Coin flips needed to simulate a one-third event

To generate an event with probability $\tfrac13$ using a fair coin, toss the coin twice: two heads means the event occurs, one head and one tail (in either order) means it does not, and two tails means you discard the pair and start again with two fresh tosses. What is the expected number of tosses used?

$\tfrac83$. Each round costs two tosses and ends the procedure with probability $\tfrac34$, so the number of rounds is geometric with mean $\tfrac43$ and the expected number of tosses is $2\cdot\tfrac43=\tfrac83$.