Waiting times for one head and for HH with bias

A coin lands heads with probability $p$. What is the expected number of tosses until the first head? What is the expected number of tosses until two heads appear back to back?

$1/p$ and $(1+p)/p^2$. The first is a geometric mean. For HH, let $E$ be the expected wait: after the first head ($1/p$ tosses on average) one more toss either completes HH (prob. $p$) or sends you back to the start, so $E=\tfrac1p+1+(1-p)E$, giving $E=\tfrac{1+p}{p^2}$ (which is $6$ for a fair coin).