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?
C. 23 45
Let E_1, E_2, and E_3 be the events that A, B, and C become managers respectively. The given probabilities are: P(E_1) = 3 10 P(E_2) = 1 2 P(E_3) = 4 5 Let S be the event that the bonus scheme is introduced. The conditional probabilities are: P(S|E_1) = 4 9 P(S|E_2) = 2 9 P(S|E_3) = 1 3 Using the law of total probability, the probability that the bonus scheme will be introduced is: P(S) = P(E_1)P(S|E_1) + P(E_2)P(S|E_2) + P(E_3)P(S|E_3) Substituting the given values: P(S) = ( 3 10 4 9 ) + ( 1 2 2 9 ) + ( 4 5 1 3 ) P(S) = 12 90 + 2 18 + 4 15 P(S) = 2 15 + 1 9 + 4 15 P(S) = 6 15 + 1 9 P(S) = 2 5 + 1 9 P(S) = 18 + 5 45 = 23 45 Answer: 23 45
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