How many streaks in twenty coin flips?

Question

You flip a fair coin 20 times in a row. A "streak" (or run) is a maximal block of the same result. On average, how many streaks appear in the 20 flips?

Accepted Answer

A new streak starts at the very first flip and at each flip that differs from the one before it, so 	ext{number of streaks} = 1 + (	ext{number of switches between neighboring flips}). There are 19 neighboring pairs among 20 flips. Put one indicator on each pair, equal to 1 when the two flips differ; for a fair coin this happens with probability 	frac12. By linearity the expected number of switches is 19 	imes 	frac12 = 	frac{19}{2}, so E[	ext{streaks}] = 1 + \frac{19}{2} = \frac{21}{2} = 10.5. Takeaway. The fair-coin streak count over n flips is 	frac{n+1}{2}; for n = 20 that is 	fra…

How many streaks in twenty coin flips?

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