Question

A college finds that 10% of students have taken a distance learning class and that 40% of students are part-time students. Of the part-time students, 20% have taken a distance learning class. Let D = the event that a student takes a distance learning class and E = the event that a student is a part-time student
a. Find P(D AND E).

b. Find P(E|D).

c. Find P(D OR E).

d. Using an appropriate test, show whether D and E are independent.

e. Using an appropriate test, show whether D and E are mutually exclusive.

Short answer

a. P(D AND E) =$0.08$

b. P(E|D) =$0.8$

c. P(D OR E) =$0.42$

d. They are not independent

e. they are not mutually exclusive

Step-by-step solution

introduction

Mutually exclusive occasions are those that can't occur simultaneously, while independent are those whose probabilities don't influence each other.

explanation part (a)

Students who took distance-learning = $10\%$or P(D) = role="math" localid="1651833127872" $0.1$

Students who are part-time = $40\%$or P(E) = $0.4$

Part-time students who took distance-learning = $20\%$or P(D/E) =$0.2$

D denote the distance learning classes event and E denotes a part-time event

$P(DandE)=P(D\mid E)P\left(E\right)$

role="math" localid="1651897467873" $P(DandE)=0.20\times 0.40=0.08$

explanation part (b)

Students who took distance-learning = $10\%$or P(D) =$0.1$

Students who are part-time = $40\%$or P(E) =$0.4$

Part-time students who took distance-learning = $20\%$or P(D/E) = $0.2$

$P\left(D\text{and}E\right)=0.08$

$P(E\mid D)=\frac{P(EandD)}{P\left(D\right)}$