You are an analyst at a large mutual fund. After examining historical data, you establish that all fund managers fall into two categories: superstars (S) and ordinaries (O). Superstars are by far the best managers. The probability that a superstar will beat the market in any given year stands at 70%. Ordinaries, on the other hand, are just as likely to beat the market as they are to underperform it. Regardless of the category in which a manager falls, the probability of beating the market is independent from year to year. Superstars are rare diamonds because only a meager 16% of all recruits turn out to be superstars.
During the analysis, you stumble upon the profile of a manager recruited three years ago, who has since gone on to beat the market every year.
A
What is the probability that this manager is a superstar?
B
What is the probability that this manager will beat the market next year?
C
What is the probability that this manager is an ordinary manager?
D
What is the probability that this manager will underperform the market next year?
E
What is the probability that this manager will beat the market in exactly two of the next three years?
F
What is the probability that this manager will beat the market in at least two of the next three years?
Explanation:
This is a Bayesian probability problem involving conditional probabilities and Bayes' theorem.
Using Bayes' Theorem: [ P(S|B^3) = \frac{P(B^3|S) \cdot P(S)}{P(B^3|S) \cdot P(S) + P(B^3|O) \cdot P(O)} ]
Where:
[ P(S|B^3) = \frac{0.343 \times 0.16}{0.343 \times 0.16 + 0.125 \times 0.84} = \frac{0.05488}{0.05488 + 0.105} = \frac{0.05488}{0.15988} \approx 0.343 ]
Answer: Approximately 34.3%
This is the weighted average of the conditional probabilities: [ P(B_{next}) = P(S|B^3) \cdot P(B|S) + P(O|B^3) \cdot P(B|O) ]
Where:
[ P(B_{next}) = 0.343 \times 0.70 + 0.657 \times 0.50 = 0.2401 + 0.3285 = 0.5686 ]
Answer: Approximately 56.9%
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