Colour of the final cherry eaten from two bowls

Bowl A has $2$ red and $5$ green cherries; bowl B has $6$ red and $3$ green. Repeat the following until every cherry is gone: if bowl B is not empty, move a randomly chosen cherry from B to A; then eat a randomly chosen cherry from A. What is the probability, rounded to five decimal places, that the very last cherry eaten is red?

About $0.55213$. Track one cherry at a time. During the $9$ transfer rounds bowl A always holds $8$ cherries when one is eaten, and the final $7$ are eaten in random order. An original A cherry is last with probability $(7/8)^9/7$; a B cherry moved in round $k$ (uniform on $1$–$9$) with probability $(7/8)^{10-k}/7$. Weighting by colour gives $\tfrac27(7/8)^9+\tfrac23\left(1-(7/8)^9\right)\approx0.55213$.