There are bottles numbered from 1 to n. Every bottle k has $\frac{1}{k}$ chance of being poisoned. A random bottle is picked (evenly distributed) and is found to be poisoned. What's the expected value for the number of the bottle that was picked?
Here's what I had in mind:
A - The bottle that was picked is poisoned
X - The number of the bottle that was picked
We're searching for:
$$\mathbb{E}[X|A]=\sum_{i=1}^n iP(X=i|A)$$
The problem is calculating the probability. I tried using Bayes' theorem but I can't calculate $P(A)$.
Any ideas?