Question

Apple products have become a household name in America with 51% of all households owning at least one Apple product (CNN, March 19, 2012). The likelihood of owning an Apple product is 61% for households with kids and 48% for households without kids. Suppose there are 1,200 households in a representative community, of which 820 are with kids and the rest are without kids. Let event A correspond to "Household with kids", and B to "Household owns an Apple product". a. Are the events "household with kids" and "household without kids" mutually exclusive and exhaustive? Yes No b. What is the probability that a household is without kids? (Round your answer to 4 decimal places.) c. What is the probability that a household is with kids and owns an Apple product? (Do not round intermediate calculations. Round your answer to 4 decimal places.) d. What is the probability that a household is without kids and does not own an Apple product? (Do not round intermediate calculations. Round your answer to 4 decimal places.)

Answer 1

The events "household with kids" and "household without kids" are mutually exclusive and exhaustive. The probability of a household without kids is 0.3167, with kids and owning an Apple product is 0.4189, and without kids and not owning an Apple product is 0.1647.

Step-by-step explanation:

The question deals with probability and statistics, focusing on the calculation of probabilities for a representative community's households. Let's tackle each part of the question systematically.

Part (a) - Mutually Exclusive and Exhaustive Events

The events "household with kids" and "household without kids" are indeed mutually exclusive and exhaustive; a household cannot be both with kids and without kids at the same time, and all households must be in one of these two categories.

Part (b) - Probability of a Household Without Kids

There are 1,200 households in total and 820 with kids, so 380 households are without kids. The probability that a household is without kids is 380 / 1200 = 0.3167.

Part (c) - Probability With Kids and Owning an Apple Product

The probability that a household is with kids and owns an Apple product is the probability of having kids multiplied by the likelihood of those households owning an Apple product, which is 0.61. Therefore, P(A and B) = P(B|A) * P(A) = 0.61 * (820/1200) = 0.4189.

Part (d) - Probability Without Kids and Not Owning an Apple Product

For households without kids, the likelihood they do not own an Apple product is 1 - 0.48 = 0.52. So, the probability that a household is without kids and does not own an Apple product is P(not B|not A) * P(not A), which is 0.52 * (380/1200) = 0.1647.