Tosses until one face appears three times

Question

You toss a fair coin until either heads or tails has appeared three times in total (not necessarily in a row). What is the expected number of tosses?

Short answer

$33/8$ . The game lasts $3$ , $4$ or $5$ tosses. It ends at $3$ with probability $\tfrac14$ (all the same), and it reaches toss $5$ only if the first four are two heads and two tails, probability $\tfrac{6}{16}=\tfrac38$ ; so $4$ tosses has probability $\tfrac38$ . Mean $=3\cdot\tfrac14+4\cdot\tfrac38+5\cdot\tfrac38=\tfrac{33}{8}$ .

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

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating

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