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 ?

Step-by-step solution

Let E₁ , E₂ , and E₃ be the events that A , B , and C become managers respectively. Given probabilities are: P(E₁) = 3 10 P(E₂) = 1 2 P(E₃) = 4 5 Let X be the event that the bonus scheme is introduced. The conditional probabilities are: P(X|E₁) = 4 9 P(X|E₂) = 2 9 P(X|E₃) = 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(E₂|X) . Using Bayes' theorem: P(E₂|X) = P(E₂)P(X|E₂) P(E₁)P(X|E₁) + P(E₂)P(X|E₂) + P(E₃)P(X|E₃) Calculating the indivi