Suppose that among a group of $n$ people, some unknown number of people $K$ know a rumor. If someone knows the rumor, there is a probability $p$ that they will tell it to us if we ask. If they don't know the rumor they will always say they don't know it.
If I go around and ask each person if they know about the rumor, and $M$ people say they do, what does that tell me about the number of people who actually know the rumor?
In particular, what is the distribution $P(K=k|M=m)$ in terms of $P(K)$?
I've been able to show that
where $b(m,k,p)$ is the binomial density function (probability of $m$ successes out of $k$ trials with probability $p$ of success). Is it possible to take this any further?