Question

When you administered the screening test last year, 45% were rated unacceptable, 5% fair, and 50% excellent. Doing further analysis, you discovered that of the 45% who rated as unacceptable on the test, 90% of them really are unacceptable, 7% are fair, and 3% are excellent. Of the 5% who were rated as fair on the test, 95% are fair, 3% are unacceptable, and 2% are excellent. Of the 50% who rated as excellent on the exam, 5% are unacceptable, 15% are fair, and 80% are excellent. Answer the following: Using Bayes formula P(H|E) = P(E)P(E|H)/P(H)

1. Given that an employee is excellent, what is the probability that the test rated them as excellent?

2. Given that an employee is fair, what is the probability that the test rated them as fair?

3. Given that an employee is unacceptable, what is the probability that the test rated them as unacceptable?

Answer 1

P(test rated as unacceptable) = 0.45

P(test rated as fair) = 0.05

P(test rated as excellent) = 0.5

P(employee is unacceptable | test rated as unacceptable) = 0.9

P(employee is fair | test rated as unacceptable) = 0.07

P(employee is excellent | test rated as unacceptable) = 0.03

P(employee is unacceptable | test rated as fair) = 0.03

P(employee is fair | test rated as fair) = 0.95

P(employee is excellent | test rated as fair) = 0.02

P(employee is unacceptable | test rated as excellent) = 0.05

P(employee is fair | test rated as excellent) = 0.15

P(employee is excellent | test rated as excellent) = 0.8

1) P(employee is excellent) = P(employee is excellent | test rated as unacceptable) * P(test rated as unacceptable) + P(employee is excellent | test rated as fair) * P(test rated as fair) + P(employee is excellent | test rated as excellent) * P(test rated as excellent)

                                            = 0.03 * 0.45 + 0.02 * 0.05 + 0.8 * 0.5

                                            = 0.4145

P(test rated as excellent | employee is excellent ) = P(employee is excellent | test rated as excellent) * P(test rated as excellent) / P(employee is excellent)

                                                                              = 0.8 * 0.5 / 0.4145

                                                                              = 0.9650

2) P(employee is fair) = P(employee is fair | test rated as unacceptable) * P(test rated as unacceptable) + P(employee is fair | test rated as fair) * P(test rated as fair) + P(employee is fair | test rated as excellent) * P(test rated as excellent)

                                    = 0.07 * 0.45 + 0.95 * 0.05 + 0.15 * 0.5

                                    = 0.154

P(test rated them as fair | employee is fair) = P(employee is fair | test rated as fair) * P(test rated as fair) / P(employee is fair)

                                                                     = 0.95 * 0.05 / 0.154

                                                                     = 0.3084

3) P(employee is unacceptable) = P(employee is unacceptable | test rated as unacceptable) * P(test rated as unacceptable) + P(employee is unacceptable | test rated as fair) * P(test rated as fair) + P(employee is unacceptable | test rated as excellent) * P(test rated as excellent)

                                                    = 0.9 * 0.45 + 0.03 * 0.05 + 0.05 * 0.5

                                                    = 0.4315

P(test rated as unacceptable | employee is unacceptable) = 0.9 * 0.45 / 0.4315 = 0.9386