Consider the following for the two (02) items that follow: The probabilities that A, B and C become managers are 3 10 , 1 2 and 4 5 respectively. The probabilities that bonus scheme will be introduced if A, B and C become managers are 4 9 , 2 9 and 1 3 respectively. If the bonus scheme has been introduced, then what is the probability that the manager appointed was B?
A. 5 23
Let E_1, E_2, and E_3 be the events that A, B, and C become managers respectively. Given probabilities are: P(E_1) = 3 10 P(E_2) = 1 2 P(E_3) = 4 5 Let X be the event that the bonus scheme is introduced. The conditional probabilities are: P(X|E_1) = 4 9 P(X|E_2) = 2 9 P(X|E_3) = 1 3 We need to find the probability that B was appointed as the manager given that the bonus scheme was introduced, which is P(E_2|X). Using Bayes' theorem: P(E_2|X) = P(E_2)P(X|E_2) P(E_1)P(X|E_1) + P(E_2)P(X|E_2) + P(E_3)P(X|E_3) Calculating the individual terms: P(E_1)P(X|E_1) = 3 10 4 9 = 12 90 = 6 45 P(E_2)P(X|E_2) = 1 2 2 9 = 1 9 = 5 45 P(E_3)P(X|E_3) = 4 5 1 3 = 4 15 = 12 45 Total probability of introducing the bonus scheme is: P(X) = 6 45 + 5 45 + 12 45 = 23 45 Substituting the values into Bayes' theorem: P(E_2|X) = 5 45 23 45 = 5 23 Answer: 5 23
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