Digital simulations and explaining differences from predictions | Year 7

01 / LEARN

The idea, step by step

Match the model to the chances

A simulation replaces a physical trial with a random-number rule. For a fair die, generate an integer from 1 to 6, with each integer equally likely. In a spreadsheet, enter =RANDBETWEEN(1,6) in A2 and fill down to A61 for 60 simulated rolls.

Check the outcome rule first. Choosing from 0 to 6 gives seven possible integers, so it does not model an ordinary die. For colour chance 3/10, choose from 1 to 10 and assign exactly three of the integers to that colour.

Preserve and interpret one run

For A2:A61, =COUNTIF(A2:A61,6) counts sixes. Divide by 60 to find their relative frequency. Random cells can recalculate when a sheet changes; copy the completed run and use Paste values if you need a fixed record.

Use the supplied table to practise interpreting a simulation without needing a spreadsheet. Each row is an example, not a guaranteed pattern. Larger runs often have proportions closer to theoretical chances, but they still vary. Report the rule, trial count, observed proportion and comparison with the prediction.

Supplied example die-simulation records; each row is a separate run
Trials	1	2	3	4	5	6
60	7	12	9	13	8	11
600	101	98	95	107	103	96

A common trap

A simulation must produce the theoretical fraction exactly.

Instead: A valid simulation follows the chance rule; its finite records can still differ from the expected fraction.

02 / WORKED EXAMPLES

See the thinking

Example 1

Compare two runs

Use the supplied records to find each even-result relative frequency and compare with the theoretical fair-die chance.

Supplied example die-simulation records; each row is a separate run
Trials	1	2	3	4	5	6
60	7	12	9	13	8	11
600	101	98	95	107	103	96

For 60 trials, the even count is $12 + 13 + 11 = 36$ , giving $36 / 60 = 0.6$ .
For 600 trials, it is $98 + 107 + 96 = 301$ , giving $301 / 600 \approx 0.502$ .
A fair die has three even faces among six: chance 0.5. The larger run is closer in this example.

Estimates 0.6 and about 0.502; theoretical chance 0.5.

Example 2

A model for unequal colour chances

Model a bag with 3 red and 7 blue equal-size counters, with replacement.

Generate an integer from 1 to 10, for example =RANDBETWEEN(1,10).
Assign 1, 2 and 3 to red; assign 4 through 10 to blue.
Three of ten equally likely integers produce red.

Choose 1–10 equally: 1–3 means red, 4–10 means blue.

Check your understanding

A model generates 1 to 8 equally and calls 1 or 2 a success. Does it model success probability $1 / 4$ ?

Show answer

Yes.

Two of the eight integers are successful: $2 / 8 = 1 / 4$ .

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Which rule models a fair die: =RANDBETWEEN(0,6) or =RANDBETWEEN(1,6)? Explain.
2 marks
Describe a random-integer model for a fair coin using integers 1 and 2.
2 marks
Use the supplied table. Find the relative frequency of 6 in the 60-trial run.
2 marks
Supplied example die-simulation records; each row is a separate run
Trials 1 2 3 4 5 6
60 7 12 9 13 8 11
600 101 98 95 107 103 96
For the 600-trial run, state observed and expected six counts, and their difference.
3 marks
Supplied example die-simulation records; each row is a separate run
Trials 1 2 3 4 5 6
60 7 12 9 13 8 11
600 101 98 95 107 103 96
A model chooses 1 to 10 equally and labels 1 to 4 as green. Find green chance and expected greens in 50 trials.
3 marks
A simulation estimate is 0.6 for an event with theoretical chance 0.5. Must its coding be wrong? State one check and one alternative explanation.
3 marks

Read the worked solutions

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

Which rule models a fair die: =RANDBETWEEN(0,6) or =RANDBETWEEN(1,6)? Explain.
2 marks
The required outcomes are exactly 1 to 6.
The other rule adds a seventh outcome, 0.
=RANDBETWEEN(1,6).
Describe a random-integer model for a fair coin using integers 1 and 2.
2 marks
Each integer has chance $1 / 2$ .
Each recorded outcome matches one integer.
Choose 1 or 2 equally; use 1 for heads and 2 for tails.
Use the supplied table. Find the relative frequency of 6 in the 60-trial run.
2 marks
Supplied example die-simulation records; each row is a separate run
Trials 1 2 3 4 5 6
60 7 12 9 13 8 11
600 101 98 95 107 103 96
The six frequency is 11.
Divide by 60; the decimal is rounded to three decimal places.
$11 / 60 \approx 0.183$ .
For the 600-trial run, state observed and expected six counts, and their difference.
3 marks
Supplied example die-simulation records; each row is a separate run
Trials 1 2 3 4 5 6
60 7 12 9 13 8 11
600 101 98 95 107 103 96
Expected six count: $600 \times 1 / 6 = 100$ .
Read the actual count 96.
$100 - 96 = 4$ .
96 observed; 100 expected; 4 fewer than expected.
A model chooses 1 to 10 equally and labels 1 to 4 as green. Find green chance and expected greens in 50 trials.
3 marks
Four of ten integers give green.
$50 \times 2 / 5 = 20$ .
Chance $4 / 10 = 2 / 5$ ; expected greens 20.
A simulation estimate is 0.6 for an event with theoretical chance 0.5. Must its coding be wrong? State one check and one alternative explanation.
3 marks
A correct model can give an unbalanced finite record.
Check that outcomes are assigned in the intended proportions.
Do not adjust recorded data to force an expected fraction.
Not necessarily. Check the random-outcome mapping; random variation can explain the difference.

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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 ↗

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