Explanation

What is the Hypergeometric Distribution?
The Hypergeometric Distribution describes the probability of drawing a specific number of successes from a finite population without replacement. It is commonly used in situations like quality control, lottery draws, and genetics, where items or subjects are not replaced after selection.

How to calculate the Hypergeometric Distribution?
The Hypergeometric Distribution is calculated by dividing the product of combinations for successes and failures by the total combinations. The formula is:
P(X = k) = (C(K, k) * C(N - K, n - k)) / C(N, n) where:

N is the total population size.
K is the total successes in the population.
n is the sample size.
k is the number of successes in the sample.

What are combinations?
Combinations are a way of counting how many different groups can be formed from a set without regard to the order. The combination formula is: C(n, r) = n! / (r! * (n - r)!) where n! represents the factorial of n, which is the product of all integers from n to 1.

How to calculate combinations?
To calculate combinations, use the formula given above. For example, if you want to know how many different groups of 3 people can be formed from a set of 10, you would use the combination formula:
C(10, 3) = 10 × 9 × 8 / (3 × 2 × 1) resulting in a total of 120 possible combinations.

Frequently Asked Questions (FAQs)

1. When should I use the Hypergeometric Distribution?

The Hypergeometric Distribution is used when you are sampling from a finite population without replacement. It applies in scenarios like quality control (e.g., defective vs. non-defective items in a batch), genetics (e.g., drawing specific gene types from a population), and lottery systems.

2. How is the Hypergeometric Distribution different from the Binomial Distribution?

The key difference is that in the Binomial Distribution, the probability of success remains constant because the sampling is done with replacement. In contrast, the Hypergeometric Distribution involves sampling without replacement, causing the probability to change after each draw.

3. Can I use a calculator for Hypergeometric Probability?

Yes! Our Hypergeometric Distribution Calculator automates the process for you, so you don’t need to manually calculate combinations and probabilities.

4. What happens when the sample size is small compared to the population?

If the sample size is significantly smaller than the population (e.g., less than 5% of the population), the Hypergeometric Distribution closely approximates the Binomial Distribution.

How to Use the Hypergeometric Distribution Calculator?

Follow these steps to calculate probabilities easily:

Enter the total population size (N).
Enter the number of successes in the population (K).
Enter the sample size (n).
Enter the desired number of successes in the sample (k).
Click the Calculate button to get the probability.

The calculator will instantly provide the probability of drawing k successes in the sample.

Hypergeometric Distribution Examples

Example 1: Quality Control

A company produces a batch of 100 items, where 10 are defective. If a quality inspector randomly selects 5 items for testing, what is the probability that exactly 2 of them are defective?

Given:

N = 100 (Total items)
K = 10 (Defective items)
n = 5 (Sample size)
k = 2 (Defective items in the sample)

Using the formula:

P(X = 2) = (C(10,2) * C(90,3)) / C(100,5)

The probability can be computed directly using the calculator.

Example 2: Lottery System

A lottery contains 50 balls, where 5 are winning numbers. If a player selects 6 balls at random, what is the probability of selecting exactly 1 winning ball?

Given:

N = 50 (Total balls)
K = 5 (Winning balls)
n = 6 (Balls drawn)
k = 1 (Winning ball in the selection)

Using the formula:

P(X = 1) = (C(5,1) * C(45,5)) / C(50,6)

The probability can be computed instantly using our online tool.

Conclusion

The Hypergeometric Distribution is an essential probability model used in various real-world applications, including quality control, genetics, and lotteries. Instead of manually calculating complex probabilities, use our Hypergeometric Calculator for instant results!