Question

The chartered financial analyst (CFA) is a designation earned after taking three annual exams (CFA I,II, and III). The exams are taken in early June. Candidates who pass an exam are eligible to take the exam for the next level in the following year. The pass rates for levels I, II, and III are 0.58, 0.75, and 0.81, respectively. Suppose that 3,000 candidates take the level I exam, 2,500 take the level II exam and 2,000 take the level III exam. A randomly selected candidate who took a CFA exam tells you that he has passed the exam. What is the probability that he took the CFA I exam? Probability =

Answer 1

A : Event of a randomly selected candidate taking level I exam

B : Event of a randomly selected candidate taking level II exam

C: Event of a randomly selected candidate taking level III exam

P(A) = Total number of candidates taking level I exam / Total number of candidates = 3000/7500=30/75=2/5

P(B) = 2500/7500=25/75=1/3

P(C) = 2000/7500=20/75=4/15

X : Event of a randomly selected candidate passing the exam

Given,

Probability of a randomly selected candidate passing the exam given that the candidate took level I exam = P(X|A) = 0.58

P(X|B) = 0.75

P(X|C) = 0.81

A randomly selected candidate who took a CFA exam tells you that he has passed the exam, Probability that he took the CFA I exam = P(A|X)

By Bayes theorem,

P(A)P(X|A) =0.58 x (2/5) = 0.232

P(B)P(X|B) = 0.75 x (1/3) = 0.25

P(C)P(X|C) =0.81 x (4/15) = 0.216

P(A)P(X|A) + P(B)P(X|B) + P(C)P(X|C) = 0.232+0.25+0.216= 0.698

A randomly selected candidate who took a CFA exam tells you that he has passed the exam, Probability that he took the CFA I exam = 0.332378223

Probability = 0.332378223