Question

In a certain community, 36 percent of the families own a dog and 22 percent of the families that own a dog also own a cat. In addition, 30 percent of the families own a cat. What is (a) the probability that a randomly selected family owns both a dog and a cat? (b) the conditional probability that a randomly selected family owns a dog given that it owns a cat?

Short answer

$P(DogandCat)=0.0792$

$P(DogICat)=0.264$

Step-by-step solution

Given Information of a & b

Given that in a certain community, 36 percent of the families own a dog and 22 percent of the families that own a dog also own a cat

We have to find the probability that a randomly selected family owns both a dog and a cat

Also, find the conditional probability that a randomly selected family owns a dog given that it owns a cat

Explanation

Explanation-a

$\text{a)}\mathrm{P}\left(\text{Dog and Cat}\right)=0.36\times 0.22$

$=0.0792$

Explanation-b

$\text{b)}P(\text{Dog}\mid \text{cat})=\frac{P\left(\text{Dog and Cat}\right)}{P\left(\text{Cat}\right)}$

$P\left(\text{cat}\right)=0.30\text{given}$

$P(\text{Dog}\mid \text{cat})=\frac{0.36\times 0.22}{0.30}$

$=0.264$

Final Answer of a & b

(a) P(Dog and Cat)=0.0792

(b) $P(Dog\mid cat)=$0.264