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In discrete time, the bounded-time optional stopping theorem says that an integrable supermartingale $X$ and bounded stopping times $S\le T$ satisfy $\mathbb{E}[X_T\mid {\mathcal{F}}_S]\le X_S$. For a martingale there is equality. The assertion at a possibly unbounded, almost-surely finite stopping time $T$ also holds with $S=0$ if $\{X_{T\wedge n}:n\ge 0\}$ is uniformly integrable: then $X_T$ is integrable and $\mathbb{E}{X}_T\le \mathbb{E}{X}_0$, with equality for a martingale. Boundedness of $T$ or the stated integrability condition cannot in general be omitted.

For the increment hypothesis here, $\mathbb{E}T<\infty$ implies $T<\infty$ almost surely, and telescoping givesThe right-hand side is an integrable random variable. Applying bounded-time optional stopping to $T\wedge n$ and then the dominated convergence theorem provesThis is bounded-increment optional stopping with integrable time; the same reasoning gives equality for a martingale.

Put $q=1-p$ and first assume $0<p<1$. Let $\tau$ be the first occurrence time of the specified seven-letter word. Its probability in a prescribed block is $r=p^4{q}^3>0$. Disjoint seven-flip blocks are independent, and a match in one forces $\tau$ to have occurred by its end. Consequently $\mathbb{P}(\tau >7m)\le (1-r{)}^m$ and $\mathbb{E}\tau \le 7/r<\infty$; this establishes the integrability needed for stopping before computing its value.

Start a new gambler with unit capital immediately before each flip. A gambler bets all current capital on the next required letter of the word, receives capital divided by $p$ for a correct head or by $q$ for a correct tail, and receives zero on failure. After completing all seven bets, the gambler retains the winnings without further betting. Each individual bet preserves its conditional expected capital. If $C_n$ is the total capital of all gamblers who have entered through flip $n$, then $C_n-n$ is a martingale.

At any time at most seven gamblers are still betting, and each one's capital is at most $L=1/(p^4{q}^3)$, since probabilities of shorter prefixes are at least the full-word probability. Completed winnings do not change. Thus the increments of $C_n-n$ have a deterministic bound, for example $14L+1$. The preceding optional-stopping argument therefore gives $\mathbb{E}(C_{\tau }-\tau )=0$.

At the first full match, no earlier gambler has completed the word. A surviving gambler who has placed $k$ successful bets corresponds to a suffix of the completed word which is also its length-$k$ prefix, namely a border of a word. The only such lengths are $3$ and $7$: the shorter common string is HHT. Their capitals are $1/(p^{2}q)$ and $1/(p^4{q}^3)$, respectively. Hence $C_{\tau }$ is this deterministic sum andFor a fair coin this is 136 flips. This is a waiting time for a word in independent nonuniform symbols; the overlap contribution is essential. If $p=0$ or $p=1$, the word cannot occur because it contains both types of flip, so the expected waiting time is infinite.