Question

Suppose that a certain college class contains 35 students. of these, 17 are juniors, 20 are social science majors, and 12 are neither. a student is selected at random from the class. (a) what is the probability that the student is both a junior and a social science major? (b) given that the student selected is a junior, what is the probability that she is also a social science major?

Answer 1

A) Suppose $J$ is the set of juniors and $S$ is the set of social science majors. There are 35 students in all, but 12 of them don't belong to either set, so $|J\cup S|=35-12=23$. Then

$|J\cap S|=|J|+|S|-|J\cup S|=17+20-23=14$

So the probability that a random student is both a junior and social science major is $\mathbb P(J\cap S)=\dfrac{14}{35}$

b) We're looking for the probability $\mathbb P(S\mid J)$. By definition, this would be


$\mathbb P(S\mid J)=\dfrac{\mathbb P(S\cap J)}{\mathbb P(J)}=\dfrac{\frac{14}{35}}{\frac{17}{35}}=\dfrac{14}{17}$