Question

Consider three students taking an examination. Their respective probabilities of obtaining the different available grades are shown in Table 1. (a) What is the probability that each will obtain a grade of A? (b) If two students were to receive an A, what is the probability that Jim would be among them?

Table 1. Probabilities of Individual Student Exam Grades*

Student    Grade   

                 A   B     C     D      F

Terry         0.5 0.3 0.1 0.06 0.04

Jim           0.4 0.4 0.1 0.07 0.03

Sam         0.3 0.3 0.3 0.05 0.05

* Assume each examination outcome to be independent.

Answer 1

a) P(each will obtain a grade of A) = P(Terry gets an A) * P(Jim gets an A) * P(Sam gets an A)

                                                       = 0.5 * 0.4 * 0.3

                                                       = 0.06

b) P(two students were to receive an A) = P(Terry and Jim gets A and Sam doesn't get A) + P(Terry and Sam gets A and Jim doesn't get A) + P(Sam and Jim gets A and Terry doesn't get A)

                                                                = 0.5 * 0.4 * (1 - 0.3) + 0.5 * 0.3 * (1 - 0.4) + 0.3 * 0.4 * (1 - 0.5)

                                                                = 0.29

P(Jim would be among them and two students were to receive an A) = P(Terry and Jim gets A and Sam doesn't get A) + P(Sam and Jim gets A and Terry doesn't get A)

                                                               = 0.5 * 0.4 * (1 - 0.3) + 0.3 * 0.4 * (1 - 0.5)

                                                               = 0.2

P(Jim would be among them | two students were to receive an A) = P(Jim would be among them and two students were to receive an A) / P(two students were to receive an A)

                                                              = 0.2 / 0.29

                                                              = 0.69