# 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 .

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