Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let E_1 be the event that the ball chosen belonged to Box I and let E_2 be the event that the ball chosen belonged to Box II. Let F_1 be the event that the ball chosen is red and let F_2 be the event that the ball chosen is green. Then which of the following statements is (are) TRUE?
C. The conditional probability P(F_1 E_1) is equal to the conditional probability P(F_1 E_2)
Total balls in Box I = 6 + 9 = 15 Total balls in Box II = 8 + 12 = 20 Total balls in the mixture = 15 + 20 = 35 Total red balls = 6 + 8 = 14 Total green balls = 9 + 12 = 21 The probabilities of the given events are: P(E_1) = 15 35 = 3 7 P(E_2) = 20 35 = 4 7 P(F_1) = 14 35 = 2 5 P(F_2) = 21 35 = 3 5 The joint probabilities are: P(E_1 F_1) = 6 35 P(E_2 F_2) = 12 35 For statement (A): P(E_1) P(F_1) = 3 7 2 5 = 6 35 = P(E_1 F_1) Thus, E_1 and F_1 are independent. Statement (A) is TRUE. For statement (B): P(E_2) P(F_2) = 4 7 3 5 = 12 35 = P(E_2 F_2) Thus, E_2 and F_2 are independent. Statement (B) is FALSE. For statement (C): P(F_1 E_1) = P(E_1 F_1) P(E_1) = 6/35 15/35 = 2 5 P(F_1 E_2) = P(E_2 F_1) P(E_2) = 8/35 20/35 = 2 5 Since P(F_1 E_1) = P(F_1 E_2), statement (C) is TRUE. For statement (D): P(F_2 E_2) = P(E_2 F_2) P(E_2) = 12/35 20/35 = 3 5 Since 2 5 Answer: The events E_1 and F_1 are independent; The conditional probability P(F_1 E_1) is equal to the conditional probability P(F_1 E_2)
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