Deriving the probability of an event from limited trial data I have a stochastic event (say a weighted coin toss) that produces a positive outcome (heads) according to some unknown probability $P$.
Given $N$ total events (coin tosses) and n positive outcomes (heads), how can I measure the likelihood that $\frac nN$ is a good approximation of $P$?
Obviously, as $N$ grows, it becomes more likely than $\frac nN$ is close to the value $P$, but how can this convergence be described mathematically?

This is the problem of confidence intervals in statistics.  Your question is about the original problem of that type, the  "confidence interval for a proportion in binomial sampling" (wikipedia http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval ) 
The asymptotic theory, when $Np(1-p)$ is very large, is described by the Central Limit Theorem, which says that $n = pN + A\sqrt{Np(1-p)}$ where the probability distribution of $A$ is the famous Gaussian (normal, "bell shaped") curve.  Qualitatively, fluctuations of $n$ around the expected value are usually within a small factor of $\sqrt{Np(1-p)}$.   
For limited data, the story is more complicated, with no perfect solution, many proposals, and a large literature.
