Explain conditional probability? Consider 3 baskets. Basket A contains 3 white and 5 red marbles. Basket B contains 8 white and 3 red marbles. Basket C contains 4 white and 4 red marbles. An experiment consists of selecting one marble from each basket at random. What is the probability that the marble selected from basket A was white, given that exactly 2 white marbles were selected in this process.
I know P(AB) = 33/176 and I understand how to get there.
However I'm not sure why P(B) = 73/176.
Here is the answer, but can you explain why?

By Bayes rule, the required probability is equal to
$$P(A_w|W=2)=\frac{P(W=2|A_w)P(A_w)}{P(W=2)}=\frac{\frac{1}{2}\frac{3}{8}}{\frac{73}{176}}=\frac{33}{73}$$ since $$P(W=2|A_w)=P(B_w)P(C_r)+P(B_r)(C_w)=\frac{1}{2}$$ and $$\begin{align*}P(W=2)&=P(A_w)P(B_w)P(C_r)+P(A_w)P(B_r)P(C_w)+P(A_r)P(B_w)P(C_w)\\\\&=\frac{3}{8}\frac{8}{11}\frac{1}{2}+\frac{3}{8}\frac{3}{11}\frac{1}{2}+\frac{5}{8}\frac{8}{11}\frac{1}{2}=\frac{73}{176}\end{align*}$$ 
