Larger simulations and variation | Year 6

01 / LEARN

The idea, step by step

A simulation copies a chance rule

A simulation is a model that produces outcomes using a specified chance rule. To model a fair coin with a fair die, use 1, 2 and 3 for heads and 4, 5 and 6 for tails. Each outcome then has three of six equally likely die faces. Record the results rather than choosing a balanced pattern yourself.

For a digital version, a random whole number from 1 to 6 can replace the die if its outcomes are equally likely. Repeating the same rule many times gives data for comparison. State the rule, trial count and results so another person can understand the model.

Compare proportions, not raw differences

In the supplied results, 14 heads from 20 trials is 70%; 54 from 100 is 54%; 260 from 500 is 52%. Each must be compared with the fair-coin prediction of 50%. A difference of ten heads in 500 trials represents a smaller proportion than a difference of four in twenty.

Larger experiments tend to give more stable relative frequencies over many repetitions. However, a particular larger experiment is not guaranteed to be closer to 50%. A sequence can move away from the prediction before moving closer again.

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

A common trap

An experiment with more trials must always have a relative frequency closer to the theoretical probability.

Instead: More trials tend to stabilise results overall, but individual results can still move either closer or further away.

02 / WORKED EXAMPLES

See the thinking

Example 1

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

For 20 trials: $14 / 20 = 70 %$ , which is 20 percentage points above 50%.
For 500 trials: $260 / 500 = 52 %$ , which is 2 percentage points above 50%.

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

Example 2

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%?

A: $13 / 25 = 52 %$ , a difference of 2 percentage points.
B: $57 / 100 = 57 %$ , a difference of 7 percentage points.

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

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

No.

It gives heads probability $1 / 6$ and tails probability $5 / 6$ , not equal one-half chances.

03 / WORKSHEET

Your turn

6 questions · 11 marks · Work at your own pace.

Worksheet PDF 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 $1 / 3$ .
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
$260 / 500 = 52 / 100 = 52 %$ .
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 $1 / 3$ .
2 marks
Two of six equally likely outcomes give $2 / 6 = 1 / 3$ .
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.

Checking saved progress…

Ready for the topic test?

Review the chapter, then choose a practice test.

Topic review

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 ↗

Back to Probability