + 33% = 66%) and lose (16.5% + 16.5% = 33%).