Question

question 13 (1 point) saved

solve the problem.

given that \\(x\\) is a hypergeometric random variable with \\(n = 10\\), \\(n = 3\\), and \\(r = 6\\), compute \\(p(x = 0)\\).

.200
.216
0
.033

Explanation:

Identify the hypergeometric parameters

To solve this problem, we first identify the parameters of the hypergeometric distribution from the given information:

• \(N = 10\): The total population size.
• \(r = 6\): The number of success states in the population.
• \(n = 3\): The number of draws (sample size).
• \(x = 0\): The number of observed successes in the sample.

We want to compute the probability \(P(x = 0)\).

State the hypergeometric probability formula

The probability mass function of a hypergeometric random variable is given by:

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

This formula calculates the probability of obtaining exactly \(x\) successes in \(n\) draws without replacement from a population of size \(N\) containing \(r\) successes.

Calculate the combinations

We need to evaluate each binomial coefficient (combination) in the formula for \(x = 0\):

1. The number of ways to choose \(0\) successes from the \(r = 6\) available successes:

$$ \binom{6}{0} = 1 $$

1. The number of ways to choose \(n - x = 3 - 0 = 3\) failures from the \(N - r = 10 - 6 = 4\) available failures:

$$ \binom{4}{3} = \frac{4!}{3!(4-3)!} = 4 $$

1. The total number of ways to choose a sample of \(n = 3\) from the population of \(N = 10\):

$$ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 $$

Compute the final probability

Now, we substitute these calculated combinations back into the hypergeometric probability formula:

$$ P(x = 0) = \frac{\binom{6}{0} \binom{4}{3}}{\binom{10}{3}} = \frac{1 \times 4}{120} = \frac{4}{120} $$

Simplifying the fraction:

$$ P(x = 0) = \frac{1}{30} \approx 0.0333 $$

Comparing this result with the given choices, the value matches \(0.033\) (rounded to three decimal places).

Answer:

• (A) .200
• (B) .216
• (C) 0
• (D) .033 (Correct answer)