High school seniors with strong academic records apply to the nation's most selective colleges in greater numbers each year. Because the number of slots remains relatively stable, some colleges reject more early applicants. Suppose that for a recent admissions class, an Ivy League college received applications for early admission. Of this group, it admitted students, early, rejected outright, and deferred to the regular admissions pool for further consideration. In the past, this school has admitted of the applicants in the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2375. Let , , and represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool.
a. Use the data to estimate P(E),P(R), and P(D).
b. Are events E and D mutually exclusive? Find P(E∩D)
c. For the 2375 students admitted to Penn, what is the probability that a randomly se- lected student was accepted during early admission?
d. Suppose a student applies to Penn for early admission. What is the probability the student will be admitted for early admission or be deferred and later admitted during the regular admission process?

Step-by-step explanation:

Hello!

For a recent admissions class, an Ivy League college received 2851 applications for early admission. Of this group, it admitted 1033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration.

In the past, this school has admitted 18% of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2375.

If the events:

E: the student applied for early admission and got it.

R: the student applied to the early admission and got outright rejected.

D: the student applied to the early admission and got deferred to regular admissions.

Total of early applications 2851

Early admitted 1033

Rejected 854

Deferred 964

Historically 18% of the deferred students get admitted.

a. To calculate the probabilities of each event, you have to divide the number of applications that correspond to the event by the total of applications received:

P(E)= 1033/2851= 0.36

P(R)= 854/2851= 0.30

P(D)= 964/2851= 0.34

b. Two events are mutually exclusive when the occurrence of one prevents the occurrence of the other in a single repetition of the experiment. This means that if we randomly choose one early application, the student can be "accepted early" or "deferred to regular admissions" but not the two of them at the same time, which means that E and D are mutually exclusive events.

Since both events are mutually exclusive and cannot happen at the same time, P(E∩D)= 0

c. In this case, the total "population" is not the number of early admissions received by the college you have to work using the total "population" of admitted students.

A: the admitted student was accepted during the early application.

P(A)= 1033/2375= 0.43

d. You need to calculate the probability of the student being admitted early or deferred and later admitted. This probability is a union of two events where "or" is symbolized with ∪.

P(E)= 0.36

Let the event B: the student was deferred and later admitted during the regular admission process. According to the text in the past 18% of the deferred students get admitted. So the probability of b is:

P(B)= 0.18

These two events are mutually exclusive, if the student was admitted early then he cannot be admitted during the regular admissions process.

P(E∪B) = P(E) + P(B)= 0.36 + 0.18= 0.54

I hope it helps!