Hypergeometric Distribution Calculator
Free hypergeometric calculator. P(X=k) for sampling without replacement. PMF, CDF, mean, variance. P
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Hypergeometric Distribution โ Sampling Without Replacement
Drawing k successes from N with K total. Poker hands, lottery odds, quality control, capture-recapture. PMF, CDF, mean nK/N.
Real-World Scenarios โ Click to Load
Inputs
PMF Bar Chart โ P(X=k)
CDF Step Chart โ P(Xโคk)
Hypergeometric vs Binomial Approximation
When n โช N, binomial with p=K/N approximates hypergeometric.
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- โข Hypergeometric models sampling without replacement from a finite population
- โข Each draw changes the composition of the remaining population
- โข Mean = nK/N; Variance includes finite population correction (N-n)/(N-1)
- โข When n โช N, binomial approximation with p=K/N works well
- โข Poker hands, lottery, quality control, capture-recapture all use hypergeometric
Did You Know?
How It Works
Sampling Without Replacement
Each draw removes an item. Probability of success changes with each draw.
PMF Formula
P(X=k) = C(K,k)รC(N-K,n-k)/C(N,n). Choose k successes from K, (n-k) failures from (N-K), divided by total ways to choose n from N.
Mean and Variance
Mean = nK/N. Variance smaller than binomial due to (N-n)/(N-1) finite population correction.
When to Use Binomial
When n/N is very small (<5%), binomial with p=K/N approximates hypergeometric well.
Valid Range for k
max(0, n-(N-K)) โค k โค min(n, K). Cannot draw more successes or failures than exist.
Expert Tips
Hypergeometric vs Binomial
Use hypergeometric when sampling without replacement; binomial when with replacement or nโชN.
Finite Population
The (N-n)/(N-1) factor reduces variance โ sampling depletes the population.
Poker Hands
Deck N=52. Hearts K=13. Draw 5: n=5. P(2 hearts) is hypergeometric.
Quality Control
Inspect n from batch N with K defectives. P(k defectives in sample) is hypergeometric.
Hypergeometric vs Binomial
| Feature | Hypergeometric | Binomial |
|---|---|---|
| Replacement | Without | With |
| Population | Finite (N) | Infinite |
| Probability p | Changes each draw | Constant |
| Variance | Smaller (FPC) | np(1-p) |
| Use case | Cards, lottery, QC | Coin flips, surveys |
Frequently Asked Questions
When hypergeometric vs binomial?
Use hypergeometric when sampling without replacement from finite population. Use binomial when trials are independent (with replacement) or n/N < 5%.
What is finite population correction?
The factor (N-n)/(N-1) in variance. Reduces variance because drawing depletes the population โ you cannot draw the same item twice.
How to calculate P(X โค k)?
Sum P(X=i) for i from max(0, n-(N-K)) to k. This is the CDF.
Can binomial approximate hypergeometric?
Yes. When n โช N (e.g., n/N < 0.05), use binomial with p=K/N. The chart shows how close they are.
Valid values for k?
max(0, n-(N-K)) โค k โค min(n, K). Need enough successes and failures in population.
Poker: 2 hearts in 5 cards?
N=52, K=13 (hearts), n=5, k=2. P(X=2) = C(13,2)รC(39,3)/C(52,5).
Lottery 6/49?
N=49, K=6 (your numbers), n=6 (drawn). P(match k) = C(6,k)รC(43,6-k)/C(49,6).
Quality control example?
Batch of 100, 5 defectives. Inspect 10. P(1 defective) = C(5,1)รC(95,9)/C(100,10).
Hypergeometric by the Numbers
Official Data Sources
Disclaimer: This calculator provides hypergeometric probabilities for educational and professional reference. For critical applications (lottery, QC, ecology), verify against established statistical software. Ensure 0 โค K โค N, 1 โค n โค N, and max(0, n-(N-K)) โค k โค min(n, K).
Related Calculators
Negative Binomial Distribution Calculator
Number of trials to achieve r successes. Generalizes geometric (r=1). Calculate P(X=k), P(Xโคk), P(aโคXโคb) with PMF bar chart, CDF step chart, and normal...
StatisticsGeometric Distribution Calculator
Calculate probabilities for the geometric distribution โ the number of trials until the first success. PMF, CDF, survival function, and memoryless property.
StatisticsPoisson Distribution Calculator
Calculate P(X=k), P(Xโคk), P(Xโฅk), and P(aโคXโคb) for the Poisson distribution with parameter ฮป. Models count of events in a fixed interval.
StatisticsBeta Distribution Calculator
Calculate PDF, CDF, mean, mode, and variance for the Beta distribution. Bayesian priors, A/B testing, and project management.
StatisticsLognormal Distribution Calculator
Calculate PDF, CDF, mean, mode, variance, and percentiles for the lognormal distribution. If ln(X) ~ N(ฮผ, ฯยฒ), then X is lognormal. Income, house prices...
StatisticsRayleigh Distribution Calculator
Calculate PDF, CDF, mean, mode, variance for the Rayleigh distribution. Models magnitude of 2D Gaussian vectors โ used in wireless communications, wind...
Statistics