Expected tosses until HHH

Question

Toss a fair coin repeatedly. What is the expected number of tosses needed before three heads have appeared back to back?

Short answer

$14$ . Each extra head in the target run doubles the wait and adds two: if $E_k$ is the expected time for $k$ heads in a row, $E_k=2E_{k-1}+2$ with $E_0=0$ , so $E_k=2^{k+1}-2$ and $E_3=14$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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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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