Flips to get three heads with a quarter-weighted coin

Question

A coin lands heads with probability $\tfrac14$ on each toss. On average, how many tosses does it take to collect three heads?

Short answer

$12$ . The wait for each head is geometric with mean $4$ , and the total is the sum of three such independent waits (a negative binomial), so the expected number of tosses is $3\cdot4=12$ .

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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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