About 15% of people in a town have both a cat and a dog. As 64% of residents have a dog, what is the probability that someone in the town owns a cat, given they have a dog?

• A. 0.23
• B. 0.15
• C. 0.64
• D. 0.096

Answer: (A)

Verified step by step guidance

1. Step 1: Recognize that this is a conditional probability problem. The formula for conditional probability is P(A|B) = P(A ∩ B) / P(B), where P(A|B) is the probability of event A given event B, P(A ∩ B) is the probability of both events A and B occurring, and P(B) is the probability of event B.
2. Step 2: Define the events in the problem. Let A represent the event 'a person owns a cat' and B represent the event 'a person owns a dog.' The problem provides P(A ∩ B) = 0.15 (probability of owning both a cat and a dog) and P(B) = 0.64 (probability of owning a dog).
3. Step 3: Substitute the given values into the conditional probability formula. Using P(A|B) = P(A ∩ B) / P(B), substitute P(A ∩ B) = 0.15 and P(B) = 0.64.
4. Step 4: Simplify the fraction to calculate P(A|B). This will give the probability that someone owns a cat, given that they own a dog.
5. Step 5: Interpret the result. The final value represents the likelihood of a person owning a cat if it is already known that they own a dog.