Waiting for the pattern HT

Question

A fair coin is tossed until, for the first time, a head is followed directly by a tail. What is the expected number of tosses?

Short answer

$4$ . First wait for a head (mean $2$ tosses). After that, any further heads change nothing and the game ends at the first tail, another mean of $2$ tosses. Total $2+2=4$ .

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Sample Spaces, Events, and the Probability Axioms

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