Question

Dogs and cats In one large city, 40% of all households own a dog, 32% own a cat, and 18% own both. Suppose we randomly select a household.

a. Make a Venn diagram to display the outcomes of this chance process using events D: owns a dog, and C: owns a cat.

b. Find P$(D\cap C^C)$.

Short answer

(a) The Venn diagram is

(b) The value of the$P\left(D\cap C^C\right)=0.22$.

Step-by-step solution

Part (a) Step 1: Given Information

We are given the values of probability of household own dog and household own cat and those who own both and we have to make a Venn diagram to show the probability of household own either cat or dog.

Part (a) Step 2: Explanation

According to the question,

The probability of a household having a dog is$0.40$, probability of a household having a cat is$32\%$and those who have both is$18\%$.

A Venn diagram showing the probability of having a dog or cat

Here,$46\%$is showing the household having either cat or dog.

Part (b) Step 1: Given Information

We are given the values of the probabilities of a household's own dog, a household's owning a cat, and those who own both and we have to find out the value of$P(D\cap C^C)$.

Part (b) Step 2: Explanation

To find out the value of the given probability, we required the probability of a household owning a dog, which is,$0.40-0.18=0.22$and the probability of a household now owning a cat, which is,$1-0.32=0.68$and the probability of a household having a dog or not having a cat, islocalid="1654001282795" $0.22+0.46=0.68$.

Now, apply the union rule of probability,

which is$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)+P(A\cap B)$

We get,$P\left(D\cap C^C\right)=0.22+0.68-0.68=0.22$.