AMC 12 2025b Problem 18

Awnik repeatedly plays a game that has a probability of winning of $\frac{1}{3}$. The outcomes of the games are independent. What is the expected value of the number of games he will play until he has both won and lost at least once?

$\textbf{(A) } \frac{5}{2} \textbf{(B) } 3 \textbf{(C) } \frac{16}{5} \textbf{(D) } \frac{7}{2} \textbf{(E) } \frac{15}{4}$