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First head strictly sooner

MediumProbabilityRandom Variables~5m

Bill and Bob each flip their own coin (heads probability 13\tfrac13) once per round, simultaneously, until each gets a first head. What is the probability that Bill gets his first head in strictly fewer flips than Bob?

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  1. By symmetry P(Bill sooner)=12(1−P(tie))\mathbb{P}(\text{Bill sooner})=\tfrac12(1-\mathbb{P}(\text{tie})).
  2. A tie is both first-heads on the same round: ∑k[(2/3)k−1(1/3)]2=1/5\sum_k[(2/3)^{k-1}(1/3)]^2 = 1/5.

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Asked at: Jane Street, Optiver

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