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Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
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Tutti i libri di testoLarson 8th EditionCh. 4 - Discrete Probability DistributionsProblema 4.Q.3
Capitolo 4, Problema 4.Q.3

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)

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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 > 5)). This is the complement of the probability of at most five successes: P(X > 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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Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the 'success' is defined as a U.S. adult who has put off medical treatment due to cost. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).
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Mean & Standard Deviation of Binomial Distribution

Probability Mass Function (PMF)

The probability mass function gives the probability of obtaining exactly k successes in n trials for a binomial distribution. It is calculated using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k). This function is essential for determining the probabilities of specific outcomes, such as exactly three adults in this scenario.
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Introduction to Probability

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain threshold. In this question, calculating the probability of 'at most four' involves summing the probabilities of getting zero, one, two, three, and four successes. This concept is crucial for understanding how to aggregate probabilities for ranges of outcomes.
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Introduction to Probability
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