Rounds until at least one of three coins shows heads

Question

Each round you flip 3 fair coins at once. You keep playing rounds until at least one coin comes up heads. On average, how many rounds does that take?

Accepted Answer

Find the chance a round produces at least one head, then treat rounds as independent trials. The chance all three coins are tails in a round is \left(	frac12ight)^3 = 	frac18, so the chance of at least one head is p = 1 - 	frac18 = 	frac78. The expected wait for the first successful round is 1/p: E = \frac{1}{7/8} = \frac{8}{7} \approx 1.14 	ext{ rounds.} Takeaway. At least one head among 3 fair coins has probability 1 - (1/2)^3 = 7/8, so the wait is 8/7 \approx 1.14 rounds.

Rounds until at least one of three coins shows heads

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