Solution
The Fund That Never Lost
Show the problem again
1 in 100 fund managers is genuinely skilled, and a skilled manager beats the market in any given year with probability $3/4$ (independently across years). The other 99 in 100 have no skill and beat the market with probability $1/2$ each year, independently. You select, from the whole population, a manager who has beaten the market in each of the last 5 years. What is the probability she is skilled? Give a reduced fraction.
Worked solution
By Bayes' theorem, using unnormalized weights: - skilled: $\frac{1}{100}\cdot\left(\frac34\right)^5 = \frac{1}{100}\cdot\frac{243}{1024}$ - unskilled: $\frac{99}{100}\cdot\left(\frac12\right)^5 = \frac{99}{100}\cdot\frac{32}{1024}$ So the posterior is $\dfrac{243}{243 + 99\cdot 32} = \dfrac{243}{3411} =$ **27/379**, about 7.1%. A 5-year streak is about 7.6 times more likely for a skilled manager, yet the base rate of skill is 99 to 1 against. Selecting on a streak from a large pool is mostly selecting luck.
Source: Base-rate reasoning for streak-selected fund managers; a standard Bayes exercise. Statement written for AxiomIQ.