A survey found that 60% of households own a dog (D), 40% own a cat (C), and 20% own both a dog and a cat (D∩C). What is the probability that a randomly selected household owns a dog, given that they own a cat?

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BlackTom AI · Accepted Answer

To start, we need to evaluate what the problem is asking: the probability that a randomly selected household owns a dog given that they own a cat, i.e., P(D|C).

First, recall the definition of conditional probability: P(D|C) = P(D ∩ C) / P(C).

From the survey data: P(D ∩ C) = 0.20 (households that own both a dog and a cat) and P(C) = 0.40 (households that own a cat).

Substituting these values into the formula gives P(D|C) = 0.20 / 0.40 = 0.5.

Thus, among households that own a cat, half also own a dog.

Why the other options aren’t correct:
- If you used P(D) = 0.60 in the numerator, you would be evaluating P(D) itself rather than the conditional event D given C, which ignores the information that the household already owns a cat.
- An answer of 0.33 would imply a ratio of 2/6 or 1/3, which would require different joint and marginal probabilities inconsistent with the given data.
- An answer of 0.6 would again reflect P(D) rather than P(D|C), mixing up unconditional and conditional probabilities.
- An answer of 0.2 would correspond to P(D ∩ C) itself, not the conditional probability given C.

Therefore, the correct conditional probability is 0.5, as calculated from P(D ∩ C) divided by P(C).

Statistics and Probability

A survey found that 60% of households own a dog (D), 40% own a cat (C), and 20% own both a dog and a cat (D∩C). What is the probability that a randomly selected household owns a dog, given that they own a cat?

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