Bag A contains 3 red and 5 black balls, white bag B contains 4 red and 4 black balls. Two balls are transferred at random from bag A to bag B and then a ball is drawn from bag B at random. If the ball drawn from bag B is found to be red, find the probability that two red balls were transferred from bag A to bag B.
Given:
Bag A contains 3 red and 5 black balls
Bag B contains 4 red and 4 black balls
It is told that two balls are transferred from bag A to bag B, the possible cases (events) will be as follows:
(i) U1 = Transferring 2 red balls from bag A to bag B
(ii) U2 = Transferring 1 red ball and 1 black ball from bag A to bag B
(iii) U3 = Transferring 2 black balls from bag A to bag B
Let us assume the event A as follows:
⇒ A = Drawing red ball from bag B
Now,
⇒ P(U1) = P(transferring 2 red balls from bag A to bag B)
⇒ 
⇒ 
⇒ 
⇒ P(U2) = P(transferring 1 red and 1 black ball from bag A to bag B)
⇒ 
⇒ 
⇒ 
⇒ P(U3) = P(transferring 2 black balls from bag A to bag B)
⇒ 
⇒ 
⇒ 
⇒ P(A|U1) = P(drawing red ball after transferring 2 red balls from bag A to bag B)
⇒ 
⇒ 
⇒ P(A|U2) = P(drawing red ball after transferring 1 red and 1 black ball from bag A to bag B)
⇒ 
⇒ 
⇒ P(A|U3) = P(drawing red ball after transferring 2 black balls from bag A to bag B)
⇒ 
⇒ 
We need to find
⇒ P(U1|A) = P(red ball is drawn after transferring two red balls from bag A to bag B)
Using Baye’s theorem,
⇒ 
⇒ 
⇒ 
⇒ 
∴ The required probability is 
.