The admissions office of a private university released the following data for the preceding academic year: From a pool of 4200 male applicants, 30% were accepted by the university, and 30% of these subsequently enrolled. Additionally, from a pool of 3300 female applicants, 35% were accepted by the university, and 30% of these subsequently enrolled. What is the probability of each of the following?

a) A male applicant will be accepted by and subsequently will enroll in the university?

b) A student who applies for admissions will be accepted by the university?

c) A student who applies for admission will be accepted by the university and subsequently will enroll?

(a) 0.09 (b) 0.322 (c) 0.0966

Step-by-step explanation:

Let's define first the following events

M: an applicant is a male

F: an applicant is a female

A: an applicant is accepted

E: an applicant is enrolled

S: the sample space

Now, we have a total of 7500 applicants, and from these applicants 4200 were male and 3300 were female. So,

P (M) = 0.56 and P (F) = 0.44, besides

P (A | M) = 0.3, P (E | A∩M) = 0.3, P (A | F) = 0.35, P (E| A∩F) = 0.3

(a) 0.09 = (0.3) (0.3) = P (A|M) P (E|A∩M) = P (E∩A∩M) / P (M) = P (E∩A | M)

(b) P (A) = P (A∩S) = P (A∩ (M∪F)) = P (A∩M) + P (A∩F) = P (A|M) P (M) + P (A|F) P (F) = (0.3) (0.56) + (0.35) (0.44) = 0.322

(c) P (A∩E) = P (A∩E∩S) = P (A∩E∩ (M∪F)) = P (A∩E∩M) + P (A∩E∩F) = 0.0504+P (E|A∩F) P (A|F) P (F) = 0.0504 + (0.3) (0.35) (0.44) = 0.0966