Question
Bag I contains 5 red, 4 blue and 3 green balls. Bag II
contains 4 red, 5 blue and 3 green balls. A fair die is thrown once. If the outcome is a prime number, two balls are drawn without replacement from Bag I. If the outcome is a composite number or 1, two balls are drawn without replacement from Bag II. What is the probability that the two balls drawn are of the same colour?Solution
Prime outcomes on a die: 2, 3, 5 Pr(prime) = 3/6 = 1/2 Pr(not prime) = 1/2 Total balls in each bag = 5 + 4 + 3 = 12 For Bag I: P(2 same colour) = P(2 red) + P(2 blue) + P(2 green) P(2 red) = C(5,2)/C(12,2) = 10/66 P(2 blue) = C(4,2)/C(12,2) = 6/66 P(2 green) = C(3,2)/C(12,2) = 3/66 So, P_same_I = (10 + 6 + 3)/66 = 19/66 For Bag II (4R, 5B, 3G): P(2 red) = C(4,2)/66 = 6/66 P(2 blue) = C(5,2)/66 = 10/66 P(2 green) = C(3,2)/66 = 3/66 P_same_II = (6 + 10 + 3)/66 = 19/66 Overall required probability: = (1/2) × P_same_I + (1/2) × P_same_II = (1/2) × 19/66 + (1/2) × 19/66 = 19/66
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