---
title: "Complement of an Event — AP Stats Definition & Formula"
description: "The complement of event E is every outcome NOT in E, with P(E') = 1 − P(E). A core AP Stats Unit 4 tool, especially for \"at least one\" probability problems."
canonical: "https://fiveable.me/ap-stats/key-terms/complement-of-an-event"
type: "key-term"
subject: "AP Statistics"
---

# Complement of an Event — AP Stats Definition & Formula

## Definition

In AP Statistics, the complement of an event E (written E' or E^C) is the set of all outcomes in the sample space that are NOT in E, and its probability is P(E') = 1 − P(E), since an event and its complement together cover the entire sample space.

## What It Is

The complement of an event E is everything in the [sample space](/ap-stats/key-terms/sample-space "fv-autolink") that isn't E. If E is "the coin lands heads," then E' (read "E complement" or "not E") is "the coin does not land heads." Because the sample space contains every possible [outcome](/ap-stats/unit-2/estimating-probabilities-using-simulation/study-guide/ABwpnnUf4VCXeVbB9q72 "fv-autolink") of a random process, an event and its complement split it perfectly in two. One of them has to happen, so their probabilities add to exactly 1.

That gives you the formula the CED spells out: **P(E') = 1 − P(E)**. It looks almost too simple, but it's one of the most-used moves in all of [AP Stats](/ap-stats "fv-autolink"). Whenever a probability is hard to compute directly but its opposite is easy, you compute the opposite and subtract from 1. Think of the sample space as a pie. E is one slice, E' is literally everything else, and the whole pie always equals 1.

## Why It Matters

This term lives in **[Unit 4](/ap-stats/unit-4 "fv-autolink"): Probability, Random Variables, and Probability Distributions**, specifically Topic 4.3 (Introduction to Probability). It directly supports learning objective **AP Stats 4.3.A**, which asks you to calculate probabilities for events *and their complements*, and it connects to **AP Stats 4.3.B**, since you also have to interpret what a probability like P(E') = 0.65 means as a long-run relative frequency. The [complement rule](/ap-stats/key-terms/complement-rule "fv-autolink") is also the engine behind "at least one" problems later in Unit 4. Calculating P(at least one success) directly can mean adding up a dozen cases, but P(at least one) = 1 − P(none) turns it into one subtraction. That shortcut shows up constantly with binomial settings and simulation questions.

## Connections

### [Sample Space (Unit 4)](/ap-stats/key-terms/sample-space)

The [complement](/ap-stats/key-terms/complement "fv-autolink") only makes sense relative to a sample space. E' is defined as "everything in the sample space that isn't E," so if you misidentify the sample space, you misidentify the complement too.

### Probability (Unit 4)

The complement rule is a direct consequence of two basic facts from [Topic 4.3](/ap-stats/unit-4/justifying-claim-about-population-mean-based-on-confidence-interval/study-guide/3bD1Y9YSWMgUnARuJxxm "fv-autolink"). Probabilities live between 0 and 1, and the whole sample space has probability 1, so whatever E doesn't claim, E' gets.

### Event (Unit 4)

A complement is itself an event, just defined by negation. Anything you can do with an event (find its [probability](/ap-stats/unit-2/intro-probability/study-guide/gfnBWfyMANOxF3vWLrbA "fv-autolink"), interpret it as a long-run frequency) you can do with its complement.

### Binomial "at least one" problems (Unit 4)

Later in Unit 4, questions like "what's the probability at least one of 10 trials is a success?" are brutal to compute directly. The complement turns them into 1 − P(zero successes), a single calculation. This is the most common place AP students actually use the complement rule under time pressure.

## On the AP Exam

Complement questions show up most often as quick multiple-choice calculations. A stem gives you P(A) = 0.7 and asks for the probability of the complement (answer: 0.3), or hands you a probability of heads like 0.35 and asks for the probability of NOT heads (0.65). Some questions flip it around and test recognition instead of computation, like asking what a student was finding when they calculated 1 − 0.37. You need to recognize that "1 minus a probability" means a complement. On FRQs, the complement rule rarely gets named directly, but it powers "at least one" calculations in probability and binomial parts. When you use it, write the rule explicitly, like P(at least one) = 1 − P(none), since clear notation earns communication credit.

## Complement of an event vs Mutually exclusive (disjoint) events

Complements are always mutually exclusive, but mutually exclusive events are not always complements. E and E' can't happen together AND one of them must happen, so their probabilities sum to exactly 1. Two disjoint events like "roll a 1" and "roll a 2" also can't happen together, but they don't cover the whole sample space, so their probabilities sum to less than 1. Complement means "opposite and exhaustive," not just "can't both happen."

## Key Takeaways

- The complement of event E, written E' or E^C, is the set of all outcomes in the sample space that are not in E.
- The complement rule says P(E') = 1 − P(E), because an event and its complement together account for the entire sample space.
- An event and its complement are always mutually exclusive and exhaustive, so their probabilities always sum to exactly 1.
- Use the complement rule whenever the direct probability is hard but the opposite is easy, especially for "at least one" questions where P(at least one) = 1 − P(none).
- Like any probability, P(E') is interpreted as a long-run relative frequency, so P(E') = 0.65 means the event fails to occur about 65% of the time over many repetitions.

## FAQs

### What is the complement of an event in AP Stats?

The complement of an event E is every outcome in the sample space that is not in E, written E' or E^C. Its probability is P(E') = 1 − P(E), which is tested under learning objective AP Stats 4.3.A in Topic 4.3.

### If P(A) = 0.7, what is the probability of the complement of A?

P(A') = 1 − 0.7 = 0.3. An event and its complement always add to exactly 1, so once you know one probability, you immediately know the other.

### Is the complement of an event the same as a mutually exclusive event?

Not quite. Complements are always mutually exclusive, but they go further, since E and E' must cover the entire sample space. Two disjoint events like "roll a 1" and "roll a 2" can't both happen, yet they aren't complements because other outcomes (3 through 6) exist outside both.

### Why does P(E) + P(E') always equal 1?

Because every outcome in the sample space is either in E or in E', never both and never neither. Since the total probability of the sample space is 1, the two events must split that 1 between them.

### When should I use the complement rule on the AP Stats exam?

Whenever computing the opposite is easier than computing the event itself, most famously in "at least one" problems. P(at least one success) = 1 − P(no successes) replaces a long sum of cases with a single subtraction, which is a huge time-saver on binomial questions.

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