Let $X,Y,Z$ be independent discrete random variables with $X\sim\text{Bin}(6,7/8),$ $Y\sim\text{Bin}(5,7/8)$ and $Z\sim\text{Geo(1/4)}$. Futhermore let $M:=\max(X+Y,Z)$.
Compute $\Pr[M>10]$.
My approach so far based on the fact that $X+Y\sim\text{Bin}(11,7/8)$:
$\Pr[M>10] = 1-\Pr[M\leq 10]=1-\Pr[X+Y\leq 10]\cdot\Pr[Z\leq 10]\\ \Pr[X+Y\leq 10] = 1-\Pr[X+Y=11]=1-(7/8)^{11}\approx 0.7698\\ \Pr[Z\leq 10]=1-(1-1/4)^{10}\approx 0.9437\\ \Longrightarrow \Pr[M>10]\approx 1-0.7698\cdot 0.9437=0.2735$
However I think this cannot be true because I haven't considered $\max$ yet. Can anyone explain me what did I miss?