Final answer:
To find the probability of 5 successes in a sample of size 8 from a finite population of 40 with 23 successes, we need to apply the hypergeometric distribution formula.
Step-by-step explanation:
The probability of observing exactly 5 successes in a sample size of 8 from a population of 40 with 23 successes follows the hypergeometric distribution, since the population is finite and without replacement. The correct formula to use in this case is:
P(X = k) = [C(K, k) * C(N-K, n-k)] / C(N, n), where:
P(X = k) is the probability of k successes,
K is the total number of successes in the population,
N is the population size,
n is the sample size,
k is the number of successes in the sample.
Thus, to find the probability of 5 successes in a sample of 8 from a population of 40 with 23 successes, we use the values K = 23, N = 40, n = 8, and k = 5:
P(X = 5) = [C(23, 5) * C(17, 3)] / C(40, 8)
After calculating the combination values, you would divide the product of the combinations of successes and failures by the combination of the total number of ways to choose the sample.
The probability is computed by dividing the product of the combinations of successes and failures by the total number of ways to choose the sample.