Expected bills drawn to post bail with half fakes

To make bail you must hand over $500 using $100 bills drawn one at a time from a bag. Each draw is real with probability $\tfrac12$ and fake with probability $\tfrac12$, independently. Whenever you hand over a fake, everything you have handed over so far is returned and you start from zero. On average, how many bills will you draw?

$62$. You need $5$ consecutive real bills. For a run of $k$ successes with probability $p=\tfrac12$ the expected number of trials is $\tfrac{p^{-k}-1}{1-p}=2^{k+1}-2$, which for $k=5$ is $62$.