Bag 1 contains 1 red and 3 blue marbles and Bag 2 contains 1

Bag 1 contains 1 red and 3 blue marbles, and Bag 2 contains 1 red and 1 blue marbles. A bag is selected at random. A marble is randomly taken from it and placed in the other bag. A marble is then randomly drawn from that bag. Find the probability that the marble chosen is red.

Solution

Let

F1 = first bag is bag 1
F2 = first bag is bag 2
S1 = second bag is bag 1
S2 = second bag is bag 2
R1, R2 = first/second ball is red
B1, B2 = first/second ball is blue

P(R2) = P(F1)P(R1|F1)P(R2|F1 n R1) + P(F1)P(B1|F1)P(R2|F1 n B1) + P(F2)P(R1|F2)P(R2|F2 n R1) + P(F2)P(B1|F2)P(R2|F2 n B1)

= (1/2)*(1/4)*(2/3) + (1/2)*(3/4)*(1/3) + (1/2)*(1/2)*(2/5) + (1/2)*(1/2)*(1/5)

= 0.358333333 [ANSWER]