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": [ "Percentage Calculation", "Hypergeometric Probability" ], "new_concepts": [ "Sampling Without Replacement" ], "current_concepts": [ "Percentage Calculation", "Hypergeometric Probability", "Sampling Without Replacement" ] } ## Calculate the number of people in each category Using the *Percentage Calculation* knowledge point $$ \begin{aligned} &\text{Total population } (N) = 200\ &\text{Percentage rating "Fair"} = 36\%\ &\text{Number of "Fair" ratings } (k) = 200 \times 0.36 = 72\ &\text{Number of "Not Fair" ratings } (N - k) = 200 - 72 = 128 \end{aligned} $$ ## Set up the probability model Using the *Hypergeometric Probability* knowledge point $$ \begin{aligned} &\text{Sample size } (n) = 18\ &\text{Target: Choose } 0 \text{ "Fair" and } 18 \text{ "Not Fair" without replacement}\ &P(X = 0) = \frac{\binom{72}{0} \binom{128}{18}}{\binom{200}{18}} = \frac{\binom{128}{18}}{\binom{200}{18}} \end{aligned} $$ ## Compute the probability We calculate the ratio of combinations: $$ \begin{aligned} P(X = 0) &= \frac{128 \times 127 \times \dots \times 111}{200 \times 199 \times \dots \times 183} \ P(X = 0) &\approx 0.000153117 \end{aligned} $$ ## Round to six decimal places Rounding $0.000153117$ to six decimal places gives: $$ P(X = 0) \approx 0.000153 $$ The probability that none of the 18 people would rate their financial shape as fair is 0.000153. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Hypergeometric Probability" ] }

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Question

Explanation:

Calculate the number of people in each category

Set up the probability model

Compute the probability

Round to six decimal places

Answer: