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. What is the probability that the bonus scheme will be introduced?

Step-by-step solution

Let E₁ , E₂ , and E₃ be the events that A , B , and C become managers respectively. The given probabilities are: P(E₁) = 3 10 P(E₂) = 1 2 P(E₃) = 4 5 Let S be the event that the bonus scheme is introduced. The conditional probabilities are: P(S|E₁) = 4 9 P(S|E₂) = 2 9 P(S|E₃) = 1 3 Using the law of total probability, the probability that the bonus scheme will be introduced is: P(S) = P(E₁)P(S|E₁) + P(E₂)P(S|E₂) + P(E₃)P(S|E₃) Substituting the given values: P(S) = ( 3 10 4 9 ) + ( 1 2 2 9 ) + ( 4 5 1 3 ) P(S) = 12 9