I have found this exercise in my notebook, from some time ago and I forgot about it. Reading it now makes no sense to me, that's why I hope I will get some explanation.
We have an urn with $n$ balls, we don't know anything about the balls. We extract a ball, what is the probability that the ball is white?
Solution: Consider $H_k$, $k=\overline{0,n}$ to be the hypothesis that in the urn there are exact $k$ white balls.
Consider $W$ the event that we extract a white ball. We don't have any evidence so we consider all the hypothesis equaly probable. $$P(W)=\sum_{k=0}^nP(W/H_k)P(H_k)=\sum_{k=0}^n \frac{k}{n}\frac{1}{n+1}$$ $$=\frac{1}{n(n+1)}\frac{n(n+1)}{2}=\frac12$$ I don't understand why $P(H_k)=\frac{1}{n+1}$, but $P(W/H_k)=\frac{k}{n}$.
Can anyone explain this to me?
$P(H_k)$ is the probability that $H_k$ is true, that is, that the urn has exactly $k$ white balls.
You've assumed "all hypotheses are equally likely", but there are exactly $n+1$ hypotheses, namely $H_0, H_1, \dots, H_n$. Hence, each is assumed to have the same probability $\frac{1}{n+1}$.
On the other hand, $P(W|H_k)$ means "the chance of drawing a white ball, in a universe where $H_k$ is true". If we live in such a universe, then your urn contains $k$ white balls out of $n$ total, so the chance of drawing a white one is $\frac{k}{n}$.
Does this make sense? If not, which part do you find to be not clear?