How many flips to collect three heads (not in a row)?

Question

Flip a fair coin until you have accumulated three heads total (they do not need to be consecutive). What is the expected number of flips? What is the standard deviation?

Accepted Answer

Getting the r-th success is a sum of r independent "wait for one success" pieces. Break the journey into stages: flips until the first head, then flips until the second head, then flips until the third. Each stage is a fresh geometric wait (by memorylessness) with mean 1/p flips, where p = 	frac12. The total number of flips is therefore negative binomial with parameters r = 3 and p = 	frac12, and its mean is simply the sum of the three geometric means: E[	ext{flips}] = \frac{r}{p} = \frac{3}{1/2} = 6. Six flips on average, intuitive, since heads come up half the time and you need three of …

How many flips to collect three heads (not in a row)?

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