Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 11F / d / Solution

Let $E_j$ be the expected additional number of tosses when the current terminal run contains $j$ consecutive heads. Then $E_k=0$, and for $0\le j<k$,For $0<p<1$, iterating this recurrence from $j=k-1$ down to $0$ givesSolving,When $p=1$, the waiting time is deterministically $k$, which is also the continuous limit of this formula as $p\to 1$.