Question

Exercise 3 (8 marks) Select a Canadian household at random, and let Abe the event that the selected household is prosperous (income exceeds $85, 000) and B the event that the household is educated (completed college). According to the Census Bureau, P(A)=0.125, P(B)=0.237, and the probability that a household is both prosperous and educated is PIA and B)=0.077. a) What is the probability that the household selected is either prosperous or educated? b) Draw a Venn diagram that shows {A and B}. c) Calculate the probability of PIA and B'), show this in a Venn diagram and describe in words what the event is. d) Calculate the probability of PLA and B'), show this in a Venn diagram and describe in words what the event is.

Answer 1

Solution:

Given:

A: The selected household is prosperous
B : the household is educated

P(A) = 0.125

P(B) = 0.237

P( A and B) = 0.077

Part a)

Find: P( Selected household is either prosperous or educated) =........?

That is:

P( A or B) = .........?

P( A or B) = P(A) + P( B) - P( A and B)

P( A or B) = 0.125 + 0.237 - 0.077

P( A or B) = 0.285

Part b)

Draw a venn diagram that shows P( A and B)

A P(A and B) 0.077

Part c)

Find: P( A and B') =.........?

P( A and B') = P(A) - P( A and B)

P( A and B') = 0.125 - 0.077

P( A and B') = 0.048

This event means occurrence of event A but not the event B

that is: selected household is prosperous but not educated.

В A P(A and B' )=0.048 P(A and B)= 0.077

Part d)

Find:
P( A' and B') = ...........?

Using De'Morgans Law:

P( A' and B') = P( A or B)'

P( A' and B') =1 - P( A or B)

P( A' and B') = 1 - 0.285

P( A' and B') = 0.715

this event means , neither A nor B.

that is:  The  selected household is neither prosperous, nor educated.

В A. P(A or B) 0.285 P(A' andB') 0.715