Coin with unknown bias flipped N times with N heads, what is p(h)? Given a coin with an unknown bias and the observation of $N$ heads and $0$ tails, what is expected probability that the next flip is a head? 
 A: This is the same as taking some unknown $p$ from $[0,1]$ and then saying that all $X_i$ uniformly independently distributed over $[0,1]$ ended up smaller than $p$ (or that $p$ was the largest of $X_i$). If all biases are equally probable (i.e. $p$ is also uniformly distributed over $[0,1]$) then the probability in question equals
$$P(X_{N+1} < p\ |\ X_1 < p, \ldots, X_N < p) = 
\frac{P(\bigcap_i^{N+1} X_i < p)}{P(\bigcap_i^{N} X_i < p)}$$
but $\{X_1, X_2, \ldots, X_{N+1}, p\}$ are independent, so the result is
$$ \frac{\int_0^1 p^{N+1} \, dp}{\int_0^1 p^N \, dp} = \frac{N+1}{N+2}$$
Maybe that will help ;-)
A: This question does not have a well-defined mathematical answer unless you have a prior probability distribution on the coin's bias.  In the 18th century, Laplace considered the probability that the sun will rise tomorrow, given that it's risen every day during the 6000 years the universe has existed.  He started with a probability distribution on the "bias", saying there is an unknown number $p$ between $0$ and $1$ such that the probability that the sun will rise on any given day, given that one knows with certainty the value of $p$, is $p$.  He said $p$ is uniformly distributed between $0$ and $1$.  And then he derived Laplace's rule of succession.  I wrote a derivation of that rule in one of my answers here; I'll see if I can find it.
Later note: I found it: Estimating a probability of head of a biased coin
A: I think this is answered in https://stats.stackexchange.com/a/23003 in the more generally section.
