Audits and Pet Ownership
According to Internal Revenue Service records, 6.42% of all household tax returns are audited. According to the Humane Society, 39% of all households own a dog. Assuming dog ownership and audits are independent events, what is the probability a randomly selected household is audited and owns a dog?

Verified step by step guidance

1. Identify the given probabilities: the probability a household is audited, \(P(A) = 0.0642\), and the probability a household owns a dog, \(P(D) = 0.39\).
2. Recall that if two events are independent, the probability that both events occur is the product of their individual probabilities.
3. Write the formula for the joint probability of independent events: \(P(A \cap D) = P(A) \times P(D)\).
4. Substitute the given values into the formula: \(P(A \cap D) = 0.0642 \times 0.39\).
5. Multiply the probabilities to find the probability that a randomly selected household is both audited and owns a dog.

Key Concepts

Probability of Independent Events

When two events are independent, the occurrence of one does not affect the probability of the other. The probability of both events happening together is the product of their individual probabilities. For example, if event A has probability P(A) and event B has probability P(B), then P(A and B) = P(A) × P(B).

Understanding Given Probabilities

The problem provides specific probabilities: 6.42% for audits and 39% for dog ownership. These percentages must be converted to decimal form (0.0642 and 0.39) before performing calculations. Recognizing and correctly interpreting these values is essential for accurate probability computation.

Application of Multiplication Rule in Probability

The multiplication rule states that for independent events, the joint probability is found by multiplying their probabilities. This rule allows us to find the likelihood of two independent events occurring simultaneously, such as a household being audited and owning a dog.