Complementary Events

Statistics • Probability

Written by STEM Calculators Team Published December 12, 2025 Updated February 24, 2026

Complementary Events

Compute the probability of an event A and its complement A (read “not A”). Complementary events are mutually exclusive and together cover all outcomes, so: P(A) + P(A) = 1.

Example

Use Fill example to populate the inputs without calculating.

Input mode

Event label (optional)

The complement will be “not (A)”.

Total outcomes (n)

Favorable outcomes for A (f)

Results

P(A): — P(A): — Check: —

P(A) + P(A) = 1 ⇒ P(A) = 1 − P(A)

Venn diagram

The rectangle is the sample space S. The circle is A. Everything outside the circle is A. The bar under the diagram shows P(A) vs P(A).

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Complementary Events

Definition

The complement of an event A, written A and read “not A,” is the event that includes all outcomes in the sample space that are not in A.

Key property

Events A and A are complementary, so they are mutually exclusive and together cover the entire sample space. Therefore:

P(A) + P(A) = 1

From this, the two practical formulas follow immediately:

P(A) = 1 − P(A) and P(A) = 1 − P(A)

Essential example

In a group of 2000 taxpayers, 400 have been audited at least once. Let A = “selected taxpayer has been audited at least once.”

P(A) = 400 / 2000 = 0.20 ⇒ P(A) = 1 − 0.20 = 0.80

Here A means “has never been audited.”

Frequently Asked Questions

What is a complementary event in probability?

The complement of an event A is the event that A does not occur. It is often written as A complement or not A.

How do you calculate the probability of not happening?

Use the complement rule: P(not A) = 1 - P(A). This works because an event either happens or it does not.

When should I use the complement rule instead of counting outcomes?

Use the complement rule when it is easier to compute P(A) than to directly compute P(not A), or when a statement is naturally phrased as not happening.

Are complementary events the same as mutually exclusive events?

No. Complementary events cover all possible outcomes together, so P(A) + P(not A) = 1. Mutually exclusive events cannot occur at the same time but may not cover all outcomes.

Complementary Events

Results

Venn diagram

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Frequently Asked Questions

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