A club basketball team will play a 60-game season. Thirty-two of these games are against class A teams, and 28 are against class B teams. The outcomes of all the games are independent. The team will win each game against a class A opponent with probability .5 and it will win each game against a class B opponent with probability .65. Let X denote its total number of victories in the season.
(a) What is the relationship between $X_A$, $X_B$, and $X$? Is $X$ a binomial random variable?
(b) Approximate the probability that the team wins 45 or more games this season.
What I did: $X=X_A+X_B \\ X_A \text{ is a binomial random variable with } n=32 \text{ and } p=0.5 \\ X_B \text{ is a binomial random variable with } n=28 \text{ and } p=0.65 \\ p_X(k) = \sum_{a=0}^k P(X_A=a)P(X_B=k-a) $ Can I express $p_X(k)$ as a binomial distribution somehow? Also, I think it is possible to approximate $P(X\ge45)$ using a normal distribution by the Central Limit Theorem, but I'm not sure how to apply it.