Q3.17 - A First Course in Probability

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? 

Step-by-Step Solution

Answer

$P(Dog and Cat)=0.0792$

$P(Dog I Cat)=0.264$

Step 1 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  

Step 2 Explanation

Step 3 Explanation-a

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

                                 $=0.0792$

Step 4 Explanation-b

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

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

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

                        $=0.264$

Step 5 Final Answer of a & b

(a) P(Dog and Cat)=0.0792

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