Run of successes

A run of successes is a consecutive block of successful trials in a binary observation sequence. For independent trials with success probability $0<p<1$, the waiting time $T_r$ until the first run of $r$ successes has meanLet $E_k$ be the remaining mean after $k$ consecutive successes. For $k<r$, $E_k=1+p{E}_{k+1}+(1-p)E_0$, while $E_r=0$. Solving backward gives the formula. At $p=1/2$, $r=2$, the mean is six, agreeing with the HH pattern waiting time. A run requires consecutive successes, unlike merely collecting $r$ successes in total.