Probabilty and baye's theorem

Question

Three identical boxes A,B, C contains 2white, 3black balls; 4white, 5black balls & 3white, 4 black balls. One ball is drawn at random from each box. If the ball is white, find the probability that it is drawn from box C?

Answer 1

P(C|W)=P(C∩W)/P(W)

P(C∩W)=P(W|C)P(C)=(3/7)*(1/3)

P(W)=P(W|A)P(A)+P(W|B)P(B)+P(W|C)P(C)=(1/3){P(W|A)+P(W|B)+P(W|C)}=(1/3){(2/5)+(4/9)+(3/7)}∴

P(C|W)=(3/7)/{(2/5)+(4/9)+(3/7)}