Complement (Probability) — Definition, Formula & Examples

The complement of an event is everything in the sample space that is not part of that event. If an event has probability

P(A)

, its complement has probability

1 - P(A)

Given a sample space $S$ and an event $A \subseteq S$ , the complement of $A$ , denoted $A'$ (or $\bar{A}$ or $A^c$ ), is the set of all outcomes in $S$ that do not belong to $A$ . By the axioms of probability, $P(A) + P(A') = 1$ .

Key Formula

P(A') = 1 - P(A)

Where:

$P(A)$ = Probability that event A occurs
$P(A')$ = Probability that event A does not occur (complement of A)

How It Works

The complement rule is especially useful when calculating the probability of an event directly would be difficult. Instead of finding

P(A)

, you find

P(A')

— the probability the event does not happen — and subtract from 1. For example, "at least one" problems are much easier to solve by first computing the probability of "none" and then taking the complement. The event and its complement are mutually exclusive and together cover every possible outcome, so their probabilities always sum to exactly 1.

Worked Example

Problem: A bag contains 3 red marbles and 7 blue marbles. You draw one marble at random. What is the probability of not drawing a red marble?

Find P(Red): There are 3 red marbles out of 10 total.

P(\text{Red}) = \frac{3}{10} = 0.3

Apply the complement rule: Subtract from 1 to find the probability of not red.

P(\text{Not Red}) = 1 - 0.3 = 0.7

Answer: The probability of not drawing a red marble is

0.7

(or

\frac{7}{10}

Why It Matters

The complement rule simplifies many real problems in AP Statistics and introductory college courses. Questions like "what is the probability that at least one person shares a birthday?" become manageable only by computing the complement. It also appears inside larger formulas such as the addition rule and Bayes' theorem.

Common Mistakes

Mistake: Confusing the complement with the "opposite" in everyday language and applying it to dependent sub-events incorrectly.

Correction: The complement is defined strictly as all outcomes not in the event. For compound events like "at least one success," the complement is "zero successes" — not "at least one failure." Identify the exact set of outcomes before subtracting.