1. What is 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 successes in sample
• \( C(a, b) \) — Combination of a items taken b at a time

Explanation: The formula calculates the probability of getting exactly k successes in n draws 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 the population size (N), number of successes in population (K), sample size (n), and desired successes in sample (k). All values must be non-negative integers with K ≤ N and n ≤ N.

5. Frequently Asked Questions (FAQ)

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

Q2: What if my sample size is large?
A: When sample size is small relative to population (n < 5% of N), binomial approximation may be appropriate.

Q3: Can I use this for continuous data?
A: No, hypergeometric distribution is for discrete counts of successes in finite populations.

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

Q5: How is the combination calculated?
A: Combinations \( C(n, k) \) are calculated using the formula \( \frac{n!}{k!(n-k)!} \), optimized for computation.