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.)

Explanation:

Determine population size and category counts

Using the Percentage Calculation knowledge point

• Total population size \(N = 400\).
• From the pie chart, the percentage of people who rate their financial shape as "fair" is \(38\%\).
• Number of people in the population who rate their financial shape as "fair" (\(k\)):

$$ k = 400 \times 0.38 = 152 $$

• Number of people in the population who do not rate their financial shape as "fair" (\(N - k\)):

$$ N - k = 400 - 152 = 248 $$

Formulate the probability model

Using the Hypergeometric Distribution and Sampling Without Replacement knowledge points

• We select a sample of size \(n = 20\) without replacement from the population of \(N = 400\).
• We want to find the probability that exactly \(x = 0\) people in the sample rate their financial shape as "fair".
• The hypergeometric probability formula is:

$$ P(X = x) = \frac{\binom{k}{x} \binom{N-k}{n-x}}{\binom{N}{n}} $$

• Substituting our values:

$$ P(X = 0) = \frac{\binom{152}{0} \binom{248}{20}}{\binom{400}{20}} = \frac{\binom{248}{20}}{\binom{400}{20}} $$

Calculate the probability

Using the Hypergeometric Distribution knowledge point

• Expand the combinations:

$$ P(X = 0) = \frac{\frac{248!}{20! \cdot 228!}}{\frac{400!}{20! \cdot 380!}} = \frac{248 \times 247 \times \dots \times 229}{400 \times 399 \times \dots \times 381} $$

• Computing this product ratio:

$$ P(X = 0) \approx 0.0000456158 $$

Round to six decimal places

Using the Hypergeometric Distribution knowledge point

• Rounding \(0.0000456158\) to six decimal places gives \(0.000046\).

Answer:

The probability that none of the 20 people would rate their financial shape as fair is <blank>0.000046</blank>.
(Round to six decimal places as needed.)