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?

1 Answer
Geoff K.
Jul 27, 2017

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%#

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