Question

Among Ph.D. students

• 65% take the probability course.

• 25% take the data science course

• 206 take both probability and data science courses

If we select randomly a student,

1. What is the probability that she takes the data science course given that she takes the probability course? 2. What is the probability that she takes the probability course given that she takes the data science course? 3. What is the probability that she doesn't take the probability course given that she takes the data science course?

Answer 1

Let $A$ be the event that the student takes the probability course and $B$ be the event that the student takes the data science course.

Given $P(A)=0.65, P(B)=0.25, P(A\cap B)=0.2$

1.

$$P(B|A)=\dfrac{P(A\cap B)}{P(A)}=\dfrac{0.2}{0.65}=\dfrac{4}{13}$$

2.

$$P(A|B)=\dfrac{P(A\cap B)}{P(B)}=\dfrac{0.2}{0.25}=0.8$$

3.

$$P(A^C\cap B)=P(B)-P(A\cap B)$$

$$=0.25-0.2=0.05$$

$$P(A^C |B)=\dfrac{P(A^C\cap B)}{P(B)}=\dfrac{0.05}{0.25}=0.2$$