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.)

excellent 5%
good 44%
fair 36%
poor 13%
other 2%

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.)

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.)

excellent 5%
good 44%
fair 36%
poor 13%
other 2%

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.)

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": [ "Combinatorics", "Hypergeometric Probability" ], "new_concepts": [], "current_concepts": [ "Combinatorics", "Hypergeometric Probability", "Pie Chart Construction" ] } ## Identify population parameters Using the *Pie Chart Construction* and *Combinatorics* knowledge points $$ \begin{aligned} &N = 200 \quad \text{(Total population size)}\ &p_{\text{fair}} = 36\% = 0.36 \quad \text{(Proportion rating "Fair")}\ &k = N \times p_{\text{fair}} = 200 \times 0.36 = 72 \quad \text{(Number of "Fair" raters)}\ &N - k = 200 - 72 = 128 \quad \text{(Number of non-"Fair" raters)}\ &n = 18 \quad \text{(Sample size chosen without replacement)} \end{aligned} $$ ## Apply hypergeometric probability formula Using the *Hypergeometric Probability* knowledge point $$ \begin{aligned} P(X = 0) &= \frac{\binom{k}{0} \binom{N-k}{n}}{\binom{N}{n}} \ &= \frac{\binom{72}{0} \binom{128}{18}}{\binom{200}{18}} \ &= \frac{\binom{128}{18}}{\binom{200}{18}} \end{aligned} $$ ## Calculate combinations and probability Using the *Combinatorics* knowledge point $$ \begin{aligned} P(X = 0) &= \frac{128! / (18! \cdot 110!)}{200! / (18! \cdot 182!)} \ &= \frac{128 \times 127 \times \dots \times 111}{200 \times 199 \times \dots \times 183} \ &\approx 0.000153112 \end{aligned} $$ ## Round to six decimal places Using the *Percentage Calculation* knowledge point $$ P(X = 0) \approx 0.000153 $$ 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 population parameters

Apply hypergeometric probability formula

Calculate combinations and probability

Round to six decimal places

Answer: