What is the addition rule for probability?

The addition rule states P(A ∪ B) = P(A) + P(B) − P(A∩B). The intersection term P(A∩B) is subtracted to avoid double-counting outcomes where both A and B occur simultaneously.

What does "mutually exclusive" mean?

Two events are mutually exclusive (disjoint) when they cannot both occur at the same time. Because P(A∩B) = 0, the addition rule simplifies to P(A ∪ B) = P(A) + P(B). For example, rolling a 1 and rolling a 2 on a single die roll are mutually exclusive.

What does "independent" mean for the union formula?

Independent events are events where the occurrence of one does not affect the probability of the other. Their joint probability is P(A∩B) = P(A)·P(B), which is then substituted into the addition rule: P(A ∪ B) = P(A) + P(B) − P(A)·P(B).

Can P(A ∪ B) exceed 1?

No. P(A ∪ B) is a probability and must be between 0 and 1 inclusive. If valid inputs cause the formula to exceed 1, the inputs are inconsistent — for example, P(A∩B) is too small relative to P(A) and P(B).

What is P(A∩B) and how do I find it?

P(A∩B) is the probability that both A and B occur. If the events are independent, use P(A)·P(B). If they are mutually exclusive, it is 0. Otherwise, P(A∩B) must be known from context or data — for instance, from a frequency table or a joint probability distribution.

Probability Union Calculator

Name: Probability Union Calculator
Availability: InStock
Author: Nham Vu

Calculate P(A ∪ B) using the addition rule. Supports independent events, mutually exclusive events, and custom P(A∩B) input.

Formula

P(A ∪ B) = P(A) + P(B)

P(A) — Probability of event A

P(B) — Probability of event B

Venn Diagram

Enter values and calculate to see the diagram

Summary

Calculate P(A ∪ B) using the addition rule. Supports independent events, mutually exclusive events, and custom P(A∩B) input.

How it works

Enter P(A) and P(B) as decimals between 0 and 1, then pick the relationship between the events.
For mutually exclusive events the intersection is 0, so the union equals P(A) + P(B). For independent events the intersection is P(A)·P(B). In custom mode, supply P(A∩B) directly.
The tool validates each input and checks that P(A∩B) does not exceed P(A) or P(B).
Read the union probability, the intersection used, and the complement P((A∪B)'), with a Venn diagram showing the overlap.

Use cases

Find the probability that at least one of two independent tests passes.
Calculate the chance that a customer buys product A or product B when purchases are mutually exclusive.
Determine the probability of at least one failure across two independent system components.
Combine probabilities from overlapping survey responses where some respondents selected both options.
Verify that a union probability stays within [0, 1] when designing probability models.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu