The hypergeometric distribution models probabilities in situations where you draw items from a finite population without replacement. This means that each selection affects subsequent probabilities because the total number of items decreases as you draw each one. It’s a commonly used distribution in probability and statistics, especially when dealing with scenarios like quality control, lotteries, or card games, where items aren’t replaced after being drawn.

Key Components of the Hypergeometric Distribution

1. Population Size (N): The total number of items in the population.
2. Successes in the Population (K): The number of items in the population that are considered "successes" (e.g., defective items, winning tickets).
3. Sample Size (n): The number of items you draw or select from the population.
4. Successes in the Sample (x): The number of items in your sample that are "successes."

Hypergeometric Probability Formula

The probability of drawing exactly $x$ successes in a sample of $n$ items from a population of $N$ items that contains $K$ successes is given by:

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

Where:

• $\binom{K}{x}$ is the number of ways to choose $x$ successes from $K$ total successes.
• $\binom{N - K}{n - x}$ is the number of ways to choose $n - x$ failures from the $N - K$ failures.
• $\binom{N}{n}$ is the number of ways to choose $n$ items from the population of $N$ items.

Explanation of the Formula

• The numerator counts the total ways to achieve the specific configuration where you draw exactly $x$ successes and $n - x$ failures.
• The denominator counts the total ways to choose $n$ items from $N$, giving us the probability of a specific arrangement.

Example Problem

Imagine a box with 10 marbles: 4 are red (successes), and 6 are blue (failures). You draw 3 marbles without replacement and want to find the probability of drawing exactly 2 red marbles.

• $N = 10$ (total marbles)
• $K = 4$ (red marbles, or successes)
• $n = 3$ (marbles drawn)
• $x = 2$ (red marbles drawn)

The probability $P(X = 2)$ would be calculated using the formula above.

Characteristics of the Hypergeometric Distribution

1. Discrete: The hypergeometric distribution is a discrete probability distribution.
2. Without Replacement: Each selection alters the probabilities of future selections, unlike the binomial distribution, where the probability remains constant across trials.
3. Non-constant Success Probability: The probability of success changes with each draw, since the population pool is reduced with each selection.

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Follow-up Questions

1. How does the hypergeometric distribution compare with the binomial distribution?
2. Can you explain the role of combinatorics in the hypergeometric formula?
3. What are some real-world applications of the hypergeometric distribution?
4. How would you calculate expected value and variance for a hypergeometric distribution?
5. Can you show an example of using the hypergeometric formula with a large population size?

Tip

To determine whether a problem fits a hypergeometric model, check if it involves a finite population with no replacement. This distinguishes it from problems suitable for the binomial distribution, where trials are independent.