Counting HHHHHHTTTTTT runs in a million flips

Question

A fair coin is flipped one million times. On average, how many times does the exact pattern of six consecutive heads immediately followed by six consecutive tails appear (overlapping occurrences counted, though this pattern cannot overlap itself)?

Short answer

About $244$ . The pattern has length $12$ , so it can start at $10^6-11$ positions, each with probability $2^{-12}$ . By linearity of expectation the mean count is $\tfrac{999{,}989}{4096}\approx244.1$ .

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q3.12, p.81; question and answer rewritten by SelfMadeQuant.com.

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