Year 10: Complementary events

Complementary events are a powerful shortcut in Year 10 probability. Instead of counting the event you want directly, it can sometimes be easier to count the opposite event first and subtract from 1.

If event

A

is the event you want, then

\Pr (A) = 1 - \Pr (not A)

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

What is a complement?

The complement of an event is everything in the sample space that is not in the event. If the event is “roll an even number”, the complement is “roll an odd number”.

Example 1: On a fair die, let event

A

be “roll a number greater than 4”.

$A = {5, 6}$ .
The complement is $not A = {1, 2, 3, 4}$ .

Skill quiz: Identify the opposite event

Use sample-space and event language to describe all outcomes outside an event.

Sample spaces and event language

Why complements are useful

Some events are awkward to count directly, but their complements are easy. This happens often with phrases like “not”, “at least one”, or “not all”.

Example 2: A bag contains 7 red counters and 3 blue counters. One counter is chosen at random. Find the probability that the counter is not blue.

$\Pr (blue) = 3 / 10$ .
So $\Pr (not blue) = 1 - 3 / 10 = 7 / 10$ .

Skill quiz: Calculate the probability of not happening

Practise simple probabilities and use one minus the event probability to check its complement.

Probability of simple events

Checking with direct counting

Complement methods should match direct counting when both are possible. That is a good Year 10 check if the numbers are small.

Example 3: For the same bag, direct counting gives

\Pr (not blue) = \Pr (red) = 7 / 10

, which matches the complement method.

Skill quiz: Compare complement and direct methods

For each suitable question, count the event directly and compare it with a complement calculation.

Probability of simple events

Complement in a compound-event setting

Complementary events also help in multi-step questions. Even if the full event is compound, the “not” version may be simpler to count.

Example 4: If a coin is tossed twice, what is the probability of getting at least one head?

The complement is “no heads”, which means $TT$ .
$\Pr (TT) = 1 / 4$ .
So $\Pr (at least one head) = 1 - 1 / 4 = 3 / 4$ .

Skill quiz: Use complements with combined outcomes

Describe the opposite of an at-least-one event and use it to simplify a compound probability calculation.

Probability of compound events

Common mistakes

Subtracting from the wrong total instead of subtracting from 1.
Using a complement when the opposite event has not been described correctly.
Forgetting that complement works on probabilities, not directly on raw counts unless the totals are handled correctly.
Mixing up “at least one” with “exactly one”.

Skill quiz: Check the opposite event

Distinguish at least one from exactly one before counting the compound outcomes.

Probability of compound events

Study routine

If a question includes the words “not”, “at least one”, or “none”, check whether a complement method is faster. Write the opposite event in words first so you know exactly what you are subtracting from 1.

Skill quizzes: Choose a useful complement

Describe the opposite event first, then choose a direct or complement calculation.