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

Use the same first-sex conditioning as in part (b). The conditional variances are $p/q^2$ after a male first capture and $q/p^2$ after a female first capture. ThusThe conditional expected values are $1+1/q$ and $1+1/p$, with probabilities $p,q$. A variable taking two values $a,b$ with probabilities $p,q$ has variance $pq(a-b{)}^2$, soThe law of total variance now givesAt $p=q=1/2$ this gives variance $2$, consistent with $N$ being one plus a geometric waiting time of parameter $1/2$.