The Complement Rule in Probability: KS3 Maths Guide

The complement of an event A is everything that is not A, written A′. Because every possible outcome must belong to either A or its complement, their probabilities always sum to 1. So P(A′) = 1 − P(A) — and this simple identity often turns a long list of outcomes into a single subtraction.

What is the complement of an event?

The complement of event A, written A′ (or sometimes Ā), is the event that A does NOT occur. Every outcome in the sample space is either in A or in A′ — never both, and never neither.

Example: Roll a standard six-sided die.

• A = rolling a 6. P(A) = 1/6.
• A′ = not rolling a 6 = rolling 1, 2, 3, 4 or 5. P(A′) = 5/6.
• P(A) + P(A′) = 1/6 + 5/6 = 1 ✓

What is the complement rule?

P(A′) = 1 − P(A)

This follows directly from the fact that all probabilities in a complete sample space sum to 1. Since A and A′ cover every possible outcome without overlap:

P(A) + P(A′) = 1 → P(A′) = 1 − P(A)

The rule works for any event in any probability scenario, whether outcomes are equally likely or not.

When is the complement rule useful?

The complement rule is most powerful when it is easier to find P(A) than to list all the outcomes of A′ directly.

Worked example 1: A bag contains 3 red, 4 blue, 2 green and 1 yellow marble. What is P(not blue)?

Method A (direct): P(not blue) = (3 + 2 + 1)/10 = 6/10 = 3/5 (adding up three separate groups)

Method B (complement): P(blue) = 4/10 = 2/5; P(not blue) = 1 − 2/5 = 3/5 ✓

The complement method needs only one probability and one subtraction.

Worked example 2: P(it rains tomorrow) = 0.35. What is P(it does not rain tomorrow)?

P(no rain) = 1 − 0.35 = 0.65

How do you use the complement for "at least one" problems?

"At least one" means one or more, which often has many possible outcomes. The complement is "none at all" — usually just one outcome.

Worked example 3: A coin is flipped 3 times. What is the probability of getting at least one head?

Direct method: count HHH, HHT, HTH, HTT, THH, THT, TTH — that is 7 outcomes out of 8. P = 7/8.

Complement method: P(at least one head) = 1 − P(no heads) = 1 − P(TTT) = 1 − (1/2)³ = 1 − 1/8 = 7/8 ✓

The complement method is much quicker when the "none" outcome is a single simple calculation.

How does the complement rule appear in tables and Venn diagrams?

In a two-way table or Venn diagram, all entries must sum to the total. So:

If P(A) = 0.42, then P(A′) = 0.58.

Table example:

In a Venn diagram with a universal set ξ, the complement A′ is the area inside ξ but outside circle A. Whatever shading or numbers are inside A, the rest of the universal set is A′.

What mistakes should you avoid?

Mistake 1 — Calculating P(A) + P(A′) as something other than 1. If your two probabilities do not add to exactly 1, at least one is wrong. Recheck the calculation before proceeding.

Mistake 2 — Confusing "not A" with "A is impossible". If P(A) = 0, then A is impossible and P(A′) = 1 (the event certainly will not happen). P(A) = 0 and P(A′) = 1 are both valid probabilities; they just describe impossible and certain events.

Mistake 3 — Using the complement when "at least one" is not in the question. The complement shortcut is most valuable for "at least one" questions and for events whose complement has fewer outcomes. Not every probability question needs the complement — sometimes direct calculation is simpler.

Frequently asked questions

Is the complement rule the same as saying the event and its opposite sum to 100%?

Yes — expressing probabilities as percentages instead of fractions, P(A) + P(A′) = 100%. The complement rule is simply the statement that "something" and "its opposite" together account for every possible outcome.

Can a probability question use the complement rule with fractions and decimals?

The complement rule works with probabilities in any form — fractions, decimals or percentages — as long as they are all in the same form when you add them. P(A) = 3/4 gives P(A′) = 1 − 3/4 = 1/4. P(A) = 0.7 gives P(A′) = 1 − 0.7 = 0.3.

What if there are more than two possible outcomes?

The complement rule still works. For a die: P(rolling a 1 or a 2) = 2/6 = 1/3. Its complement is P(not rolling a 1 or 2) = P(rolling 3, 4, 5 or 6) = 4/6 = 2/3. Check: 1/3 + 2/3 = 1 ✓. The complement of any event is all the outcomes NOT in that event.

How does the complement relate to mutually exclusive events?

An event and its complement are always mutually exclusive (they cannot both occur at the same time) and exhaustive (together they cover every possible outcome). This is the defining combination that forces P(A) + P(A′) = 1. Two mutually exclusive events that are not exhaustive would sum to less than 1 and would not be complements of each other.

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