# Range of binomial probability, given a certain number of observations?

Let's say I am given $n$ flips of a coin, $k$ of which are heads. These are iid flips.

Can I say, with probability $p > 1/2$, that the true probability of heads is in range $[p_1, p_2]$ ? What is that range?

How do I integrate prior knowledge of the binomial distribution? What if I have no prior knowledge?

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There is a whole field around questions like this called bayesian statistics, it's been a while since I looked at this stuff but if I remember right.

Sadly you do need to have some pre-determined view of what p is. That is some before flipping the n coins you have a distribution in mind for the value of p (called the prior distribution). This distribution changes as you flip the coins (you get a posterior distribution).

For example you might start out believing that the coin has a 50% chance of being fair and a 50% chance of coming up heads 2/3rds of the time. (You might believe this if you know the person you got the coin from has both types of coins and there is a 50% chance he's trying to trick you).

The interesting (or at least nice) case is when your prior distribution is a "conjugate prior". Which basically means that your posterior distiribution for p is of the same parametric family as your prior distribution. I believe the conjugate prior for this is the beta distribution, but you might want to google "conjugate prior" and "bayesian statistics".

Hope that helps.

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Some very sharp bounds for questions like these are provided by something called a Chernoff bound. The example in the wikipedia article will give you what you need.

Edit: Oh, I forgot to say that you need an estimator for the "true" probability, but I guess that the one you are using is just the average over the samples.

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