Three first-heads in non-decreasing order

HardProbability Random Variables~9m

Three people flip fair coins independently, each until their first head; let $T, G, P$ be the flip counts. Find $\mathbb{P}[\,T \le G \le P\,]$ (ties allowed). Writing it as $\tfrac{p}{q}$ in lowest terms, report $p + q$ .

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Each count is Geometric $(1/2)$ . Write the weak-order probability as a triple sum over $t \le g \le p$ .
Collapse from the inside out. The ties make the answer exceed $1/6$ ; you should get $8/21$ .

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Three first-heads in non-decreasing order

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