Total Probability Theorem Calculator

The Law of Total Probability states that if {B₁, B₂, ..., Bₙ} is a partition of the sample space (mutually exclusive and collectively exhaustive events), then for any event A: P(A) = P(A|B₁)·P(B₁) + P(A|B₂)·P(B₂) + ... + P(A|Bₙ)·P(Bₙ). It essentially decomposes the probability of A into weighted conditional probabilities across every possible scenario.

A set of events {B₁, B₂, ..., Bₙ} forms a partition of the sample space when two conditions are met: (1) the events are mutually exclusive — no two can occur simultaneously — and (2) they are collectively exhaustive — one of them must always occur. This means P(B₁) + P(B₂) + ... + P(Bₙ) = 1. The calculator checks this sum and flags it if your inputs don't add up to 1.

Select the number of partition events you have, then enter P(Bᵢ) — the prior probability for each partition — and P(A|Bᵢ) — the conditional probability of your event A given each Bᵢ. Make sure your P(Bᵢ) values sum to 1. The calculator instantly applies the formula and shows the total probability P(A), each weighted term, and a bar chart of contributions.

Conditional probability P(A|B) gives the probability of event A given that a specific event B has already occurred. Total probability, on the other hand, combines multiple conditional probabilities across all possible scenarios (partition events Bᵢ) to compute the overall, unconditional probability P(A). Think of total probability as a weighted average of the conditional probabilities.

The Law of Total Probability requires the partition events to be exhaustive, so their probabilities must sum to exactly 1. If your values don't sum to 1, the calculated P(A) will be mathematically incorrect. The calculator displays the sum of your P(Bᵢ) inputs as a secondary output so you can verify this before trusting the result.

Bayes' Theorem and the Law of Total Probability are closely linked. Bayes' Theorem uses the total probability P(A) — computed from the Law of Total Probability — as the denominator to update beliefs: P(Bᵢ|A) = P(A|Bᵢ)·P(Bᵢ) / P(A). In other words, you typically need the total probability calculation as a prerequisite for applying Bayes' Theorem.

Yes. This calculator supports up to 6 partition events. Simply select the desired number from the dropdown and fill in the prior and conditional probabilities for each. The formula generalises naturally: P(A) = Σ P(A|Bᵢ)·P(Bᵢ) for all i from 1 to n, regardless of how many partitions you have.

Suppose a factory has three machines (B₁, B₂, B₃) that produce 50%, 30%, and 20% of total output respectively. Each machine has a known defect rate: 2%, 5%, and 10%. The probability that a randomly chosen item is defective is P(Defective) = 0.02×0.5 + 0.05×0.3 + 0.10×0.2 = 0.01 + 0.015 + 0.02 = 0.045, or 4.5%. This is a classic application of the total probability theorem.