Suppose there are $n$ prisoners, and each day every prisoner independently has a probability $p$ of escaping. What is the expected length of time before all prisoners have escaped?
Someone asked me this question a while back, and I thought it would be relatively straightforward, but so far I've been struggling to find a solution.
My initial approaches have been with setting up recurrences, but I've been unable to arrive at something solvable (at least by methods I know). I was also told there were many ways to solve the problem, so seeing other approaches would be interesting.
Let $Y$ be the time until they have all escaped. Then $$E(Y)=\sum_{i=1}^\infty \Pr(Y\ge i).$$ So let us calculate $\Pr(Y\ge i)$.
This is $1$ minus the probability that $Y\le i-1$.
The probability that a particular prisoner is gone at time $i-1$ or earlier is $1$ minus the probability she is not gone, that is, $1-(1-p)^{i-1}$. It follows that $$E(Y)=\sum_{i=1}^\infty \left[1-\left(1-(1-p)^{i-1}\right)^n\right].$$
We can expand using the Binomial Theorem, and sum the various related infinite geometric series. So we end up with a sum of $n$ terms. That sum may have a pleasant closed form.