A three-way race to the first head

HardProbability Random Variables~8m

Three people independently flip fair coins until each gets their first head. Let $T$ , $G$ , $P$ be the number of flips each needs. What is $\mathbb{P}[T < G < P]$ ? The answer is $\tfrac{1}{M}$ for an integer $M$ ; find $M$ .

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Each flip count is Geometric $(1/2)$ and they're independent. Write $\mathbb{P}[T<G<P]$ as a triple sum over $t < g < p$ .
Collapse it from the inside out with geometric series. Beware: the answer is not $1/6$ , because ties have positive probability.

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A three-way race to the first head

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