Question

According to the record of the registrar's office at a state university, 35% of the students arefreshman, 25% are sophomore, 16% are junior, and the rest are senior. Among the freshmen,sophomores, juniors, and seniors, the portion of students who live in the dormitory are,respectively, 80%, 60%, 30%, and 20%. What is the probability that a randomly selected studentis a freshman who lives in a dormitory?

A) 0.28
B) 0.32
C) 0.52
D) 0.38

Answer 1

The probability that a randomly selected student is a freshman who lives in a dormitory is 0.28. So the correct option is A.

Step-by-step explanation:

To find the probability that a randomly selected student is a freshman who lives in a dormitory, we need to multiply the probability that a student is a freshman by the probability that a freshman lives in the dormitory. According to the given information, 35% of the students are freshmen, and among them, 80% live in the dormitory.

We calculate this probability using the formula:

Probability of (Freshman who lives in dorm) = Probability of (Freshman) × Probability of (Dorm | Freshman)

Probability of (Freshman who lives in dorm) = 0.35 × 0.80

Probability of (Freshman who lives in a dorm) = 0.28

Thus, the correct answer is A) 0.28.

Answer 2

the probability is 0.28

Step-by-step explanation:

using Bayes's theorem

P(A|B)=P(A∩B)/P(B)

where

P(A∩B) = probability that events A and B happen

P(A|B) = probability that event A happen if B already happened

P(B)= probability of event B

therefore

P(A∩B)=P(A|B)*P(B)

if event A= selection of a student that lives in a dormitory and event B = selection of a freshmen student

P(A|B) = 0.8 (live in a dormitory knowing that is a freshmen student )

P(B) = 0.35 (freshmen student)

P(A∩B)=P(A|B)*P(B) = 0.8* 0.35 =0.28