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?

Using the total probability theorem: P(Bonus) = P(A)·P(Bonus|A) + P(B)·P(Bonus|B) + P(C)·P(Bonus|C). Substituting: P(Bonus) = (3/10)(4/9) + (1/2)(2/9) + (4/5)(1/3). Computing each term: (3/10)(4/9) = 12/90 = 2/15; (1/2)(2/9) = 2/18 = 1/9; (4/5)(1/3) = 4/15. Converting to common denominator 45: 2/15 = 6/45; 1/9 = 5/45; 4/15 = 12/45. Sum = 6/45 + 5/45 + 12/45 = 23/45. Therefore the probability that the bonus scheme will be introduced is 23/45.