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I have a problem where I flip 200 coins and I want to find the expected number of runs of 7-tails in a row, where a ``run of $q$ tails in a row" is $q$-consecutive flips that are all tails. For example, TTHTTT has three runs of 2-tails in a row. I'm having some issues with this because the event space seems so huge. Should I maybe make an indicator random variable that will be 1 if a given interval of coins are all tails and 0 otherwise, and create $\frac{200}{7}$ intervals? I'm not completely sure if that is the best wait to go about it.

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  • $\begingroup$ Probability that any given consecutive $q$ throws are a $q$-run is $(\frac 1 2)^q$. There are $N+1-q$ such sequences in a run, so expected number of q-runs is $$(N+1-q)(\frac 1 2)^q\approx 1.5$$ by linearity of expectation. $\endgroup$ – A.S. Nov 20 '15 at 3:48
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For $i=1$ to $194$, let $X_i=1$ if there is a run of $7$ tails in a row starting with the $i$-th toss. Let $Y=X_1+\cdots+X_{194}$.

By the linearity of expectation we have $E(Y)=E(X_1)+\cdots +E(X_{194})$.

But $E(X_i)=\Pr(X_i=1)=\frac{1}{128}$, so $E(Y)=\frac{194}{128}$.

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