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.)

Explanation:

Identify population parameters

Using the Pie Chart Construction and Combinatorics knowledge points

We are given a finite population of \(N = 200\) people.
The pie chart shows that \(36\%\) of the population rate their financial shape as "Fair".
We calculate the number of people in the population who rate their shape as "Fair" (\(k\)):

$$ k = 200 \times 0.36 = 72 $$

The number of people who do not rate their shape as "Fair" (\(N - k\)) is:

$$ N - k = 200 - 72 = 128 $$

Set up the probability model

Using the Hypergeometric Probability knowledge point

Since we are choosing a sample of \(n = 18\) people at random without replacement from a finite population of \(N = 200\), we use the hypergeometric distribution.
We want to find the probability that \(x = 0\) people in our sample rate their financial shape as "Fair".
The formula for hypergeometric probability is:

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

Substituting our values:

$$ P(X = 0) = \frac{\binom{72}{0} \binom{128}{18}}{\binom{200}{18}} = \frac{\binom{128}{18}}{\binom{200}{18}} $$

Calculate the probability

Using the Hypergeometric Probability knowledge point

We expand the combinations to compute the ratio:

$$ P(X = 0) = \frac{\frac{128!}{18! \times 110!}}{\frac{200!}{18! \times 182!}} = \frac{128! \times 182!}{110! \times 200!} $$

This can be written as the product of 18 sequential fractions:

$$ P(X = 0) = \frac{128}{200} \times \frac{127}{199} \times \frac{126}{198} \times \dots \times \frac{111}{183} $$

Calculating this product:

$$ P(X = 0) \approx 0.000153115 $$

Rounding to six decimal places gives \(0.000153\).

Answer:

The probability that none of the 18 people would rate their financial shape as fair is <blank>0.000153</blank>
(Round to six decimal places as needed.)