the pie chart to the right shows how adults rate their financial shape. suppose 18 people are chosen at random from a group of 300. what is the probability that none of the 18 people would rate their financial shape as fair? (make the assumption that the 300 people are represented by the pie chart.)

excellent 5%
good 45%
fair 36%
poor 13%
other 1%

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 300. what is the probability that none of the 18 people would rate their financial shape as fair? (make the assumption that the 300 people are represented by the pie chart.)

excellent 5%
good 45%
fair 36%
poor 13%
other 1%

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

excellent 5%
good 45%
fair 36%
poor 13%
other 1%

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": [ "Hypergeometric Probability", "Sampling Without Replacement", "Combinatorics" ], "new_concepts": [], "current_concepts": [ "Hypergeometric Probability", "Sampling Without Replacement", "Combinatorics", "Classical Probability" ] } ## Identify population parameters Using the *Sampling Without Replacement* and *Combinatorics* knowledge points $$ \begin{aligned} &N = 300 \quad \text{(Total population size)}\ &n = 18 \quad \text{(Sample size chosen without replacement)}\ &p_{\text{fair}} = 0.36 \quad \text{(Proportion rating "Fair" from pie chart)}\ &K = 300 \times 0.36 = 108 \quad \text{(Number of "Fair" raters in population)}\ &N - K = 300 - 108 = 192 \quad \text{(Number of non-"Fair" raters)} \end{aligned} $$ ## Formulate hypergeometric probability Using the *Hypergeometric Probability* knowledge point $$ \begin{aligned} &P(X = 0) = \frac{\binom{K}{0} \binom{N-K}{n}}{\binom{N}{n}} = \frac{\binom{108}{0} \binom{192}{18}}{\binom{300}{18}} = \frac{\binom{192}{18}}{\binom{300}{18}} \end{aligned} $$ ## Calculate combinations ratio Using the *Combinatorics* knowledge point $$ \begin{aligned} P(X = 0) &= \frac{192! / (18! \cdot 174!)}{300! / (18! \cdot 282!)} = \frac{192! \cdot 282!}{174! \cdot 300!}\ P(X = 0) &= \prod_{i=0}^{17} \frac{192 - i}{300 - i} = \frac{192}{300} \times \frac{191}{299} \times \dots \times \frac{175}{283}\ P(X = 0) &\approx 0.000122467 \end{aligned} $$ ## Round to six decimal places Using the *Hypergeometric Probability* knowledge point $$ \begin{aligned} P(X = 0) \approx 0.000122 \end{aligned} $$ The probability that none of the 18 people would rate their financial shape as fair is 0.000122. { "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

Formulate hypergeometric probability

Calculate combinations ratio

Round to six decimal places

Answer: