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The hypergeometric distribution is a statistical tool used to determine the probability of a specific outcome when sampling from a finite population without replacement. This is different from other probability distributions, such as the binomial distribution, where each draw is independent, and sampling is done with replacement. In our example, the population size is 100, the success centers around drawing items from a subgroup with 20 successes, and we draw 4 samples.To find the probability of obtaining exactly a certain number of successes, we use the hypergeometric formula:

• Calculate the number of ways to choose "x" successes from "K" total successes, denoted by \( \binom{K}{x} \).
• Calculate the number of ways to choose the remaining successes from the non-successful population, denoted by \( \binom{N-K}{n-x} \).
• Calculate the total number of ways to choose "n" items from the total population, denoted by \( \binom{N}{n} \).

This formula gives the probability \( P(X = x) = \frac{\binom{K}{x} \binom{N-K}{n-x}}{\binom{N}{n}} \). For instance, calculating \( P(X = 1) \) involves:

• Calculating \( \binom{20}{1} = 20 \)
• Calculating \( \binom{80}{3} = 82160 \)
• Calculating \( \binom{100}{4} = 3921225 \)

Thus, the probability is approximately 0.418.Similarly, calculating \( P(X = 4) \) and \( P(X = 6) \) follows the same principle, although for \( P(X = 6) \), it's important to recognize that such an outcome is not possible as it exceeds the number of draws.

Determining the mean and variance in a hypergeometric distribution provides insights into the expected outcomes and variability within a finite sampling setup. These calculations are essential for understanding the central tendency and spread of the distribution data.**Mean** The mean of the hypergeometric distribution, represented by \( \mu \), depicts the expected number of successes, calculated as:\[\mu = n \left( \frac{K}{N} \right)\]Substitute the values into the formula:\[\mu = 4 \times \frac{20}{100} = 0.8\]This tells us that, on average, we expect 0.8 successes in each sample draw. This value acts as a central point in the distribution.**Variance** Variance \( \sigma^2 \) measures the distribution's spread from the mean. It's calculated using the formula:\[\sigma^2 = n \left( \frac{K}{N} \right) \left( \frac{N-K}{N} \right) \left( \frac{N-n}{N-1} \right)\]Plugging in the known values:\[\sigma^2 = 4 \times 0.2 \times 0.8 \times \frac{96}{99} \approx 0.619\]This quantifies the distribution's variability and shows that while we expect around 0.8 successes, there's room for some fluctuation.

Finite population sampling is a critical element when working with hypergeometric distributions. This concept is about drawing samples or subsets from a limited set or population and is significant in situations where the population size is small or controlled. Key aspects of finite population sampling include:

• **No Replacement:** Unlike infinite population scenarios (like drawing repeatedly from a deck with replacement), once an item is drawn, it's not replaced, affecting future sampling probabilities.
• **Effect on Probability:** Every draw modifies the pool's composition, changing the likelihood of subsequent outcomes. This creates the need for more intricate probability calculations, as seen with hypergeometric distributions.
• **Practical Applications:** This method is used in inventory audits, quality control processes, and any situation where checking a subset gives insights into the larger whole.

By further understanding finite population sampling, it becomes clear why the hypergeometric distribution can provide more precise probabilities and expectations when sampling in such settings.