Question

In the past year, thirty-three percent of U.S. adults have put off medical treatment because of the cost. You randomly select nine U.S. adults. Find the probability that the number who have put off medical treatment because of the cost in the past year is (a) exactly three, (b) at most four, and (c) more than five. (Source: Gallup)

Accepted Answer

Step 1: Recognize that this is a binomial probability problem. The problem involves a fixed number of trials (n = 9), two possible outcomes (putting off medical treatment or not), a constant probability of success (p = 0.33), and independent trials. The binomial probability formula is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'k' is the number of successes. Step 2: For part (a), calculate the probability of exactly three successes (k = 3). Use the binomial formula: P(X = 3) = (9 choose 3) * $$(0.33)^3 * (1-0.33)^(9-3). $$Compute the binomial coefficient (9 choose 3) = 9! / [3!(9-3)!], then substitute the values into the formula. Step 3: For part (b), calculate the probability of at most four successes (P(X ≤ 4)). This is the sum of probabilities for X = 0, 1, 2, 3, and 4. Use the binomial formula for each value of k (0 through 4) and sum the results: P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4). Step 4: For part (c), calculate the probability of more than five successes (P(X \u003e 5)). This is the complement of the probability of at most five successes: P(X \u003e 5) = 1 - P(X ≤ 5). First, calculate P(X ≤ 5) by summing the probabilities for X = 0 through 5 using the binomial formula, then subtract this value from 1. Step 5: Use a calculator or statistical software to compute the binomial probabilities for each part. Alternatively, use a binomial probability table or a cumulative distribution function (CDF) for the binomial distribution to simplify the calculations for parts (b) and (c).

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Key Concepts

Binomial Distribution

Probability Mass Function (PMF)

Cumulative Probability