Understand the idea.
Equal chances do not force equal counts
Variation means that repeated experiments can give different results. A fair coin gives heads and tails equal chances on each toss, but it does not promise equal counts in every group of ten tosses.
In the table, all groups used ten tosses, yet their head counts are 4, 7 and 5. These are different observations from the same fair procedure; none must be changed to make all the groups agree.
| Group | Heads | Tails |
|---|---|---|
| A | 4 | 6 |
| B | 7 | 3 |
| C | 5 | 5 |
Compare fairly and explain cautiously
Compare batches with the same number of trials, and state both the outcome counts and trial totals. Eight heads has a different meaning in ten tosses than in twenty tosses because there were different numbers of opportunities.
A short uneven batch does not by itself prove that a coin is unfair. More observations and a check of the method can help investigate. Past counts also cannot guarantee the exact counts in a future batch.
See the method in action.
Compare equal batches
Using the supplied table, which group recorded the most heads?
| Group | Heads | Tails |
|---|---|---|
| A | 4 | 6 |
| B | 7 | 3 |
| C | 5 | 5 |
Worked solution
- 1
Read the heads counts for all three groups.
All counts come from batches of ten tosses, so the batch sizes match.
- 2
Compare the counts.
Seven is the largest number of heads recorded.
- 3
Name the group beside seven.
Group B recorded seven heads, the most in the supplied table.
Group B recorded the most heads.
A claim about the future
Group C recorded five heads and five tails. Does that guarantee its next ten tosses will also split five and five?
Worked solution
- 1
Identify what Group C’s record tells us.
The record describes the completed batch only.
- 2
Consider the next batch’s tosses.
Each new fair-coin toss has the same chances, regardless of the previous batch’s counts.
- 3
Reject the guarantee.
Different sequences of heads and tails are possible, so the next ten tosses can have different totals.
No. The next batch can differ.
Try it yourself.
Check your understanding
Can two groups using the same fair coin rule get different results?
Show answer & reasoning
Yes.
Yes. Each group records its own chance outcomes, so the frequencies can differ even when both groups use the same fair procedure and the same number of tosses.
Keep practising
6 worksheet questions · 9 marks
Read the questions online
- Using the supplied table, how many more heads did B record than A?1 mark
Practice data: three groups each toss a fair coin ten times Group Heads Tails A 4 6 B 7 3 C 5 5 - Check the total trials for Group C in the table.1 mark
Practice data: three groups each toss a fair coin ten times Group Heads Tails A 4 6 B 7 3 C 5 5 - Group D records nine heads in ten tosses. Is ten heads impossible in another batch?1 mark
- A bag trial gives red 8 times in 20 draws; another gives red 12 times in 20 draws. Describe the difference.2 marks
- Why is comparing “6 heads from 10 tosses” with “8 heads from 20 tosses” using the head counts alone unfair?2 marks
- A student says a fair coin is broken because ten tosses gave 6 heads and 4 tails. Is this enough evidence? Explain.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Using the supplied table, how many more heads did B record than A?1 mark
Practice data: three groups each toss a fair coin ten times Group Heads Tails A 4 6 B 7 3 C 5 5 .
3. - Check the total trials for Group C in the table.1 mark
Practice data: three groups each toss a fair coin ten times Group Heads Tails A 4 6 B 7 3 C 5 5 .
10. - Group D records nine heads in ten tosses. Is ten heads impossible in another batch?1 mark
Every toss can land heads even though that batch is unusual.
No. Ten heads is possible. - A bag trial gives red 8 times in 20 draws; another gives red 12 times in 20 draws. Describe the difference.2 marks
; the counts vary between batches.
The second batch has 4 more red outcomes. - Why is comparing “6 heads from 10 tosses” with “8 heads from 20 tosses” using the head counts alone unfair?2 marks
More trials give more opportunities for heads; compare like-sized batches or state both totals.
The groups used different trial totals. - A student says a fair coin is broken because ten tosses gave 6 heads and 4 tails. Is this enough evidence? Explain.2 marks
Repeat the experiment and examine the method rather than expecting every short batch to be equal.
No. That result can occur by chance with a fair coin.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum Mathematics 2026
Ordering likelihood; independent/dependent events; repeated trials; comparing variation.
Official curriculum source ↗