Question

(1 point in a certain community, 28% of the families own a dog, 36% of the families own a cat, and 25% of the familles that own a dog also own a cat. (Type all your answers as decimals. If rounding is necessary, round to 3 decimal places.) (a) What is the probability that a randomly selected family owns both a dog and a cat? (b) What is the probability that a randomly selected family owns only a cat? (c) What is the probability that a randomly selected family owns a dog given that it owns a cat?

Answer 1

The probability of a randomly selected family owning both a dog and a cat is 0.25. The probability of a randomly selected family owning only a cat is 0.11. The probability of a randomly selected family owning a dog given that it owns a cat is approximately 0.227.

Step-by-step explanation:

To find the probability that a randomly selected family owns both a dog and a cat, we need to use the given information. Among the families that own a dog, 25% also own a cat. Therefore, the probability of a random family owning both a dog and a cat is 0.25.

To find the probability that a randomly selected family owns only a cat, we subtract the probability of owning both a dog and a cat from the probability of owning only a cat. The probability of owning only a cat is 36% minus 25%, which equals 0.11.

To find the probability that a randomly selected family owns a dog given that it owns a cat, we use conditional probability. The probability of owning both a dog and a cat is 25%, and the probability of owning only a cat is 11%. Therefore, the probability of owning a dog given that the family owns a cat is 0.25 divided by 0.11, which equals approximately 0.227 (rounded to 3 decimal places).

Answer 2

The probability that a randomly selected family owns both a dog and a cat is 0.07. The probability that a randomly selected family owns only a cat is 0.64. The probability that a randomly selected family owns a dog given that it owns a cat is 0.694.

Step-by-step explanation:

To find the probability that a randomly selected family owns both a dog and a cat, we can use the formula for the intersection of two events:

P(A ∩ B) = P(A) * P(B|A)

Using the given information, P(A) = 28% = 0.28 and P(B|A) = 25% = 0.25. Therefore, the probability that a randomly selected family owns both a dog and a cat is:

P(A ∩ B) = 0.28 * 0.25 = 0.07

To find the probability that a randomly selected family owns only a cat, we can use the formula for the complement of an event:

P(A') = 1 - P(A)

Using the given information, P(A) = 36% = 0.36. Therefore, the probability that a randomly selected family owns only a cat is:

P(A') = 1 - 0.36 = 0.64

To find the probability that a randomly selected family owns a dog given that it owns a cat, we can use the formula for conditional probability:

P(A|B) = P(A ∩ B) / P(B)

Using the given information, P(A ∩ B) = 25% = 0.25 and P(B) = 36% = 0.36. Therefore, the probability that a randomly selected family owns a dog given that it owns a cat is:

P(A|B) = 0.25 / 0.36 = 0.694