The Complement Rule
Simple Explanation
The complement of an event E, written E', is "E does not happen." Since an event either happens or does not (there is no third option), P(E) and P(E') must always add up to 1: P(E') = 1 − P(E).
Why Do We Need It?
Sometimes the complement of an event is much easier to calculate than the event itself — the complement rule lets you solve the easy version and subtract from 1, instead of tackling the hard version directly.
Formula
The Complement Rule
P(E') = 1 − P(E)
The probability that an event does NOT happen equals 1 minus the probability that it does happen.
- P(E')
- — the probability of the complement of E — E not happening
- P(E)
- — the probability of E happening
When to use it: Whenever finding "the probability that something does not happen" is easier than finding the probability that it does.
Worked Example
Use the complement rule
Three coins are tossed. Find the probability of getting at least one head.
Why Does This Work?
An event and its complement together cover every single possible outcome in the sample space exactly once, with no overlap — so their probabilities, which are fractions of the same whole sample space, must add to exactly 1.
Real-Life Example
Quality control "at least one defect" checks
A factory wants the probability that a batch of 10 items has at least one defective item.
Computing "no defects at all" directly is far simpler than checking every possible combination with at least one defect, so the complement rule (1 − P(no defects)) is the practical approach.
Practice
The probability that it rains tomorrow is 0.35. What is the probability it does NOT rain?
MediumCommon mistake
Forgetting that P(E) + P(E') must equal exactly 1 — trying to compute both directly and getting numbers that do not add to 1 signals a calculation error somewhere.
Quick Review
- P(E') = 1 − P(E). An event and its complement always add to 1.
- Useful when the complement (opposite) is easier to calculate directly.
- "At least one" problems are a classic case where the complement ("none at all") is simpler.