Quote:
Originally posted by ModelPerfect
It's also interesting to note that your odds are also 66.7/33.3 if the host RANDOMLY selects which box to open out of the 2 left. Again, this is because he gives more information to you and thus throws off the randomness. i.e. The difference comes in when he randomly opens C to reveal the cash...you obviously will choose this one.
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Wrong.
If he *randomly* (!!!) selects which box to open of the boxes you didn't pick, and the one he selects happens to be empty, the ones that are left both have a 50% chance of containing the prize.
You see, in the original problem, you don't get any new information, because he will never reveal the box containing the prize, and there is always a box to open which doesn't contain a prize.
However, if a box is randomly selected, there is a 1/3 chance that it's the real one. If the box happens to be empty, that does not magically transfer 1/3 chance to the other box you didn't pick. It's just eliminated as a genuine possibility.
If a genuine possibility gets eliminated, chances get redistributed entirely.
The actual problem is:
X = A (1/3 chance) or Y (2/3 chance)
Y= (B or C)
Y will always contain at least 1 empty box, and an empty box will always be revealed. No genuine possibility will get eliminated by revealing a box.
The problem when used with randomness is:
X = A (1/3) or B (1/3) or C (1/3)
if X = B or C it might get revealed. A genuine possibility is confirmed or eliminated.
Non-random revealing: 1/3 vs 2/3
Random revealing: 1/2 vs 1/2