22 coins against 20: who gets more heads

Question

Tim tosses $20$ fair coins and Jeff tosses $22$ fair coins. Find the probability, to three decimal places, that Jeff ends up with strictly more heads than Tim.

Short answer

About $0.561$ . Compare Tim's coins with Jeff's first $20$ : Jeff leads, ties or trails, with lead and trail equally likely. Jeff wins if he already leads, or ties and gets at least one head in his last two ( $\tfrac34$ ), or trails by exactly one and gets two heads ( $\tfrac14$ ). Plugging in the binomial tie and off-by-one probabilities gives $\approx0.5612$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating
Continuous Random Variables and Probability Density Functions

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