Red cherry after moving one between bowls

Bowl $A$ holds $4$ red and $5$ purple cherries; bowl $B$ holds $6$ red and $3$ purple. Jenny moves one cherry chosen at random from $A$ into $B$, and then eats a cherry chosen at random from $B$. What is the probability that the cherry she eats is red?

$\tfrac{29}{45}$. Bowl $B$ now has $10$ cherries, one of which came from $A$. The eaten cherry is the transferred one with probability $\tfrac1{10}$ (red with chance $\tfrac49$) and an original $B$ cherry otherwise (red with chance $\tfrac23$): $\tfrac1{10}\cdot\tfrac49+\tfrac9{10}\cdot\tfrac23=\tfrac{29}{45}$.