the pie chart to the right shows how adults rate their financial shape. suppose 20 people are chosen at random from a group of 400. what is the probability that none of the 20 people would rate their financial shape as fair? (make the assumption that the 400 people are represented by the pie chart.)

excellent 6%
good 40%
fair 38%
poor 14%
other 2%

the probability that none of the 20 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 20 people are chosen at random from a group of 400. what is the probability that none of the 20 people would rate their financial shape as fair? (make the assumption that the 400 people are represented by the pie chart.)

excellent 6%
good 40%
fair 38%
poor 14%
other 2%

the probability that none of the 20 people would rate their financial shape as fair is .
(round to six decimal places as needed.)

excellent 6%
good 40%
fair 38%
poor 14%
other 2%

the probability that none of the 20 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": [ "Hypergeometric Distribution", "Sampling Without Replacement", "Combinatorics" ], "new_concepts": [], "current_concepts": [ "Hypergeometric Distribution", "Sampling Without Replacement", "Combinatorics", "Classical Probability" ] } ## Identify population parameters Using the *Sampling Without Replacement* and *Hypergeometric Distribution* knowledge points $$ \begin{aligned} &N = 400 \quad \text{(Total population size)}\ &n = 20 \quad \text{(Sample size chosen at random)}\ &p_{\text{fair}} = 0.38 \quad \text{(Proportion rating financial shape as "fair" from pie chart)}\ &K = N \times p_{\text{fair}} = 400 \times 0.38 = 152 \quad \text{(Number of people rating "fair" in population)}\ &N - K = 400 - 152 = 248 \quad \text{(Number of people not rating "fair" in population)} \end{aligned} $$ ## Set up hypergeometric probability Using the *Combinatorics* and *Hypergeometric Distribution* knowledge points $$ \begin{aligned} &P(X = 0) = \frac{\binom{K}{0} \binom{N-K}{n}}{\binom{N}{n}} = \frac{\binom{152}{0} \binom{248}{20}}{\binom{400}{20}} = \frac{\binom{248}{20}}{\binom{400}{20}} \end{aligned} $$ ## Calculate the probability value Using the *Hypergeometric Distribution* knowledge point $$ \begin{aligned} &P(X = 0) = \frac{248 \times 247 \times \dots \times 229}{400 \times 399 \times \dots \times 381} \approx 0.00003233 \end{aligned} $$ ## Round to six decimal places Using the *Hypergeometric Distribution* knowledge point $$ \begin{aligned} &P(X = 0) \approx 0.000032 \end{aligned} $$ The probability that none of the 20 people would rate their financial shape as fair is 0.000032. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Hypergeometric Distribution" ] }

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QUESTION IMAGE

Question

Explanation:

Identify population parameters

Set up hypergeometric probability

Calculate the probability value

Round to six decimal places

Answer: