Q. Box A contains 3 red and 2 white marbles. Box B contains

Long one so bear with me...
If Ba = Event of Box a being choosen
If Bb = Event of Box b being choosen
If Wa = Event of white ball being choosen from box A
If Wb' = Event of white ball being choosen from box B, after a white is picked from A and put in B

Define corresponding events for Red ball from box A, box A, and the events from Box B.


Prob(Success) = P(Ba)x [P(Wa)xP(Wb') +P(Ra)xP(Rb')] +
P(Bb)x[P(Wb)xP(Wa') + P(Rb)xP(Ra')]

P(Ba) = P(Bb) = 1/2
P(Wa) = 2/5
P(Ra) = 3/5
P(Wa') = 3/6 {After white picked from B and put in A}
P(Ra') = 4/6 {After red picked from B and put in A}
P(Wb) = 5/7
P(Rb) = 2/7
P(Wb') = 6/8 {After white picked from A and put in B}
P(Rb') = 3/8 {After red picked from A and put in B}

Solving using above equations and values
P(Success) = 3604/6720 = 0.54