Hypergeometric Calculator Cards Multiple Cards

This hypergeometric calculator helps you determine probabilities when drawing multiple cards from a finite population without replacement. It's particularly useful in probability theory, quality control, and sampling scenarios.

Introduction

The hypergeometric distribution describes the probability of k successes in n draws from a finite population without replacement. This calculator provides a practical way to compute these probabilities when dealing with card games, quality inspection, or any scenario involving finite populations.

The key parameters are:

Population size (N): Total number of items in the population
Number of successes in population (K): Number of successful items in the population
Sample size (n): Number of items drawn from the population
Number of observed successes (k): Number of successful items in the sample

How to Use the Calculator

Enter the total population size (N)
Enter the number of successful items in the population (K)
Enter the sample size (n)
Enter the number of observed successes (k)
Click "Calculate" to get the probability
Review the result and interpretation

Note: The calculator assumes that the population is finite and that sampling is done without replacement.

The Hypergeometric Formula

The probability mass function for the hypergeometric distribution is:

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 = population size
K = number of success states in population
n = number of draws
k = number of observed successes

The formula accounts for the probability of drawing exactly k successes in n draws from a finite population without replacement.

Worked Example

Suppose you have a deck of 52 playing cards (N = 52). You want to know the probability of drawing exactly 3 aces (K = 4) in a 5-card hand (n = 5).

Using the calculator:

Set N = 52
Set K = 4 (number of aces in a deck)
Set n = 5 (hand size)
Set k = 3 (desired number of aces)
Calculate the probability

The calculator will show the probability of drawing exactly 3 aces in a 5-card hand from a standard deck.

Frequently Asked Questions

What is the difference between hypergeometric and binomial distribution?

The hypergeometric distribution applies to sampling without replacement from a finite population, while the binomial distribution applies to sampling with replacement or an infinite population. The hypergeometric distribution accounts for the changing probabilities as items are removed from the population.

When should I use the hypergeometric calculator?

Use this calculator when dealing with finite populations and sampling without replacement, such as card games, quality control sampling, or any scenario where the probability changes with each draw.

What if my sample size is larger than the population?

The calculator will alert you if the sample size exceeds the population size, as this would be impossible in a finite population without replacement scenario.