Examining goats until both sexes have appeared

A farmer needs one male and one female goat. He examines goats one at a time; each is male with probability $\tfrac34$, independently of the others. He stops as soon as the goats he has examined include both a male and a female. Let $N$ be the number of goats examined. Find $\mathbb{E}[N]$.

$\tfrac{13}{3}$. Condition on the first goat. If it is male (probability $\tfrac34$), he waits a geometric time with mean $4$ for a female, so $5$ in total; if female, he waits a mean of $\tfrac43$ for a male, so $\tfrac73$. Averaging, $\tfrac34\cdot5+\tfrac14\cdot\tfrac73=\tfrac{13}{3}$.