Three first-head times in weakly increasing order

Ty, Guy and Psy each toss their own fair coin until it first lands heads; let $T$, $G$ and $P$ be the respective numbers of tosses. Find $\mathbb{P}(T\le G\le P)$.

$\tfrac{8}{21}$. The three are i.i.d. $\text{Geom}(\tfrac12)$. Summing the joint pmf, $\mathbb{P}(P\ge g)=2^{-(g-1)}$, then $\sum_{g\ge t}2^{-g}\cdot 2^{1-g}=\tfrac83\cdot4^{-t}$, and finally $\sum_{t\ge1}2^{-t}\cdot\tfrac83\cdot 4^{-t}=\tfrac83\cdot\tfrac17$.