Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 9F / b / Solution

This is the waiting time to observe both Bernoulli outcomes. Write $q=1-p$, and let $S$ record the sex of the first captured animal. That first capture is always needed. If $S$ is male, every subsequent trial independently produces the required female with probability $q$, so $N-1$ has a geometric distribution with parameter $q$. If $S$ is female, $N-1$ has a geometric distribution with parameter $p$. By the law of total expectation,Since $p+q=1$, this simplifies to $\boxed{EN=1/(pq)-1}$, orThe expectation includes both the first capture and the final capture completing the pair.