Draws with replacement until four jacks in a row

Bobby repeatedly draws a card from a standard 52-card deck, putting it back each time. What is the expected number of draws until he has drawn jacks on four consecutive draws?

$30{,}940$. Each draw is a jack with probability $p=\tfrac1{13}$, and any non-jack resets the streak. For a run of $k$ successes the expected waiting time is $\sum_{i=1}^{k}p^{-i}$, so the answer is $13+13^2+13^3+13^4=30{,}940$.