Question 145
Two teams are in a best-two-out-of-three playoff: the teams will play at most $3$ games, and the winner of the playoff is the first team to win $2$ games. The first game is played on Team $A$'s home field, and the remaining games are played on Team $B$'s home field. Team $A$ has a $\frac{2}{3}$ chance of winning at home, and its probability of winning when playing away from home is $p$. Outcomes of the games are independent. The probability that Team $A$ wins the playoff is $\frac{1}{2}$. Then $p$ can be written in the form $\frac{1}{2}(m - \sqrt{n})$, where $m$ and $n$ are positive integers. What is $m + n$?
(A) 10 (B) 11 (C) 12 (D) 13 (E) 14
(A) 10 (B) 11 (C) 12 (D) 13 (E) 14