Answer to Question #233140 in Statistics and Probability for Neel

Question

Two students, Alice and Bob, forgot to put their names on their exam papers. The professor knows that Alice does well with probability 0.8, and Bob does well with probability 0.4, independently of each other.

After grading, the professor notices that Alice and Bob forgot to put their names on their exams. One of their exams was done well and the other was done poorly. Given this information, and assuming that they worked independently of each other, what is the probability that the good exam belongs to Alice?

Answer 1

Probability that exactly one of Alice and Bob does well on the test

=P(Alice well and Bob poor)+P(Alice poor and Bob well) $=0.8 \times (1-0.4)+(1-0.8) \times 0.4=0.56$

P( good test belongs to Alice |exactly one does well) $=\frac{P(Alice \;well \;and \;Bob \;poor\;)}{P(exactly\; one\; does \;well)}$

$=0.8\times \frac{(1-0.4)}{0.56}=0.8571$