Hypergeometric Probability Calculator N R N

The hypergeometric probability calculator helps you determine the probability of drawing specific items from a finite population without replacement. This calculator uses the parameters N (population size), K (number of success states in the population), n (number of draws), and k (number of observed successes).

What is Hypergeometric Probability?

Hypergeometric probability is used when you want to find the probability of drawing a certain number of success items from a finite population without replacement. This is different from binomial probability which assumes sampling with replacement.

Common applications include quality control, genetic testing, and sampling without replacement scenarios. The hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws without replacement from a finite population of size N that contains exactly K successes.

Key characteristics of hypergeometric probability:

Finite population size (N)
Fixed number of successes in population (K)
Sampling without replacement
Discrete outcomes

The 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(n, k) is the combination of n items taken k 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(n, k) = n! / (k! * (n-k)!)

Where "!" denotes factorial, the product of all positive integers up to that number.

How to Use the Calculator

Using the hypergeometric probability calculator is straightforward:

Enter the total population size (N)
Enter the number of success states 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 drawing exactly k successes in n draws from a population of size N with K successes.

Important notes:

N must be greater than or equal to K
n must be less than or equal to N
k must be less than or equal to both K and n

Worked Example

Let's calculate the probability of drawing exactly 3 defective items from a sample of 5 when there are 10 defective items in a batch of 50.

Example Calculation

Given:

N (population size) = 50
K (success states) = 10
n (draws) = 5
k (observed successes) = 3

Using the formula:

P(X = 3) = [C(10, 3) * C(40, 2)] / C(50, 5)

Calculating each combination:

C(10, 3) = 120
C(40, 2) = 780
C(50, 5) = 2,118,760

Final calculation:

P(X = 3) = (120 * 780) / 2,118,760 ≈ 0.0448 or 4.48%

This means there's approximately a 4.48% chance of drawing exactly 3 defective items in a sample of 5 from this batch.

Frequently Asked Questions

What's the difference between hypergeometric and binomial probability?

Hypergeometric probability is used for sampling without replacement from a finite population, while binomial probability assumes sampling with replacement. The hypergeometric distribution accounts for the changing probabilities as items are removed from the population.

When should I use the hypergeometric calculator?

Use the hypergeometric calculator when you need to calculate probabilities for scenarios like quality control testing, genetic testing, or any situation where items are drawn without replacement from a finite population.

What happens if I enter invalid parameters?

The calculator will display an error message if you enter parameters that violate the basic rules (e.g., n > N, k > K, etc.). Make sure your inputs satisfy N ≥ K, n ≤ N, and k ≤ min(K, n).

Can I calculate cumulative probabilities with this calculator?

This calculator currently calculates the probability of exactly k successes. For cumulative probabilities, you would need to sum the probabilities for all values from 0 to k.