the pie chart to the right shows how adults rate their financial shape. suppose 18 people are chosen at random from a group of 200. what is the probability that none of the 18 people would rate their financial shape as fair? (make the assumption that the 200 people are represented by the pie chart.)

the probability that none of the 18 people would rate their financial shape as fair is
(round to six decimal places as needed.)

Question

the pie chart to the right shows how adults rate their financial shape. suppose 18 people are chosen at random from a group of 200. what is the probability that none of the 18 people would rate their financial shape as fair? (make the assumption that the 200 people are represented by the pie chart.)

the probability that none of the 18 people would rate their financial shape as fair is
(round to six decimal places as needed.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Sampling Without Replacement", "Combinatorics" ], "new_concepts": [], "current_concepts": [ "Sampling Without Replacement", "Combinatorics", "Hypergeometric Probability" ] } ## Identify the population parameters Using the **Sampling Without Replacement** and **Combinatorics** knowledge points $$ \begin{aligned} &N = 200 \quad \text{(Total population size)}\ &n = 18 \quad \text{(Sample size chosen without replacement)}\ &P(\text{Fair}) = 38\% = 0.38 \quad \text{(From the pie chart)} \end{aligned} $$ ## Calculate the number of successes and failures in the population Using the **Sampling Without Replacement** knowledge point $$ \begin{aligned} &k = 200 \times 0.38 = 76 \quad \text{(Number of people who rate financial shape as "Fair")}\ &N - k = 200 - 76 = 124 \quad \text{(Number of people who do not rate financial shape as "Fair")} \end{aligned} $$ ## Set up the hypergeometric probability formula Using the **Combinatorics** knowledge point $$ \begin{aligned} &P(X = 0) = \frac{\binom{76}{0} \binom{124}{18}}{\binom{200}{18}} = \frac{\binom{124}{18}}{\binom{200}{18}} \end{aligned} $$ ## Compute the probability value Using the **Sampling Without Replacement** knowledge point $$ \begin{aligned} &P(X = 0) = \frac{124}{200} \times \frac{123}{199} \times \frac{122}{198} \times \dots \times \frac{107}{183}\ &P(X = 0) \approx 0.000152865 \approx 0.000153 \end{aligned} $$ The probability that none of the 18 people would rate their financial shape as fair is 0.000153 (Round to six decimal places as needed.) { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Hypergeometric Probability" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the population parameters

Calculate the number of successes and failures in the population

Set up the hypergeometric probability formula

Compute the probability value

Answer: