CFA Level IQuantitative MethodsMedium
A mutual fund analyst estimates that in a bull market (probability 0.60), the fund beats its benchmark with probability 0.70. In a bear market (probability 0.40), the fund beats its benchmark with probability 0.30. Using the total probability rule, what is the unconditional probability that the fund beats its benchmark?
- A0.54
- B0.60
- C0.42
- D0.70
Show answer & explanationAnswer & explanation
Correct answer: A. 0.54
Total probability = P(Bull) x P(Beat|Bull) + P(Bear) x P(Beat|Bear) = (0.60 x 0.70) + (0.40 x 0.30) = 0.42 + 0.12 = 0.54.
Why the other options are wrong
- B. This is simply the probability of a bull market, not the unconditional probability of beating the benchmark.
- C. This is only the bull-market joint probability, omitting the bear-market contribution.
- D. This is the conditional probability of beating the benchmark given a bull market only.
Total Probability Rule
The total probability rule calculates the unconditional probability of an event by summing the probabilities of that event occurring within each mutually exclusive and exhaustive scenario, weighted by the scenario's probability.
- P(A) = sum of P(S_i) x P(A|S_i) across all scenarios S_i
- Scenarios must be mutually exclusive and exhaustive
- Widely used to combine conditional probabilities into an overall probability
Memory trick: Weight each conditional outcome by its scenario's chance, then add.