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Statistics · Distributions

Poisson Binomial Distribution Calculator

Calculate exact and cumulative success-count probabilities when independent Bernoulli trials have different success probabilities.

Statistics · Distributions

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Private in-browser calculation · explicit assumptions
Discrete support uses separate whole-number outcomes; probability is carried by the individual masses.
0support10
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
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Enter plain numbers without measurement units. Datasets accept commas, spaces, semicolons, or line breaks and are limited to 10,000 values. Results stay in this browser.

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Understand the distribution

What the poisson binomial 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.

Probability mass rule

The Poisson-binomial mass is the coefficient of z^k in ∏[(1−pᵢ)+pᵢz].

A probability mass is a probability at an allowed discrete outcome.

Interpretation and limits

The model preserves unequal trial probabilities instead of replacing them with a single average probability.

Model boundary: Trials must be independent and each listed probability must correspond to one Bernoulli trial; dependence requires a different model.

Quick guide

How to use this calculator

  1. Enter the observations, probabilities, model parameters, or summary statistics requested by the visible labels.
  2. Keep every value on the same scale and confirm that the selected sampling relationship, distribution, and tail convention match the question you are investigating.
  3. 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 poisson binomial distribution calculator works

The Poisson-binomial mass is the coefficient of z^k in ∏[(1−pᵢ)+pᵢz].

The model preserves unequal trial probabilities instead of replacing them with a single average probability.

Worked example

Poisson Binomial Distribution example

Probabilities 0.2, 0.5, and 0.8 give P(exactly 2 successes)=0.42.

The Poisson-binomial mass is the coefficient of z^k in ∏[(1−pᵢ)+pᵢz].

Supported inputs

Precision and limits

Model and design

Trials must be independent and each listed probability must correspond to one Bernoulli trial; dependence requires a different model.

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

The model preserves unequal trial probabilities instead of replacing them with a single average probability.

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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