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More coins, more heads?

HardProbabilityCombinatorics~9m

Tim flips 2020 fair coins and Jeff flips 2222 fair coins. What is the probability that Jeff gets strictly more heads than Tim? Round to the nearest thousandth.

Show hints (2)+
  1. Recall: with exactly one extra coin, the bigger player has probability exactly 1/21/2 of strictly more heads.
  2. Jeff has two extra coins, so the answer is a bit above 1/21/2. Convolve Bin(22,12)\text{Bin}(22,\tfrac12) against Bin(20,12)\text{Bin}(20,\tfrac12) to get ≈0.561\approx0.561.

Answer

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0.561 (± 0.002)

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Asked at: Jane Street, Citadel

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