Expected flips for a streak of n heads

Question

A fair coin is flipped until it lands heads $n$ times consecutively. What is the expected number of flips?

Short answer

$2^{n+1}-2$ . If $E_k$ is the expected flips to reach a run of $k$ heads, then $E_k=E_{k-1}+1+\tfrac12E_k$ (after reaching $k-1$ , one more flip either finishes or resets), so $E_k=2E_{k-1}+2$ with $E_0=0$ .

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 5.2 Martingale and Random Walk, 'Coin sequence', p.118; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

The Optional Stopping Theorem
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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