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 and learn that the household owns a cat. Find the probability that the household owns a dog.

Short answer

The probability that the household owns a dog is $0.5625$

Step-by-step solution

Given information

$40\%$of the households own a dog.
$32\%$of the households own a cat.
$18\%$ of the households own both pets.

Explanation

Let,

$D$:own a dog

$C$: own a cat

The following probabilities correspond:

Probability for owing a dog,

$P\left(\mathrm{D}\right)=0.40$

Possibility of having a cat,

$P\left(\mathrm{C}\right)=0.32$

Possibility of owning two pets

$P\left(D\text{and}\mathrm{C}\right)=0.18$

Because the household that was chosen at random owns a cat.

Use the conditional probability formula:

$P(\mathrm{D}\mid \mathrm{C})=\frac{P\left(\text{Dand}\right)}{P\left(\mathrm{C}\right)}=\frac{0.18}{0.32}=\frac{9}{16}=0.5625$

Therefore, the probability that the randomly selected household owns a dog is $0.5625$