Q.3.20 - A First Course in Probability

Question

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

Step-by-Step Solution

Answer

The conditional probability  shows that,

a) $40\%$

b) $3.846\%$

Step 1: Given Information (part a)

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

Step 2: Explanation (part a)

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.

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

Step 3: Final Answer (part a)

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

Step 4: Given Information (part b)

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

Step 5: Explanation (part b)

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

$P_S(C\mid F)=\frac{P_S\left(FC\right)}{P_S\left(F\right)}=\frac{0.02}{0.52}=0.03846=3.846\mathrm{\%}$

Step 6: Final Answer (part b)

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