Problem 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

(a) The probability that a randomly selected family owns both a dog and a cat is 0.0792 or 7.92%. (b) The conditional probability that a randomly selected family owns a dog given that it owns a cat is 0.264 or 26.4%.

Step 1: (Step 1: Finding the joint probability of owning a dog and a cat)

The problem states that 22 percent of the families that own a dog also own a cat. Since 36 percent of the families own a dog, we can find the joint probability of owning a dog and a cat by multiplying the percentage of families that own a dog by the percentage of dog owners who also own a cat: Joint probability = \(0.36 * 0.22 = 0.0792\) So, 7.92 percent of the families own both a dog and a cat.

Step 2: (Step 1a: Answer for part (a))

The probability that a randomly selected family owns both a dog and a cat is equal to the joint probability we just found, which is: P(A and B) = 0.0792 (a) Answer: The probability that a randomly selected family owns both a dog and a cat is 0.0792 or 7.92%.

Step 3: (Step 2: Finding the conditional probability of owning a dog given the family owns a cat)

To find the conditional probability of owning a dog given the family owns a cat, we will use the formula for conditional probability: P(Dog | Cat) = \(\frac{P(Dog \cap Cat)}{P(Cat)}\) We already found the joint probability P(Dog and Cat) and we know that 30 percent of the families own a cat. So, we can compute the conditional probability as: \(P(Dog | Cat) = \frac{0.0792}{0.30} = 0.264\)

Step 4: (Step 2a: Answer for part (b))

The conditional probability that a randomly selected family owns a dog given that it owns a cat is: P(Dog | Cat) = 0.264 (b) Answer: The conditional probability that a randomly selected family owns a dog given that it owns a cat is 0.264 or 26.4%.