Probability update on an object being in a room? Say there's a room with $n>1$ boxes. You know there's a 50/50 chance of your keys being inside this room. If the keys are here, they are equally likely to be in any one of these boxes. The remaining boxes are empty.
You go in and open the first box, finding it empty. Question: should your confidence of the keys being in the room decrease?
Bob says: "Yes, your confidence should go down. This is because it's more likely to see an empty box when the keys aren't in the room versus when they are."
Alice says: "No, your confidence shouldn't change. Going into the room, you already knew that one of the boxes was going to be empty. So seeing that box 1 was empty isn't any new information on whether the keys are in the room, it only changes the probability distribution between the boxes. Your confidence should still be 0.5."
Who's right?
 A: Bob the Bayesian is correct. Although both answers seemed plausible to me on intuitive grounds, a simple application of Bayes' theorem gives us the answer.
Let $K$ be the event of the keys being in the room. Let $E$ be the event "box 1 is opened and found empty".
Then what we're asking for is $P(K|E)$. Which is
$$P(K|E) = \frac{P(E|K)\cdot P(K)}{P(E)}$$
$$= \frac{P(E|K)\cdot P(K)}{P(E|K)\cdot P(K) + P(E|\overline{K})\cdot P(\overline{K})}$$
$$= \frac{\frac{n-1}{n} \cdot \frac{1}{2}}{\frac{n-1}{n} \cdot \frac{1}{2} +\frac{1}{2}}$$
$$= \frac{n-1}{2n-1} < \frac{1}{2}$$
So the confidence level decreases. The only cases where it wouldn't decrease would be where $P(K) \in \{0,1\}$. Otherwise it will decrease.
Intuitively, what was wrong about Alice's argument is that if the keys are in the room, then opening a box and finding it empty is new information regarding the presence of the keys; we placed a nonzero probability of the keys being in that box, and we aren't sure that they keys are in the room to begin with, so finding that it was empty gives us some more reason to believe the keys aren't there. Bayes' theorem basically explains that it's more likely to see an empty box when there aren't keys in the room vs when there are, so we should decrease our confidence of there being keys if we open an empty box.
Note that this basically becomes monty hall with some changes: if there was someone in the room who knew where the keys were (if they were there) and opened a box for you which he knew was empty. In this case, Alice's answer is correct.
