Hypergeometric Calculator N N R X

The hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws from a finite population without replacement. This calculator helps you compute probabilities for scenarios where sampling is done without replacement.

What is Hypergeometric Distribution?

The hypergeometric distribution is used when you have a finite population of size N, containing exactly K items of interest (successes), and you want to know the probability of drawing exactly k successes in n draws without replacement.

Key characteristics of the hypergeometric distribution include:

Finite population size (N)
Number of successes in population (K)
Number of draws (n)
Number of observed successes (k)

This distribution is commonly used in quality control, genetics, and sampling problems where items are not replaced after selection.

How to Use the Calculator

To use the hypergeometric calculator:

Enter the total population size (N)
Enter the number of successes in the population (K)
Enter the number of draws (n)
Enter the number of observed successes (k)
Click "Calculate" to get the probability

The calculator will display the probability of getting exactly k successes in n draws from a population of size N with K successes.

Hypergeometric Formula

The probability mass function for the hypergeometric distribution is given by:

P(X = k) = [C(K, k) × C(N-K, n-k)] / C(N, n)

Where:

C(a, b) is the combination of a items taken b at a time
N = total population size
K = number of success states in the population
n = number of draws
k = number of observed successes

The combination formula is C(a, b) = a! / (b! × (a-b)!)

Worked Example

Suppose you have a lot of 50 light bulbs, of which 10 are defective. You randomly select 5 bulbs for testing. What is the probability that exactly 2 of them are defective?

Using the calculator:

N (total bulbs) = 50
K (defective bulbs) = 10
n (bulbs drawn) = 5
k (defective bulbs found) = 2

The calculator would compute this probability using the hypergeometric formula.

Applications of Hypergeometric Distribution

The hypergeometric distribution is used in various fields including:

Quality control to estimate defect rates
Genetics to model inheritance patterns
Sampling problems in finite populations
Lottery probability calculations
Medical testing for disease prevalence

It's particularly useful when dealing with scenarios where items are not replaced after selection, making it different from the binomial distribution which assumes replacement.

FAQ

What's the difference between hypergeometric and binomial distributions?

The binomial distribution assumes independent trials with replacement, while the hypergeometric distribution models dependent trials without replacement from a finite population.

When should I use the hypergeometric calculator?

Use this calculator when you're dealing with sampling from a finite population without replacement, such as quality control testing or genetic studies.

What if my numbers don't make sense for the calculation?

The calculator will validate your inputs to ensure they're logically consistent (e.g., n ≤ N, k ≤ K, etc.).