# A fair die is rolled up. If 1 turns up, a ball is picked up at random from bag A. If 2 0r 3 turns up, a ball is picked up at random from bag B, otherwise a ball is picked up from bag C. Bag A contains 3 red and 2 white balls, bag B contains 3 red and 5 white balls and bag C contains 4 red and 5 white balls. The die is rolled, a bag is picked up and a ball is drawn from it. If the ball drawn is red, what is the probability that bag B was picked up?

E1 - Selection of Bag A

E2 - Selection of Bag B

E3 - Selection of Bag C

A - red ball is selected

since Bag A is chosen when , 1 turns up  when a fair die is rolled

P(E1) -  1/6,

since Bag B is chosen when ,2 0r 3 turns up when a fair die is rolled

P(E2) - 2/6

and since Bag C is chosen when 4 or 5 or 6 occurs when a fair die is rolled

P(E3) - 3/6

Bag A contains 3 red and 2 white balls - P(A/E1) =3/5                                                           Bag Bcontains 3 red and 5white balls - P(A/E2)=3/7                                                                       Bag C contains 4 red and 5 white balls-P(A/E3)=4/9

Formula:- P(E2/A)= P(E2).P(A/E2)/ P(E2).P(A/E2)+P(E1).P(A/E1)+P(E3).P(A/E3)

substitute

P(E2/A)=(2/6.3/7)/2/6.3/7+1/6.3/5+3/6.4/9

=90/293 .