= Run of successes
{wiki}
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 mean
$$
\mathbb E T_r=\frac{1-p^r}{(1-p)p^r}.
$$
Let $E_k$ be the remaining mean after $k$ consecutive successes. For $k<r$, $E_k=1+pE_{k+1}+(1-p)E_0$, 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.
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