Matching head counts

Two players each flip a fair coin 3 times. What is the probability they end up with the same number of heads? Give a decimal to four places.

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They can match at 0, 1, 2, or 3 heads; independence makes each level's agreement $p_k^2$ .
$\sum_k p_k^2 = \tfrac{1}{64}\sum_k\binom{3}{k}^2$ , and $\sum_k\binom{3}{k}^2=\binom{6}{3}$ (Vandermonde).

Answer

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0.3125 (± 0.0005)

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Asked at: Game-Based Aptitude, ETF Market-Making