The Complement of an Event

The Complement of an Event — Moosa Academy

The complement of an event $A$ , written $A'$ , is the event that $A$ does not happen. Since $A$ either happens or it doesn't, their probabilities always add to 1.

Complement One minus the probability

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

Either $A$ happens or it doesn't — there is no third case — so the whole bar equals 1 and $P(A) + P(A') = 1$ .

Strategy When to use the complement

Reach for the complement when the question contains the words "not" or "no" — especially when the unwanted outcomes are fewer and easier to count than the wanted ones.

Step 1 — identify the original event

A

Step 2 — find

P(A)

Step 3 — subtract:

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

Step 4 — check by counting the opposite outcomes directly.

Example Three worked cases

8 red and 12 blue balls — not drawing red

P(\text{red}) = \tfrac{8}{20} = \tfrac{2}{5}

, so

P(A') = 1 - \tfrac{2}{5}

⟹ P(not red) = 3/5 = 60%.

Two dice — the sum is not 7

There are 6 ways to make 7, so

P(7) = \tfrac{6}{36} = \tfrac{1}{6}

P(A') = 1 - \tfrac{1}{6}

⟹ P(not 7) = 5/6 ≈ 83.3%.

6 red + 4 blue + 5 green — not drawing blue

P(\text{blue}) = \tfrac{4}{15}

, so

P(A') = 1 - \tfrac{4}{15}

Check:

6 + 5 = 11

balls are not blue.

⟹ P(not blue) = 11/15 ≈ 73.3%.

Note Properties of the complement

The complement of the complement is the original:

(A')' = A

The complement of a certain event (probability 1) is impossible (probability 0).

The complement of an impossible event (probability 0) is certain (probability 1).

Reference Complements on a die

Event A	P(A)	Complement A'	P(A')
Rolling a 3	1/6	not a 3	5/6
An even number	1/2	an odd number	1/2
Greater than 4	1/3	4 or less	2/3
A multiple of 3	1/3	not a multiple of 3	2/3

Summary

The complement $A'$ is the event that $A$ does not happen.
Rule: $P(A') = 1 - P(A)$ ; the two always add to 1.
Use the complement when a question says "not" or "no", especially when the unwanted outcomes are fewer.
The complement of a complement is the original: $(A')' = A$ .