Larger simulations and variation | Year 6

01 / LEARN

Understand the idea.

A simulation copies a chance rule

A simulation uses a random process to copy a chance rule. To model a fair coin with a fair die, assign faces $1, 2, 3$ to heads and $4, 5, 6$ to tails. Each group then contains three equally likely faces, matching a probability of $\frac{1}{2}$ .

Keep that rule for every trial and record each result. Equally likely random whole numbers from $1$ to $6$ can replace the die. Choosing an alternating heads-tails pattern yourself would force balance and would fail to model random variation.

Compare proportions, not raw differences

Compare relative frequencies, because trial totals differ. The supplied heads proportions are $70 %$ , $54 %$ and $52 %$ . Compare each with the fair-coin prediction of $50 %$ , using the distance in percentage points rather than just the number of extra heads.

Over many repetitions, larger experiments tend to show more stable proportions. This does not guarantee that every larger result lies closer to $50 %$ . A result may move towards or away from the prediction as new random trials are added.

Supplied simulated fair-coin results
Trials	Heads	Tails
20	14	6
100	54	46
500	260	240

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Compare two sample sizes

Using the supplied results, compare 20 trials with 500 trials.

Supplied simulated fair-coin results
Trials	Heads	Tails
20	14	6
100	54	46
500	260	240

Worked solution

1
Find the heads percentage for twenty trials.
$\frac{14}{20} = \frac{70}{100} = 70 %$
Divide the observed heads count by that experiment's own trial total.
2
Find the heads percentage for five hundred trials.
$\frac{260}{500} = \frac{52}{100} = 52 %$
Using a proportion allows a fair comparison with the much smaller experiment.
3
Measure each distance from fifty percent.
$\begin{aligned} 70 - 50 & = 20 \\ 52 - 50 & = 2 \end{aligned}$
The distances are twenty and two percentage points above the fair-coin prediction.
4
Compare the distances.
$2 < 20$
The five-hundred-trial result is closer to fifty percent in these supplied results.

Answer

The 500-trial result is closer to 50% in this supplied example.

EXAMPLE 02

A larger result can move away

Experiment A gives 13 heads in 25 tosses. Experiment B gives 57 heads in 100 tosses. Which is closer to 50%?

Worked solution

1
Find Experiment A's heads percentage.
$\frac{13}{25} = \frac{52}{100} = 52 %$
Multiplying both parts by four expresses the twenty-five-trial result per hundred.
2
Find Experiment B's heads percentage.
$\frac{57}{100} = 57 %$
With one hundred trials, the heads count directly gives the percentage.
3
Calculate the distances from fifty percent.
$\begin{aligned} A: 52 - 50 & = 2 \\ B: 57 - 50 & = 7 \end{aligned}$
A differs by two percentage points, while B differs by seven.
4
Choose the smaller distance.
$2 < 7$
A is closer even though it has fewer trials; larger sample size does not guarantee a closer individual result.

Answer

A is closer, although it has fewer trials. Larger does not mean guaranteed closer.

03 / YOUR TURN

Try it yourself.

Check your understanding

To simulate a fair coin using a fair die, is using only face 6 for heads and faces 1 to 5 for tails suitable?

Show answer & reasoning

No.

One face for heads gives probability $\frac{1}{6}$ , while five faces for tails gives $\frac{5}{6}$ . These are unequal, so the rule cannot model a fair coin's two equal chances of $\frac{1}{2}$ .

Keep practising

6 worksheet questions · 11 marks

Open worksheet Worked solutions

Read the questions online

For the supplied 100-trial simulation, check that heads and tails add to the trial count.
1 mark
Supplied simulated fair-coin results
Trials Heads Tails
20 14 6
100 54 46
500 260 240
Find the heads percentage in the 500-trial simulation.
2 marks
Supplied simulated fair-coin results
Trials Heads Tails
20 14 6
100 54 46
500 260 240
Which supplied simulation has a heads relative frequency closest to $50 %$ ?
2 marks
Supplied simulated fair-coin results
Trials Heads Tails
20 14 6
100 54 46
500 260 240
A simulation cycles heads, tails, heads, tails without any random choice. Why is that unsuitable for demonstrating random variation?
2 marks
Suggest a fair-die rule to simulate an event with probability $\frac{1}{3}$ .
2 marks
A larger experiment records $51 %$ successes after a smaller experiment recorded $50 %$ . Does this contradict the idea that larger experiments tend to be more stable?
2 marks

Read the worked solutions

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

For the supplied 100-trial simulation, check that heads and tails add to the trial count.
1 mark
Supplied simulated fair-coin results
Trials Heads Tails
20 14 6
100 54 46
500 260 240
The two outcomes account for every trial.
They do: $54 + 46 = 100$ .
Find the heads percentage in the 500-trial simulation.
2 marks
Supplied simulated fair-coin results
Trials Heads Tails
20 14 6
100 54 46
500 260 240
$\frac{260}{500} = \frac{52}{100} = 52 %$ .
$52 %$ .
Which supplied simulation has a heads relative frequency closest to $50 %$ ?
2 marks
Supplied simulated fair-coin results
Trials Heads Tails
20 14 6
100 54 46
500 260 240
Their heads percentages are $70 %$ , $54 %$ and $52 %$ .
Their distances from $50 %$ are 20, 4 and 2 percentage points.
The 500-trial simulation.
A simulation cycles heads, tails, heads, tails without any random choice. Why is that unsuitable for demonstrating random variation?
2 marks
The rule forces balance rather than modelling independent random choices.
It cannot show the variation of a genuine chance experiment.
The outcomes are fixed in advance.
Suggest a fair-die rule to simulate an event with probability $\frac{1}{3}$ .
2 marks
Two of six equally likely outcomes give $\frac{2}{6} = \frac{1}{3}$ .
For example, success on faces 1 or 2; no success on 3, 4, 5 or 6.
A larger experiment records $51 %$ successes after a smaller experiment recorded $50 %$ . Does this contradict the idea that larger experiments tend to be more stable?
2 marks
Tend to is not a guarantee for each pair of experiments.
Random variation can move a result slightly away from the prediction.
No.

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Curriculum reference & lesson scope

Australian Curriculum v9.0; SA Mathematics Curriculum 2026

Numerical likelihood; fractions/decimals/percentages; expected versus observed outcomes; larger simulations and variation.

Official curriculum source ↗

In this lesson

Understand the idea.

A simulation copies a chance rule

Compare proportions, not raw differences

See the method in action.

Compare two sample sizes

Find the heads percentage for twenty trials.

Find the heads percentage for five hundred trials.

Measure each distance from fifty percent.

Compare the distances.

A larger result can move away

Find Experiment A's heads percentage.

Find Experiment B's heads percentage.

Calculate the distances from fifty percent.

Choose the smaller distance.

Try it yourself.

Check your understanding

Keep practising