Exams › JEE Advanced › Maths
Bag U1 contains 3 white and 2 red balls; Bag U2 contains only 1 white ball. A fair coin is tossed: on Heads, 1 ball is drawn at random from U1 and placed in U2; on Tails, 2 balls are drawn at random from U1 and placed in U2. After this, 1 ball is drawn from U2. What is the probability that the ball drawn from U2 is white?
- 13/30
- 23/30
- 19/30
- 11/30
Correct answer: 23/30
Solution
Case H (prob 1/2): 1 ball transferred to U2 (which then has 2 balls). P(transferred W) = 3/5, P(W from U2|W transferred) = 2/2 = 1. P(transferred R) = 2/5, P(W from U2|R transferred) = 1/2. P(W|H) = (3/5)*1 + (2/5)*(1/2) = 3/5 + 1/5 = 4/5. Case T (prob 1/2): 2 balls transferred to U2 (which then has 3 balls). Sub-cases: 2W: C(3,2)/C(5,2) = 3/10, P(W|2W) = 3/3 = 1. 1W1R: C(3,1)*C(2,1)/C(5,2) = 6/10, P(W|1W1R) = 2/3. 2R: C(2,2)/C(5,2) = 1/10, P(W|2R) = 1/3. P(W|T) = (3/10)*1 + (6/10)*(2/3) + (1/10)*(1/3) = 3/10 + 4/10 + 1/30 = 9/30 + 12/30 + 1/30 = 22/30 = 11/15. Total: P(W) = (1/2)*(4/5) + (1/2)*(11/15) = 2/5 + 11/30 = 12/30 + 11/30 = 23/30.
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