Equally likely versus unequal outcomes | Year 7

01 / LEARN

The idea, step by step

Equal labels do not mean equal chances

Outcomes are equally likely when the chance process gives them the same chance. The six numbers on a fair die are equally likely. In a well-mixed bag of four same-size counters, each counter may be equally likely even when the recorded colours are not.

With three blue counters and one red counter, there are three opportunities for blue and one for red. The recorded colour sample space is ${B, R}$ , but its two outcomes are not equally likely. Count the counters before assigning probability fractions.

Use the sector size

For an ideal spinner, every direction is equally likely. A colour’s chance is the fraction of the circle it occupies. A half-circle has twice the chance of a quarter-circle. Count sectors only when they really have equal angles.

A fair selection procedure does not necessarily give all colours equal chances. Mixing and choosing without looking avoids deliberate selection, but it does not change the numbers of the different colours.

A common trap

A spinner has three colours, so each colour must have probability one third.

Instead: Use the fraction of the circle occupied by each colour. The colour names need not have equal chances.

02 / WORKED EXAMPLES

See the thinking

Example 1

Two colours, unequal chances

A well-mixed bag holds three blue and one red equal-size counters. One is chosen at random. Are the colour outcomes equally likely?

There are four counters altogether.
Three give blue, while one gives red.
The blue chance is three times the red chance.

No.

P (B) = \frac{3}{4}

and

P (R) = \frac{1}{4}

Example 2

Read an unequal spinner

Use the spinner shown. Find the chance of blue and compare it with red.

Blue occupies one quarter of the circle.
Red occupies one half of the circle.
A half is twice a quarter.

P (blue) = \frac{1}{4}

; red is twice as likely.

Check your understanding

A well-mixed bag holds five yellow and five purple equal-size counters. Are the colours equally likely?

Show answer

Yes.

Each colour occupies five of the ten equally selectable counters, so each chance is $1 / 2$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

A fair six-sided die is rolled. Are its face numbers equally likely? State the assumption.
2 marks
A counter is chosen at random from a bag with two red and six blue equal-size counters. Which colour is more likely, and how many times as likely?
2 marks
An ideal spinner is half orange and half purple. Are the colours equally likely? Explain.
2 marks
Use the spinner shown. Which two colours have equal chances?
2 marks
Four equal spinner sectors are labelled A, A, A and B. Are A and B equally likely? Explain.
2 marks
A student looks into a bag and chooses a favourite colour. Can you use the equal-counter random-selection model? Explain.
2 marks

Read the worked solutions

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

A fair six-sided die is rolled. Are its face numbers equally likely? State the assumption.
2 marks
Fair means no face is favoured.
Each face then has chance $1 / 6$ .
Yes, because the die is fair.
A counter is chosen at random from a bag with two red and six blue equal-size counters. Which colour is more likely, and how many times as likely?
2 marks
Compare six blue opportunities with two red opportunities.
$6 \div 2 = 3$ .
Blue is three times as likely.
An ideal spinner is half orange and half purple. Are the colours equally likely? Explain.
2 marks
The colours occupy equal half-circles.
Each colour has chance $1 / 2$ .
Yes.
Use the spinner shown. Which two colours have equal chances?
2 marks
Each colour occupies a quarter of the circle.
The sectors have the same angle.
Blue and green.
Four equal spinner sectors are labelled A, A, A and B. Are A and B equally likely? Explain.
2 marks
Each sector is equally likely.
Three sectors give A and one gives B.
No. A has chance $3 / 4$ and B has chance $1 / 4$ .
A student looks into a bag and chooses a favourite colour. Can you use the equal-counter random-selection model? Explain.
2 marks
Deliberately choosing can favour particular counters.
Mix and choose without looking before using that model.
No.

Checking saved progress…

Continue learning

Theoretical probability

Next lesson

Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 7 single-stage sample spaces, equally likely outcomes, probabilities, expected frequencies, chance experiments, simulations and relative frequencies.

Official curriculum source ↗

Back to Probability