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 distribution formula:

Where:

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

Explanation: The formula calculates the probability of getting exactly k successes in n draws from a finite population without replacement.

3. When to Use Hypergeometric Distribution

Details: Use this distribution when dealing with: quality control, lottery probabilities, ecological studies, and any scenario with sampling without replacement from a finite population.

4. Using the Calculator

Tips: Enter the lower and upper bounds for k, population size (N), number of successes in population (K), and sample size (n). All values must be positive integers with K ≤ N and n ≤ N.

5. Frequently Asked Questions (FAQ)

Q1: How is this different from binomial distribution?
A: Hypergeometric accounts for sampling without replacement, while binomial assumes sampling with replacement (or infinite population).

Q2: What are typical applications?
A: Quality control (defective items), wildlife sampling (tagged animals), card games probabilities, and any finite population sampling.

Q3: What if my sample size is large?
A: For large N and small n/N ratio (<5%), binomial approximation may be reasonable.

Q4: Can I calculate cumulative probabilities?
A: Yes, this calculator computes P(lower ≤ X ≤ upper) by summing individual probabilities.

Q5: Why does my TI-84 give slightly different results?
A: Differences may occur due to rounding methods or computational precision limitations in different implementations.