Understand the distribution
What the multinomial distribution calculator models
Assigning probability to separate outcomes. A discrete probability mass function assigns probability directly to each allowed count or category combination. All mutually exclusive masses across the full support sum to one.
1Define the count support and parameters2Evaluate the requested mass or cumulative range3Check the trial, replacement, and dependence assumptions
Probability mass rule
P(X₁=x₁,…,Xₘ=xₘ)=n!∏pᵢ^xᵢ/xᵢ!, with Σxᵢ=n and Σpᵢ=1.
A probability mass is a probability at an allowed discrete outcome.
Interpretation and limits
Category counts are dependent because they must sum to the fixed total even when trials are independent.
Model boundary: Trials must be independent with stable mutually exclusive exhaustive category probabilities, and the aligned probability list must total one.
Quick guide
How to use this calculator
- Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
- Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
- Read the result together with its assumptions and interpretation. Statistical output summarizes uncertainty under a model; it does not repair biased data or establish causation.
Calculation method
How the multinomial distribution calculator works
P(X₁=x₁,…,Xₘ=xₘ)=n!∏pᵢ^xᵢ/xᵢ!, with Σxᵢ=n and Σpᵢ=1.
Category counts are dependent because they must sum to the fixed total even when trials are independent.
Worked example
Multinomial Distribution example
Counts 2, 1, 1 with probabilities 0.5, 0.25, 0.25 have joint probability 0.1875.
P(X₁=x₁,…,Xₘ=xₘ)=n!∏pᵢ^xᵢ/xᵢ!, with Σxᵢ=n and Σpᵢ=1.
Supported inputs
Precision and limits
Model and design
Trials must be independent with stable mutually exclusive exhaustive category probabilities, and the aligned probability list must total one.
Numerical scope
Inputs use double-precision numerical methods with guarded domains. Datasets accept up to 10,000 finite plain-decimal values. Extremely large parameters or probabilities deep in a numerical tail may require specialist statistical software.
Interpretation
Category counts are dependent because they must sum to the fixed total even when trials are independent.
Decision boundary
The calculator does not validate how data were collected, diagnose dependence or bias, choose a scientifically meaningful effect, or replace review by a qualified statistician for consequential research, medical, regulatory, safety, or policy decisions.
Privacy
Entered values and calculated results stay in this browser and are not sent to an analytics service.
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