Any athlete who fails the Enormous State University\'s women\'s soccer fitness t

Question

Any athlete who fails the Enormous State University's women's soccer fitness test is automatically dropped from the team. Last year, Mona Header failed the test, but claimed that this was due to the early hour. (The fitness test is traditionally given at 5 AM on a Sunday morning.) In fact, a study by the ESU Physical Education Department suggested that 54% of athletes fit enough to play on the team would fail the soccer test, although no unfit athlete could possibly pass the test. It also estimated that 41% of the athletes who take the test are fit enough to play soccer. Assuming these estimates are correct, what is the probability that Mona was justifiably dropped? (Round your answer to four decimal places.)

Explanation / Answer

Let

P = fit to play
F = failed the test

Hence, if Mona was justifiably dropped, then she was not fit to play.

Hence, we look for P(P'|F).

As by Bayes' Rule,

P(F) = P(P) P(F|P) + P(P') P(F|P') = 0.41*0.54 + (1-0.41)*1 = 0.8114

Thus,

P(P'|F) = P(P') P(F|P')/P(F) = (1-0.41)*1/0.8114 = 0.72713828 [ANSWER]

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