Hypergeometric Distribution Calculator

Hypergeometric Distribution Formula:

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

Number of Items in Population (N):

Number of Successes in Population (K):

Number of Items in Sample (n):

Number of Successes in Sample (x):

Hypergeometric Probability:

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1. What is the Hypergeometric Distribution?

The hypergeometric distribution describes the probability of drawing a specific number of successes in a sample drawn without replacement from a finite population containing a known number of successes.

2. How Does the Calculator Work?

The calculator uses the hypergeometric distribution formula:

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

Where:

\( N \) — Total population size
\( K \) — Number of successes in population
\( n \) — Sample size
\( x \) — Number of successes in sample
\( \binom{a}{b} \) — Binomial coefficient (a choose b)

Explanation: The formula calculates the probability of exactly x successes in n draws from a finite population of size N containing exactly K successes, without replacement.

3. Importance of Hypergeometric Distribution

Details: The hypergeometric distribution is crucial in quality control, statistical sampling, genetics, and various fields where sampling is done without replacement from finite populations.

4. Using the Calculator

Tips: Enter the total population size (N), number of successes in population (K), sample size (n), and desired number of successes in sample (x). All values must be valid integers with appropriate constraints.

5. Frequently Asked Questions (FAQ)

Q1: When should I use hypergeometric distribution instead of binomial?
A: Use hypergeometric when sampling without replacement from a finite population. Use binomial when sampling with replacement or from an infinite population.

Q2: What are the constraints for valid inputs?
A: K ≤ N, n ≤ N, x ≤ n, x ≤ K, and all values must be non-negative integers.

Q3: Can this calculator handle large numbers?
A: The calculator uses efficient computation methods, but extremely large numbers may cause computational limitations.

Q4: What real-world applications use hypergeometric distribution?
A: Quality control (defective items), biological sampling (species counts), card games probabilities, and genetic inheritance studies.

Q5: How is this different from the binomial coefficient calculation?
A: While both use binomial coefficients, hypergeometric distribution accounts for the changing probabilities when sampling without replacement from finite populations.