Imagine there is a prize in one of three boxes. You chose one of the boxes, and then someone opens one of the other two boxes, revealing nothing inside. You then switch your guess to the other unopened box. What percent chance do you have of this being the box with the prize?The problem is that, in this scenario, it is unknown whether the person revealing the empty box had any chance of revealing a box with a prize, since presumably if they had opened a box with the prize, they wouldn't have let me switch my guess to that box.

If they might have revealed a box with the prize, then the probability breaks down this way: there's a 1/3 chance that I chose the right box, and they reveal an empty box; there's a 1/3 chance that I chose the wrong box, and they reveal an empty box; and there's a 1/3 chance that I chose the wrong box, and they reveal the prize (and don't let me change my guess). Hence, if they reveal an empty box, then there is a 50% chance that the prize is in each of the remaining boxes.

If they never reveal the box with the prize, then the probability breaks down this way: there's a 1/3 chance that I chose the right box, and they reveal an empty box; and there's a 2/3 chance that I chose the wrong box, and they reveal an empty box, meaning that the prize is in the remaining box.

So I've asked turkers again, which this phrasing:

Imagine a game with a prize in one of three boxes. The game works like this: you choose a box, the host reveals an empty box that you didn't choose, and then you have a chance to switch your guess to the remaining box. What percent chance do you have of getting the prize if you switch to the remaining box?I'm not totally satisfied with this phrasing either. Though, I suspect the outcome on turk will be the same, since I think people typically only get this problem right if they've heard it before, and if they've heard it before, I think they'll assume it's the correct version of the problem, without reading it carefully.

I'm also curious, of the people who have heard it before, how many would believe that it matters whether the host was definitely going to show an empty box, since if the host does show an empty box, it seems equivalent to a host that was definitely going to show an empty box -- especially since we're just asking about a particular instance of the game.

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