Expected number of HT patterns in 100 coin flips

Question

A fair coin is flipped 100 times. What is the expected number of positions where a head is immediately followed by a tail (the pattern HT)?

Accepted Answer

Look at each adjacent pair of positions (i, i+1). There are 99 such pairs in 100 flips. Let Ii indicate that pair i is exactly heads-then-tails, so the total count is X = \sum{i=1}^{99} Ii. Each pair is HT with probability \frac12 	imes \frac12 = \frac14 (the two flips are independent): E[Ii] = \frac14. By linearity of expectation (adjacent pairs overlap and are not independent, but linearity does not require independence): E[X] = 99 	imes \frac14 = \frac{99}{4} = 24.75. Takeaway. For "expected number of occurrences of a short pattern," place indicators on each window where the pattern coul…

Expected number of HT patterns in 100 coin flips

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