More coins, more heads?

HardProbability Combinatorics~9m

Tim flips $20$ fair coins and Jeff flips $22$ fair coins. What is the probability that Jeff gets strictly more heads than Tim? Round to the nearest thousandth.

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Recall: with exactly one extra coin, the bigger player has probability exactly $1/2$ of strictly more heads.
Jeff has two extra coins, so the answer is a bit above $1/2$ . Convolve $\text{Bin}(22,\tfrac12)$ against $\text{Bin}(20,\tfrac12)$ to get $\approx0.561$ .

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0.561 (± 0.002)

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More coins, more heads?

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