100. 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.

Let D = the event that a student takes a distance learning class and E = the event that a student is a part-time student

No. of students that have taken a distance learning class is$10\%$

No. of students that are part-time students is $40\%$

No. of part-time students that have taken a distance learning class $20\%$

Let D = the event that a student takes a distance learning class and E = the event that a student is a part-time student

now,

No. of students that have taken a distance learning class is$P\left(D\right)=0.10$

No. of students that are part-time students is$P\left(E\right)=0.40$

No. of part-time students that have taken a distance learning class$P(D\mid E)=0.20$

Calculating using $P(DANDE)=P(D\mid E)P\left(E\right)$

$$\begin{gathered} P(DANDE)=0.20\times 0.40 \\ =0.08 \end{gathered}$$

now,

No. of students that have taken a distance learning class is $P\left(D\right)=0.10$

No. of students that are part-time students is $P\left(E\right)=0.40$

$P(DANDE)=0.08$

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

$P(E\mid D)=\frac{0.08}{0.10}=0.8$

No. of students that have taken a distance learning class is $P\left(D\right)=0.10$

No. of students that are part-time students is $P\left(E\right)=0.40$

$P(DANDE)=0.08$

Calculating $P(DORE)=P\left(D\right)+P\left(E\right)-P(DANDE)$

$P(DORE)=0.10+0.40-0.08=0.42$

Two events are considered independent if $P(D\mid E)=P\left(D\right)$

In this case,

$P(D\mid E)=0.20$and $P\left(D\right)=0.10$are not equal.

hence they are not independent.

Two events are mutually exclusive if $P(DANDE)=0$

But we have,

$P(DANDE)=0.08\ne 0$

Hence they are not mutually exclusive