Average flips used by a rejection-based three-way pick

Question

To choose one of three children at random with a fair coin, you flip the coin in pairs: HH, HT and TH each select a different child, while TT means you discard that pair and flip another pair. On average, how many individual flips does this procedure use?

Short answer

$8/3$ . Each pair succeeds with probability $\tfrac34$ , so the number of pairs is geometric with mean $\tfrac43$ , and each pair costs two flips: $2\cdot\tfrac43=\tfrac83$ .

Adapted from Crack, Heard on the Street, Q4.9, p.38; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Derivation — Means and Variances of the Named Distributions

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