How many streaks in ten coin flips?

Question

You flip a fair coin 10 times in a row. A "streak" (or run) is a maximal block of the same result, so the sequence HHTHT has 4 streaks: HH, T, H, T.

On average, how many streaks appear in the 10 flips?

Accepted Answer

Counting streaks directly is awkward, but there is a clean trick: a new streak begins exactly at the first flip and at every flip that differs from the one before it. So 	ext{number of streaks} = 1 + (	ext{number of places where a flip differs from the previous one}). There are 9 neighboring pairs among 10 flips. Put one indicator on each pair, equal to 1 when the two flips differ. For a fair coin the second flip differs from the first with probability 	frac12: P(	ext{a given pair differs}) = \frac12. By linearity the expected number of "switches" is 9 	imes 	frac12 = 	frac92, so E[	e…

How many streaks in ten coin flips?

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