Question

In a certain community, 25% of the families own a dog, and 20% of the families that own a dog also own a cat. It is also know that 28% of all the families own a cat. What is the probability that a randomly selected family owns a cat?

What is the conditional probability that a randomly selected family owns a dog given that it doesn't own a cat?

Answer 1

The probability that a randomly selected family owns a cat is 28%.

The conditional probability that a randomly selected family owns a dog, given that it doesn't own a cat, is approximately 27.8%.

Explanation:

The first question can be answered straight from the given information. If 28% of all the families own a cat, then the probability of randomly selecting a family that owns a cat is 28%.

Let `D` be the event that a randomly selected family owns a dog.
Let `C` be the event that a randomly selected family owns a cat.

Then:
`P(D)=25%`
`P(C)=28%`
`P(C|D)=20%`

We want to find:
`P(D|notC)=(P(D nn notC))/(P(notC))`

`color(white)(P(D|notC))=(P(D)-P(D nn C))/(1-P(C))`

`color(white)(P(D|notC))=(P(D)-P(D)P(C|D))/(1-P(C))`

`color(white)(P(D|notC))=(P(D)[1-P(C|D)])/(1-P(C))`

`color(white)(P(D|notC))=((0.25)[1-0.20])/(1-0.28)`

`color(white)(P(D|notC))=((0.25)[0.80])/(0.72)`

`color(white)(P(D|notC))=(0.20)/(0.72)`

`color(white)(P(D|notC))=5/18" "~~" "0.278" "=" "27.8%`