= 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. Thus
$$
E[\operatorname{Var}(N\mid S)]=\frac{p^2}{q^2}+\frac{q^2}{p^2}.
$$
The 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> $pq(a-b)^2$, so
$$
\operatorname{Var}(E[N\mid S])=pq\left(\frac1q-\frac1p\right)^2
=\frac{(p-q)^2}{pq}.
$$
The <law of total variance> now gives
$$
\boxed{\operatorname{Var}N=\frac{p^2}{q^2}+\frac{q^2}{p^2}
+\frac{(p-q)^2}{pq}
=\frac1{p^2q^2}-\frac3{pq}-2,\qquad q=1-p.}
$$
At $p=q=1/2$ this gives <variance> $2$, consistent with $N$ being one plus a geometric waiting time of parameter $1/2$.
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