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Heads then tails

Expected valueEasy

A fair coin is flipped repeatedly. What is the expected number of flips until a heads is immediately followed by a tails (the pattern $H T$ )?

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Hint

Split the wait into $2$ stages: first get a heads, then get a tails right after it.

Solution

Reaching the first heads takes $2$ flips on average.
From then on, each flip is tails with probability $\frac{1}{2}$ and ends the game; a heads simply leaves you in the same state, so the second stage also takes $2$ flips on average.
The total is $2 + 2 = 4$ .

Answer: $4$