Flipping until two successive results agree

Question

Keep flipping a fair coin and stop the first time a flip shows the same face as the flip just before it. On average, how many flips will you make in total?

Short answer

$3$ . The first flip can never end the game. Every later flip matches its predecessor with probability $\tfrac12$ , so the number of additional flips is geometric with mean $2$ , giving $1+2=3$ .

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q3.7, p.81; 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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