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Head-count duel

StatisticsHard

Alice and Bob each flip their own fair coin $3$ times, with all flips independent. What is the probability that Alice gets strictly more heads than Bob?

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Hint

Find the probability of a tie, then use the symmetry between the players.

Solution

The number of outcome pairs with equal head counts is $1^{2} + 3^{2} + 3^{2} + 1^{2} = 20$ , using the coefficients $1$ , $3$ , $3$ , $1$ .
There are $2^{6} = 64$ equally likely outcome pairs, so the tie probability is $\frac{20}{64} = \frac{5}{16}$ .
By symmetry, Alice leads and Bob leads with equal probability.
So Alice leads with probability $\frac{1 - \frac{5}{16}}{2} = \frac{11}{32}$ .

Answer: $\frac{11}{32}$

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