Waiting time for two consecutive successes ($\mathbb{E}T=(1+p)/p^2$)

For independent trials of success probability $p\in (0,1]$, let $T$ be the first index completing two consecutive successes. If $e_0,e_1$ are the mean remaining times with no trailing success and with one trailing success, conditioning gives $e_0=1+(1-p)e_0+p{e}_1$ and $e_1=1+(1-p)e_0$. Solving yields $e_0=(1+p)/p^2$, so fair trials have mean waiting time six. Disjoint pairs give a geometric tail bound and justify finiteness before solving the equations. Overlapping pairs are not independent trials.