Here's the solution to yesterday's Price is Right brain-teaser:
The proper course of action is to switch. Most people, including Marilyn vos Savant, think that the answer is that it doesn't make a difference: after you take one choice away it's fifty-fifty which door the grand prize is behind. As it happens, that's not the case, because of the ground rules under which the game is played--specifically the fact that Bob will never choose to reveal what's behind the door you've picked before showing one of the other doors, and will never show the grand prize (creating an anti-climax) before offering you a chance to switch. If you switch you'll win two-thirds of the time; contrariwise if you stay, you'll only win one-third of the time.
Proof:
Let's call the doors A, B, and C. Without loss of generality, let's assume that the grand prize is behind door C, and you pick the first door randomly. Watch what happens in each case:
| Chance: | You pick: | Bob Reveals: | You switch to: |
| 1/3 | A | B | C - you win |
| 1/3 | B | A | C - you win |
| 1/6 | C | A | B - you lose |
| 1/6 | C | B | A - you lose |
If you end up on C, you win the grand prize.
It's important to note that in the first two cases (2/3 chance total) where you picked a door behind which there was Rice-A-Roni, Bob has no choice--he can't reveal the grand prize, so he must reveal whichever Rice-A-Roni door you didn't pick. Switching will then guarantee that you end up on the door with the grand prize. It's only if you happened to pick the grand-prize door at the start that you will end up switching off of it and onto a Rice-A-Roni door, but a priori there's only 1/3 chance of that. Ergo, the strategy of switching results in a 2/3 chance of ending up with the grand prize.