Question

A college finds that 15% of students have taken a distance learning class and 30% of the students are part time students. Of the part time students, 25% have taken a distance learning class. Let D = event that a student takes a distance learning class and E = 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 indepenent.

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

*Sho all work / formulas

Answer 1

We can divide the sample space into 4 regions:

① part-time students only

② distance learning students only

③ D and E students

④ other students

①+②+③+④=100%

Since E=30%=①+③, ③=25% of 30%=7.5% therefore ①=22.5%.

Also D=15%=②+③, so ②=7.5%. (Although not needed, ④=100-(22.5+7.5+7.5)=100-37.5=62.5%.)

(A) P(D AND E)=③=7.5%.

(B) P(E | D)=50% because given D=15%, E (=7.5%) is 50% of 15%.

(C) P(D OR E)=P(D)+P(E)-P(D AND E)=15+30-7.5=37.5%.

(D) The events D and E are independent because the number of part-time students could, for example, be zero, without affecting the number of distance learning students. However, the percentages are not independent.

(E) If the events were mutually exclusive there could be no intersection (D AND E), in contradiction to the given information.