Question 110
$A$ and $B$ are playing ping-pong, with the agreement that the winner of a game will get $1$ point and the loser $0$ point; the match ends as soon as one of the players is ahead by $2$ points or the number of games reaches $6$. Suppose that the probabilities of $A$ and $B$ winning a game are $\frac{2}{3}$ and $\frac{1}{3}$, respectively, and each game is independent. Then the expectation $E\xi$ for the match ending with $\xi$ games is ( ).
(A) $\frac{241}{81}$
(B) $\frac{266}{81}$
(C) $\frac{274}{81}$
(D) $\frac{670}{243}$
(A) $\frac{241}{81}$
(B) $\frac{266}{81}$
(C) $\frac{274}{81}$
(D) $\frac{670}{243}$