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?