One max of exponentials beating twice another

Mario throws twice with distances $M_1,M_2\sim\text{Exp}(1)$, and Luigi throws twice with distances $L_1,L_2\sim\text{Exp}(2)$ (rate parameters, so means $1$ and $\tfrac12$); all four are independent. With $M=\max(M_1,M_2)$ and $L=\max(L_1,L_2)$, find $P(M>2L)$.

$1/2$. If $L_i$ has rate $2$ then $2L_i$ has rate $1$, so $2L$ is the maximum of two independent $\text{Exp}(1)$ variables, exactly like $M$. Two i.i.d. continuous variables are equally likely to be the larger, so the probability is $\tfrac12$.