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.)

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.)

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" ], "new_concepts": [], "current_concepts": [ "Hypergeometric Distribution", "Sampling Without Replacement", "Combinatorics" ] } ## Identify population parameters Using the *Hypergeometric Distribution* and *Sampling Without Replacement* knowledge points $$ \begin{aligned} &N = 400 \quad \text{(Total population size)}\ &n = 20 \quad \text{(Sample size chosen without replacement)}\ &p_{\text{fair}} = 0.38 \quad \text{(Proportion rating financial shape as "fair")}\ &K = N \times p_{\text{fair}} = 400 \times 0.38 = 152 \quad \text{(Number of "fair" raters in population)}\ &N - K = 400 - 152 = 248 \quad \text{(Number of non-"fair" raters in population)} \end{aligned} $$ ## Formulate hypergeometric probability Using the *Hypergeometric Distribution* and *Sampling Without Replacement* 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 combinations ratio Using the *Hypergeometric Distribution* and *Sampling Without Replacement* knowledge points $$ \begin{aligned} P(X = 0) &= \frac{248 \times 247 \times \dots \times 229}{400 \times 399 \times \dots \times 381} \ &= \prod_{i=0}^{19} \frac{248 - i}{400 - i} \ &\approx 0.00005218 \end{aligned} $$ ## Round to six decimal places Using the *Hypergeometric Distribution* and *Sampling Without Replacement* knowledge points $$ \begin{aligned} P(X = 0) \approx 0.000052 \end{aligned} $$ The probability that none of the 20 people would rate their financial shape as fair is 0.000052. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Hypergeometric Distribution" ] }

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Question

Explanation:

Identify population parameters

Formulate hypergeometric probability

Calculate combinations ratio

Round to six decimal places

Answer: