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 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

J
240 points

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?

In a
jeffp

8.7k points

Remember the relation,

$$P(E | F) = \frac{P(EF)}{P(F)}$$

In this problem let C be the event that the family owns a cat, and D be the event that the family owns a dog. Then we have,

$P(D)=0.36$

$P(C)=0.30$

$P(C|D)=0.22$

the probability that a randomly selected family owns both a dog and a cat

from the above relation we have,

$$P(CD)=P(C|D)P(D)=0.22 \cdot 0.36 = 0.0792$$

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

$$P(D|C) = \frac{P(CD)}{P(C)}=\frac{0.0792}{0.3}=0.264$$

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