At least one head in three tosses

Question

You toss three fair coins. A friend checks and tells you, truthfully, that at least one of them came up heads. Given that, what is the probability that all three are heads?

Accepted Answer

There are 2^3 = 8 equally likely outcomes for three coins. The clue "at least one head" simply rules out the single all-tails outcome, TTT. So 8 - 1 = 7 outcomes remain in play, all equally likely: 	ext{HHH, HHT, HTH, THH, HTT, THT, TTH.} Exactly one of these seven, HHH, has all three heads. Therefore P(	ext{all heads} \mid 	ext{at least one head}) = \frac{1}{7} \approx 0.143. A tempting wrong answer is 	frac18, the unconditional chance of HHH. But the clue removed a losing outcome (TTT) from the pool, which nudges the probability of HHH up from 	frac18 to 	frac17. Another tempting wron…

At least one head in three tosses

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