Question

Suppose that a certain college class contains

40

students. Of these,

24

are sophomores,

20

are mathematics majors, and

9

are neither. A student is selected at random from the class.

(a) What is the probability that the student is both a sophomore and a mathematics major?

(b) Given that the student selected is a mathematics major, what is the probability that he is also a sophomore?

Write your responses as fractions.

Answer 1

solution:

let A be the event that a student is a sophomore and B be th event that the student is a mathemetics major.

so according to the given information

total students = n = 40

number of sophomore = A = 24, so P(A) = 24/40 = 0.6

number of mathematics major = B = 20, so P(B) = 20/40 = 0.5

number of neither sophomore nor mathematics = 9,

so P(A' n B') = 9/40 = 0.225

P(A u B) = 1 - P(A' n B') = 1 - 0.225 = 0.775

a)

we know that

P(A u B) = P(A) + P(B) - P(A n B)

so, P(A n B) = P(A) + P(B) - P(A u B)

P(A n B) = 0.6 + 0.5 - 0.775 = 0.325

so, the probability that the student is both a sophomore and mathematics major = P(A n B ) = 0.325

b)

P(A | B) = P(A n B) / P(B)

P(A | B) = 0.325 / 0.5 = 0.65

the venn diagram of the given problem as follows: