If a household has at least one cat, there is a 30% probability that the household has at least one dog. Thirty percent of households have at least one cat and 33% of households have at least one dog. Given that a household does not have at least one dog, the probability that the same household does not have at least one cat is --------.
A)0.54
B)0.61
C)0.37
D)0.69

Question

Quizplus · Accepted Answer

To find the probability that a household does not have at least one cat given it does not have at least one dog, we can use the formula for conditional probability. Let's denote C as the event that a household has at least one cat, and D as the event that a household has at least one dog. We are given:- P(C) = 0.30 (probability of having at least one cat)- P(D) = 0.33 (probability of having at least one dog)- P(D|C) = 0.30 (probability of having at least one dog given that the household has at least one cat)We want to find P(C'|D'), where C' and D' are the complements of C and D, respectively, meaning the household does not have a cat or a dog.First, find P(C and D), the probability of a household having both a cat and a dog. Since P(D|C) = 0.30, it means 30% of the 30% of households with cats also have dogs, so:P(C and D) = P(D|C) * P(C) = 0.30 * 0.30 = 0.09.Now, find P(D'), the probability of not having a dog:P(D') = 1 - P(D) = 1 - 0.33 = 0.67.Next, we need to find P(C' and D'), the probability of a household having neither a cat nor a dog. Since we know the probabilities of having a cat, a dog, and both, we can find the probability of having neither by subtracting from 1 the probabilities of having just a cat, just a dog, and both:P(C' and D') = 1 - P(C) - P(D) + P(C and D) = 1 - 0.30 - 0.33 + 0.09 = 0.46.Finally, use the formula for conditional probability to find P(C'|D'):P(C'|D') = P(C' and D') / P(D') = 0.46 / 0.67 ≈ 0.69.Therefore, the correct answer is D) 0.69.

If a Household Has at Least One Cat, There Is