Solution
= Solution
By the <inclusion-exclusion principle for probabilities>,
$$
P(B_1\cup A_2)=P(B_1)+P(A_2)-P(B_1\cap A_2).
$$
After an initial banana, five of the nine remaining pieces are apples, so the intersection <probability> is $(1/2)(5/9)=5/18$. Consequently
$$
\boxed{P(B_1\cup A_2)=\frac12+\frac12-\frac5{18}=\frac{13}{18}.}
$$
As a check, the complementary event is an apple followed by a banana, which also has <probability> $5/18$.