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Question

Bag I contains 3 red and 4 black balls and Bag II contains 4 red and 5 black balls. One ball is transferred from Bag I and Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be red in colour. Find 310 times the probability that the transferred ball is black.

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Solution

Let E1 and E2 respectively denote the events that a red ball is transferred from bag I to II and a black ball is transferred from bag I to II.
P(E1)=37 and P(E2)=47
Let A be the event that the ball drawn is red.
When a red ball is transferred from bag I to II,
P(A|E1)=510=12
When a black ball is transferred from bag I to II,
P(A|E2)=410=25
P(E2|A)=P(E2)P(A|E2)P(E1)P(A|E1)+P(E2)P(A|E2)
=47×2537×12+47×25
=1631=0.51

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