Fifty-two percent of the students at a certain college are females. Five percent of the students in this college are majoring in computer science. Two percent of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that

(a) the student is female given that the student is majoring in computer science;

(b) this student is majoring in computer science given that the student is female

The conditional probability that the student is female given that the student is majoring in computer science

Considered events:

S - a person is a student in the college

C - a person is studying computer science

F - a person is female

Given conditional probabilities:

$P(F\mid S)=0.52\Rightarrow P_S\left(F\right)=0.52$

$P(C\mid S)=0.05\Rightarrow P_S\left(C\right)=0.05$

$P(CF\mid S)=0.02\Rightarrow P_S\left(CF\right)=0.02$

Calculate:

$\text{a)}{P}_S(F\mid C)=P(F\mid CS)=\text{?b)}{P}_S(C\mid F)=P(C\mid FS)=\text{?}$

Note that conditional probability satisfies all axioms, so treat it like a probability.

Theses are obtained directly from the definition of conditional probability.

$P_S(F\mid C)=\frac{P_S\left(FC\right)}{P_S\left(C\right)}=\frac{0.02}{0.05}=0.4=40\mathrm{\%}$

The student is female given that the student is majoring in computer science is$40\%$

The conditional probability that this student is majoring in computer science given that the student is female.

Note that conditional probability satisfies all axioms, so treat it like a probability.

This student is majoring in computer science given that the student is female is $3.846\%$.