r/probabilitytheory • u/Negative-Bowl-3678 • 18h ago

[Education] Bayes' Theorem Practice Problem

Historical data shows that fund managers at a financial firm are classified into two categories:

  • Elite Managers: These top performers have a 75% chance of beating the market in any given year.
  • Standard Managers: These average performers have a 50% chance of beating the market (they are equally likely to outperform or underperform it).

For both groups, performance is completely independent from one year to the next. Truly elite managers are rare, making up only 16% of the total talent pool.

Three years ago, you hired a new manager who has successfully beaten the market in each of the last three consecutive years.

  1. What was the initial probability that this manager belonged to the elite category when they were first hired?
  2. Given their three-year streak of outperforming the market, what is the updated probability that this manager is elite?
  3. Based on their past three years of performance, what is the probability that this manager will beat the market again next year?
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u/wittgenstein1312 17h ago
  1. 16%
  2. ~ 36%
  3. ~ 59%

1

u/mirlaca 5h ago

I'm learning, so may you confirm 1. certainly 0.16, because it's stated in the problem

2. because of law of total probability, P(E3) = P(E|O1) + P(E|O2) + P(E|O3)= 0.16 * 0.75 * 3 = 0.36

3. P(O4) = P(O4|E) + P(O4|S) = 0.75 * 0.36 + 0.5 * (1-0.36) = 0.59

It matches the answers above, but is it all correct?