Expected number of head-head pairs

Question

You flip a fair coin ten times in a row. Every time a head is immediately followed by another head, you score a point (so the run HHH counts as two overlapping pairs, positions 1-2 and 2-3).

On average, how many points do you score?

Accepted Answer

Trying to track runs of heads directly is painful. Linearity of expectation makes it easy: the expected value of a sum is the sum of the expected values of its parts, even when those parts overlap and are not independent. Look at each of the 9 adjacent slots where a pair can sit: flips (1,2), (2,3), \ldots, (9,10). For each such slot, define an indicator that is 1 when both those flips are heads. The chance both are heads is P(	ext{both heads}) = \frac12 	imes \frac12 = \frac14. The score is the sum of these 9 indicators, so its expected value is simply E[	ext{score}] = 9 	imes \frac14 = …

Expected number of head-head pairs

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