Understand the idea.
Equal labels do not mean equal chances
Equally likely outcomes have the same chance under the selection procedure. A fair die gives each face the same chance. For a well-mixed bag of equal-size counters chosen without looking, each individual counter has the same chance of selection.
Recorded colours can still have unequal chances. Three blue counters and one red counter give four equally selectable objects, but three objects produce blue. The colour list has two labels, not two equally likely choices.
Use the sector size
On an ideal spinner, every direction is equally likely. A colour's probability is the fraction of the full turn covered by that colour. A half-circle therefore has twice the chance of a quarter-circle, even though each is one coloured region.
Count regions only when their angles are equal. For a bag, count equally selectable counters instead. Mixing or choosing without looking makes the procedure random; it does not make unequal colour quantities equal.
See the method in action.
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?
Worked solution
- 1
Count all equally selectable counters.
The well-mixed equal-size counters supply four individual possibilities.
- 2
Find each colour's share of the counters.
Three counters produce blue, while only one produces red.
- 3
Compare the probabilities.
Blue is three times as likely as red, so the two colour outcomes are not equally likely.
No. and .
Read an unequal spinner
Use the spinner shown. Find the chance of blue and compare it with red.
Worked solution
- 1
Read blue's fraction of the full circle.
The blue sector occupies one quarter of all possible spinner directions.
- 2
Express red's chance in quarters.
The red half-circle covers two quarters of the circle.
- 3
Compare the sizes on the same scale.
Red covers twice as many directions as blue, so red is twice as likely.
; red is twice as likely.
Try it yourself.
Check your understanding
A well-mixed bag holds five yellow and five purple equal-size counters. Are the colours equally likely?
Show answer & reasoning
Yes.
There are equally selectable counters. Yellow and purple each account for of them, so their chances are equal.
Keep practising
6 worksheet questions · 12 marks
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 .
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.
.
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 .
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 and B has chance . - 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.
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 ↗