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.)

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", "Sampling Without Replacement" ], "new_concepts": [], "current_concepts": [ "Combinatorics", "Hypergeometric Probability", "Sampling Without Replacement" ] } ## Identify population parameters Using the **Sampling Without Replacement** and **Combinatorics** knowledge points $$ \begin{aligned} &N = 300 \quad \text{(Total population size)} \ &n = 18 \quad \text{(Sample size chosen at random)} \ &p_{\text{fair}} = 0.36 \quad \text{(Proportion rating financial shape as "Fair" from the pie chart)} \ &K = N \times p_{\text{fair}} = 300 \times 0.36 = 108 \quad \text{(Number of people who rate financial shape as "Fair")} \ &N - K = 300 - 108 = 192 \quad \text{(Number of people who do not rate financial shape as "Fair")} \end{aligned} $$ ## Set up the probability formula Using the **Hypergeometric Probability** and **Combinatorics** knowledge points $$ \begin{aligned} P(X = 0) &= \frac{\binom{K}{0} \binom{N-K}{n}}{\binom{N}{n}} \ P(X = 0) &= \frac{\binom{108}{0} \binom{192}{18}}{\binom{300}{18}} = \frac{\binom{192}{18}}{\binom{300}{18}} \end{aligned} $$ ## Calculate the combination terms Using the **Combinatorics** knowledge point $$ \begin{aligned} \frac{\binom{192}{18}}{\binom{300}{18}} &= \frac{\frac{192!}{18! \times 174!}}{\frac{300!}{18! \times 282!}} \ &= \frac{192 \times 191 \times \dots \times 175}{300 \times 299 \times \dots \times 283} \ &= \prod_{i=0}^{17} \frac{192 - i}{300 - i} \end{aligned} $$ ## Compute the final probability value Using the **Hypergeometric Probability** knowledge point $$ \begin{aligned} P(X = 0) &= \frac{192}{300} \times \frac{191}{299} \times \frac{190}{298} \times \dots \times \frac{175}{283} \ &\approx 0.000130285 \dots \end{aligned} $$ Rounding to six decimal places yields $0.000130$. The probability that none of the 18 people would rate their financial shape as fair is 0.000130. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Hypergeometric Probability" ] }

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Question

Explanation:

Identify population parameters

Set up the probability formula

Calculate the combination terms

Compute the final probability value

Answer: