Hypergeometric Distribution
Successes when sampling without replacement.
Key idea
Successes when sampling without replacement.
In depth
Counts successes in n draws without replacement from a population with K successes among N: P(X = k) = C(K, k)C(N − K, n − k)/C(N, n). It is the binomial’s counterpart when draws affect each other.
Worked example
Drawing 5 cards, the chance of exactly 2 hearts: C(13,2)C(39,3)/C(52,5) ≈ 0.274.
Real-world uses
- Working out the chance of a defective item in a sample taken from a small batch without replacement
Common mistake
Using the binomial when sampling without replacement from a small population.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Hypergeometric Distribution?
Successes when sampling without replacement. It is a topic at the expert level in Probability & Statistics.
Where is Hypergeometric Distribution used in real life?
Real-world examples: Working out the chance of a defective item in a sample taken from a small batch without replacement.
What should I learn before Hypergeometric Distribution?
Make sure you are comfortable with Poisson Distribution, the previous topic in Probability & Statistics.
What is a common mistake with Hypergeometric Distribution?
Using the binomial when sampling without replacement from a small population.
How can I practise Hypergeometric Distribution?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Hypergeometric Distribution?
Move on to Uniform Distribution next.
Related expert topics in Probability & Statistics
- Skewness and Kurtosis – Measures of asymmetry and tail heaviness.
- Residuals and R² – Error of a fit and proportion of variance explained.
- Simpson’s Paradox – A trend reverses when groups are combined.
- Law of Total Probability – Split an event over a partition.
- Bayes’ Theorem – Update beliefs with evidence.
- Medical Tests and Base Rates – Why positive results can still be probably false.