Solution
= Solution
{c}
Recognize the <mixture distribution>
$$
M_Y(t)=\tfrac12\frac1{1-t}+\tfrac12\frac2{2-t}.
$$
It is a half-and-half mixture of <exponential distributions> with rates one and two. Uniqueness of a <moment-generating function> finite near zero identifies the law. Therefore
$$
\boxed{\mathbb P(Y\leq1)=\tfrac12(1-e^{-1})+\tfrac12(1-e^{-2})
=1-\tfrac12(e^{-1}+e^{-2}).}
$$