Question

Assume the box contains 9 balls:
4 red, 2 blue, and 3 yellow. A ball is drawn and its color noted. If the ball is yellow, it is replaced; otherwise, it is not. A second ball is then drawn and its color is noted.
What is the probability that the first ball was yellow, given that the second was red?

Answer 1

P(Y1 | R2) = P(Y1R2)/P(R2)

P(R2 | Y1) = 4/9 ..........P(Y1) = 1/3
P(R2 | B1) = 1/2 ..........P(B1) = 2/9
P(R2 | R1) = 3/8 ..........P(R1) =4/9

P(R2) = [4/9*1/3] + [1/2*2/9] + 3/8*4/9] = 23/54

P(Y1R2) = P(Y1)*P(R2|Y1) = 1/3*4/9 = 4/27
P(Y1 | R2) = 4/27*54/23 =8/23

hope this help