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Statistics · Probability

Law of Total Probability Calculator

Combine an exhaustive list of case probabilities and case-specific conditional probabilities into an auditable total event probability.

Statistics · Probability

Enter your statistical inputs

Private in-browser calculation · explicit assumptions
Follow each branch by multiplying conditional probabilities; add mutually exclusive terminal paths when the question combines them.
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example

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.

Statistics result

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Understand the probability

What the law of total probability calculator is calculating

Following conditional branches. Conditional probability changes the reference group. A branch probability must be read inside the branch condition, while Bayes’ theorem reverses the conditioning direction using prior probabilities.

Probability rule

For exhaustive disjoint cases Aᵢ, P(B)=ΣP(Aᵢ)P(B|Aᵢ).

Every probability must remain between 0 and 1, and overlapping regions must be jointly feasible. The calculator rejects combinations that violate the stated model.

How to interpret it

Each path contribution shows how prevalence and conditional likelihood jointly determine the overall event rate.

Model boundary: Case probabilities must be nonnegative, align with conditionals, total one, and form mutually exclusive exhaustive cases.

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 law of total probability calculator works

For exhaustive disjoint cases Aᵢ, P(B)=ΣP(Aᵢ)P(B|Aᵢ).

Each path contribution shows how prevalence and conditional likelihood jointly determine the overall event rate.

Worked example

Law of Total Probability example

Cases with priors 0.4 and 0.6 and conditional probabilities 0.2 and 0.5 give P(B)=0.38.

For exhaustive disjoint cases Aᵢ, P(B)=ΣP(Aᵢ)P(B|Aᵢ).

Supported inputs

Precision and limits

Model and design

Case probabilities must be nonnegative, align with conditionals, total one, and form mutually exclusive exhaustive cases.

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

Each path contribution shows how prevalence and conditional likelihood jointly determine the overall event rate.

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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