Question

in which of the following should the random variable x not be modeled with a geometric distribution?
a according to a recent study, approximately 8% of adults in the country have a master’s degree. let x represent the number of randomly selected adults in the country surveyed to find someone with a master’s degree.
b suppose it is known that 5% of the light bulbs manufactured at a particular company are defective. let x represent the number of randomly selected light bulbs that are inspected to find one defective light bulb.
c a particular basketball player is known to consistently make 90% of her free throws, and the outcomes of her free - throw attempts are independent. let x represent the number of attempted free - throws to get one missed free - throw.
d in a bag of 30 different colored candies, about 20% are red. one candy will be selected one at a time without replacement, and its color will be recorded. let x represent the number of candies selected before red is selected.
e it is believed that about 40% of people in the country have purchased a certain product. let x represent the number of people randomly selected to find the first one who has purchased the product.

Explanation:

A geometric distribution models the number of independent trials until the first success, with a constant probability of success \( p \) (and failure \( 1 - p \)) for each trial. Let's analyze each option:

• Option A: Trials (surveying adults) are independent (assuming large population, so sampling without replacement is approximately independent), probability of success (having a master’s degree) is constant (\( p = 0.08 \)). Geometric is appropriate.
• Option B: Trials (inspecting light bulbs) are independent (assuming large production, so sampling without replacement is approximately independent), probability of success (defective bulb) is constant (\( p = 0.05 \)). Geometric is appropriate.
• Option C: Trials (free - throw attempts) are independent, probability of success (missing a free - throw, \( p = 0.10 \)) is constant. Geometric is appropriate (success here is defined as a miss).
• Option D: The candies are selected without replacement from a bag of 30 (a finite population). As we select candies, the probability of selecting a red candy changes (since the population size is small and we are not replacing), violating the "constant probability" and "independent trials" assumptions of the geometric distribution.
• Option E: Trials (selecting people) are independent (assuming large population), probability of success (purchased the product) is constant (\( p = 0.40 \)). Geometric is appropriate.

Answer:

D. In a bag of 30 different colored candies, about 20% are red. One candy will be selected one at a time without replacement, and its color will be recorded. Let \( X \) represent the number of candies selected before red is selected.