1. What is the Hypergeometric Distribution?

The hypergeometric distribution describes the probability of k successes in n draws without replacement from a finite population of size N containing exactly K successes. It's used when sampling without replacement from a finite population.

2. How Does the Calculator Work?

The calculator uses the hypergeometric probability formula:

Where:

• \( N \) — Population size
• \( K \) — Number of successes in population
• \( n \) — Sample size
• \( k \) — Number of observed successes in sample
• \( \binom{n}{k} \) — Combination (n choose k)

Explanation: The formula calculates the probability of getting exactly k successes in a sample of size n drawn without replacement from a population of size N containing K successes.

3. When to Use Hypergeometric Distribution

Details: Use this distribution when sampling without replacement from a finite population. Common applications include quality control, lottery probabilities, and biological sampling.

4. Using the Calculator

Tips: Enter all positive integer values where K ≤ N, n ≤ N, and k ≤ min(K, n). The calculator will compute the exact probability.

5. Frequently Asked Questions (FAQ)

Q1: How is this different from binomial distribution?
A: Binomial distribution assumes sampling with replacement, while hypergeometric assumes sampling without replacement from a finite population.

Q2: What if my sample is more than 10% of the population?
A: Hypergeometric should be used when sample size is >10% of population, as binomial approximation becomes inaccurate.

Q3: Can I use this for very large populations?
A: For very large N (typically >100,000), binomial distribution becomes a good approximation.

Q4: What are typical applications?
A: Quality control (defective items), ecological studies (species counts), and card game probabilities.

Q5: How accurate is the calculator?
A: It calculates exact probabilities using combinatorial mathematics, limited only by PHP's numerical precision.