Model Answer
0 min readIntroduction
Probability theory is a fundamental tool in decision-making under uncertainty, widely used in management and risk assessment. This question presents a scenario where the introduction of a Bonus Scheme is contingent upon the appointment of a specific manager – X, Y, or Z. Understanding the probabilities associated with each manager’s appointment and their respective likelihood of introducing the scheme allows for a comprehensive evaluation of the overall probability of the scheme’s implementation. Furthermore, it explores the reverse probability: given the scheme is implemented, what is the probability that Y was the manager responsible?
Part 1: Probability of Bonus Scheme Introduction
Let B be the event that the Bonus Scheme is introduced. We are given the following probabilities:
- P(X) = 4/9 (Probability of X becoming manager)
- P(Y) = 2/9 (Probability of Y becoming manager)
- P(Z) = 1/9 (Probability of Z becoming manager)
- P(B|X) = 3/10 (Probability of Bonus Scheme given X is manager)
- P(B|Y) = 2/5 (Probability of Bonus Scheme given Y is manager)
- P(B|Z) = 1/2 (Probability of Bonus Scheme given Z is manager)
We can use the law of total probability to find P(B):
P(B) = P(B|X)P(X) + P(B|Y)P(Y) + P(B|Z)P(Z)
P(B) = (3/10)(4/9) + (2/5)(2/9) + (1/2)(1/9)
P(B) = 12/90 + 4/45 + 1/18
P(B) = 12/90 + 8/90 + 5/90 = 25/90 = 5/18
Therefore, the probability that the Bonus Scheme will be introduced is 5/18.
Part 2: Probability that Manager was Y, given Bonus Scheme was Introduced
We need to find P(Y|B), the probability that Y was the manager given that the Bonus Scheme was introduced. We can use Bayes' theorem:
P(Y|B) = [P(B|Y)P(Y)] / P(B)
We already know:
- P(B|Y) = 2/5
- P(Y) = 2/9
- P(B) = 5/18
So:
P(Y|B) = [(2/5)(2/9)] / (5/18)
P(Y|B) = (4/45) / (5/18)
P(Y|B) = (4/45) * (18/5)
P(Y|B) = (4 * 18) / (45 * 5) = 72 / 225 = 8/25
Therefore, if the Bonus Scheme has been introduced, the probability that the manager appointed was Y is 8/25.
Conclusion
In conclusion, the probability of the Bonus Scheme being introduced is 5/18, and given its introduction, the probability that manager Y was responsible is 8/25. This demonstrates the application of fundamental probability principles – the law of total probability and Bayes’ theorem – in a practical managerial scenario. These calculations provide valuable insights for decision-making and risk assessment within organizations.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.