Complementary events | Year 8

01 / LEARN

Understand the idea.

What “not” includes

The complement of an event contains all outcomes that do not meet its condition. An event and its complement cannot occur together, and every possible outcome belongs to one or the other. Be clear about the whole sample space first.

For a die, the complement of rolling 6 includes 1, 2, 3, 4 and 5. For a coloured spinner, 'not blue' includes every other colour, not just one chosen alternative. 'Not' reverses the event's condition across the entire sample space.

Probabilities total one

Because an event and its complement cover all possibilities without overlap, their probabilities add to 1. Thus $P (not A) = 1 - P (A)$ . Use the same whole consistently: 1 for fractions or decimals, and 100% for percentages.

An impossible event has a certain complement, while a certain event has an impossible complement. After subtracting, check that the probability lies between 0 and 1. Equal-outcome counting is valid only when those outcomes really are equally likely.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

Find P(not 6) for a fair die.

Worked solution

1
Find the probability of the excluded result.
$P (6) = \frac{1}{6}$
A fair die has six equally likely faces and exactly one is 6.
2
Subtract it from the whole probability.
$P (not 6) = 1 - \frac{1}{6} = \frac{5}{6}$
Rolling 6 and not rolling 6 are complementary events.
3
Check by counting the other faces.
${1, 2, 3, 4, 5} \Rightarrow \frac{5}{6}$
Five of the six equally likely faces satisfy not 6.

Answer

5/6.

EXAMPLE 02

Worked example 2

A spinner has 10 equal sectors: 3 blue,4 red,3 yellow. Find P(not blue).

Worked solution

1
Identify all sectors outside blue.
$4 + 3 = 7$
Every red and yellow sector belongs to the complement of blue.
2
Divide by the total equal sectors.
$P (not blue) = \frac{7}{10}$
The ten equal sectors are equally likely on the stated spinner model.
3
Express and check the complement.
$\begin{aligned} \frac{7}{10} & = 70 % \\ 1 - \frac{3}{10} & = \frac{7}{10} \end{aligned}$
The non-blue share is the whole minus the three blue sectors.

Answer

7/10=70%.

03 / YOUR TURN

Try it yourself.

Check your understanding

If P(win)=0.28, find P(not win).

Show answer & reasoning

0.72.

Winning and not winning cover all outcomes without overlap. Therefore $P (not win) = 1 - 0.28 = 0.72$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Find $P (not 1)$ for a fair standard die.
2 marks
If $P (rain) = 0.35$ , find $P (no rain)$ .
2 marks
A bag has 4 red,3 blue,5 green tokens. Find $P (not green)$ for a random draw.
2 marks
An event has probability 72%. Find the complement probability.
2 marks
Why are even and odd outcomes complements on a standard die?
3 marks
Find the complement probability of an impossible event.
1 mark

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find $P (not 1)$ for a fair standard die.
2 marks
Five of six equally likely outcomes are not 1.
$\frac{5}{6}$ .
If $P (rain) = 0.35$ , find $P (no rain)$ .
2 marks
$1 - 0.35 = 0.65$ .
0.65.
A bag has 4 red,3 blue,5 green tokens. Find $P (not green)$ for a random draw.
2 marks
There are 12 tokens and 7 are not green.
$\frac{7}{12}$ .
An event has probability 72%. Find the complement probability.
2 marks
$100 % - 72 % = 28 %$ .
28%.
Why are even and odd outcomes complements on a standard die?
3 marks
Every die outcome belongs to exactly one of the two sets.
They do not overlap and together cover all six outcomes.
Find the complement probability of an impossible event.
1 mark
$1 - 0 = 1$ .
1.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Complementary events; two-stage sample spaces; tables, trees and Venn diagrams; probabilities of combined outcomes; experimental versus theoretical results; compound-event simulations and describing results. Formal conditional probability is Year 10.

Official curriculum source ↗

In this lesson

Understand the idea.

What “not” includes

Probabilities total one

See the method in action.

Worked example 1

Find the probability of the excluded result.

Subtract it from the whole probability.

Check by counting the other faces.

Worked example 2

Identify all sectors outside blue.

Divide by the total equal sectors.

Express and check the complement.

Try it yourself.

Check your understanding

Keep practising