Chapter 5.2, Problem 51E

(a)

To determine

Venn diagram to display the outcomes of chance process.

Explanation of Solution

Given information:

C: owns a cat

D: owns a dog

Draw two partially overlapping ellipses.

Denote right ellipse as D.

Denote left ellipse as C.

We know that

18% own both a cat as well as a dog.

Thus,

Write 18% in the area where two ellipses are overlapping.

According to the statement,

40% own a dog, while 18% own a dog and a cat.

This implies

Remaining 22% own a dog but not a cat.

Thus,

Write 22% in the part of ellipse D that does not overlap with ellipse C.

According to the statement,

32% own a cat, while 18% own both a dog and a cat.

This implies

Remaining 14% own a cat but not a dog.

Thus,

Write 14% in the part of ellipse C that does not overlap with ellipse D.

We also know

Venn diagram requires 100% representation.

Until now,

The total amount in Venn diagram is 54%.

That means

54% own either a single pet or both the pets.

This implies

Remaining 46% do not own a single pet.

Thus,

Write 46% in the area outside both the ellipses.

  

(b)

To determine

Probability for the event $\text{D}\cap {\text{C}}^{\text{c}}$ .

Answer to Problem 51E

Probability for the event,

  $P\left(\text{D}\cap {\text{C}}^{\text{c}}\right)=22\%=0.22$

Explanation of Solution

Given information:

C: owns a cat

D: owns a dog

  ${\text{C}}^{\text{c}}$ : does not own a cat

  ${\text{D}}^{\text{c}}$ : does not own a dog

Note that

In the Venn diagram,

The part of ellipse D that does not overlap with the corresponding ellipse C comprises of 22%.

That implies

22% do not own a cat

Thus,

The probability for the event,

  $P\left(\text{D}\cap {\text{C}}^{\text{c}}\right)=22\%=0.22$