Solution
= Solution
Rewrite the function as
$$
M_Y(t)=\frac12\frac1{1-t}+\frac12\frac2{2-t}.
$$
These are equal mixture weights on the moment-generating functions of exponential distributions of rates $1$ and $2$. Moment-generating functions finite near zero determine the distribution, so $Y$ has that mixture law. Consequently
$$
\boxed{P(Y\leq1)=\frac12(1-e^{-1})+\frac12(1-e^{-2})
=1-\frac{e^{-1}+e^{-2}}2.}
$$