Understand the idea.
Match the random mechanism
A simulation must reproduce the original chance mechanism. For two independent fair coins, HH, HT, TH and TT are equally likely labels. For two fair dice, generate two independent uniform integers from 1 to 6 and then add them.
Generating a uniform sum from 2 to 12 would describe a different experiment. There is only one ordered pair giving sum 2 but six giving sum 7. Preserve these unequal chances by modelling each stage fairly before summarising its outcome.
Specify one complete trial
Define the successful event and one complete trial before starting. State the stage rules and reset rule: with replacement, return the token before the second draw; without replacement, keep it out until the complete trial finishes, then restore the bag.
Record every trial, including failures, and divide successes by the full trial count. Compare that estimate with theory while allowing random variation. The supplied table is practice simulation data used to analyse a method, rather than evidence of a real survey.
| Outcome | Count |
|---|---|
| HH | 5 |
| HT | 7 |
| TH | 6 |
| TT | 6 |
See the method in action.
Worked example 1
Use the supplied simulation table to estimate P(exactly one head).
| Outcome | Count |
|---|---|
| HH | 5 |
| HT | 7 |
| TH | 6 |
| TT | 6 |
Worked solution
- 1
Choose the outcomes with exactly one head.
HH has two heads and TT has none, so neither is successful.
- 2
Total the successful trials and all trials.
Every table row contributes to the denominator, including unsuccessful outcomes.
- 3
Calculate the simulation estimate.
The estimate is observed successes divided by the complete trial count.
- 4
Compare with the fair-coin model.
Two of four equally likely outcomes are successful; the observed thirteen successes are close to the twelve expected in 24 trials.
Experimental 13/24 versus theoretical 1/2.
Worked example 2
Use random integers 1–4 to simulate blue with probability 1/4 and red with probability 3/4.
Worked solution
- 1
Specify a fair random generator.
The four integer labels must be equally likely to represent quarter probabilities.
- 2
Assign one label to blue.
One of the four equally likely labels produces blue.
- 3
Assign the remaining labels to red.
Three distinct labels represent the three-quarter red probability.
- 4
Repeat stages independently when required.
Generate a fresh independent integer for each independent stage rather than reusing the previous result.
One blue label and three red labels.
Try it yourself.
Check your understanding
A simulation has 45 successes in 180 trials. Find its estimate.
Show answer & reasoning
1/4=0.25.
The estimate uses all 180 trials in the denominator: . Only the 45 successful trials belong in the numerator.
Keep practising
6 worksheet questions · 16 marks
Read the questions online
- Map equally likely digits 0–9 to and .2 marks
- Two tokens are drawn without replacement in a trial. When is the full bag restored for repeated trials?2 marks
- Why does a uniform random integer from 2 to 12 not simulate a fair two-dice sum?3 marks
- Use the table to estimate .2 marks
Original practice simulation: 24 trials of two fair coins Outcome Count HH 5 HT 7 TH 6 TT 6 - A student deletes all TT trials as uninteresting. Explain the problem.3 marks
- Describe a reproducible simulation of two dice and sum 7, including a check.4 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Map equally likely digits 0–9 to and .2 marks
Four labels versus six gives the required probabilities.
For example, 0–3 blue and 4–9 red. - Two tokens are drawn without replacement in a trial. When is the full bag restored for repeated trials?2 marks
Restoring between draws changes the experiment.
After both draws, before the next complete trial. - Why does a uniform random integer from 2 to 12 not simulate a fair two-dice sum?3 marks
The required outcome multiplicities differ.
It treats sums 2 and 7 equally although they have 1 and 6 physical pairs. - Use the table to estimate .2 marks
Original practice simulation: 24 trials of two fair coins Outcome Count HH 5 HT 7 TH 6 TT 6 Count HH,HT,TH:.
. - A student deletes all TT trials as uninteresting. Explain the problem.3 marks
All complete trials must be recorded.
Selective removal changes the denominator and distribution. - Describe a reproducible simulation of two dice and sum 7, including a check.4 marks
Each ordered pair is modelled fairly and the trial count is reported.
Generate two independent integers 1–6 each trial, count sum 7, divide by trial count, and compare with .
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum
Complementary events; two-stage sample spaces; tables, trees and Venn diagrams; probabilities of combined outcomes; experimental versus theoretical results; compound-event simulations and describing results. Formal conditional probability is Year 10.
Official curriculum source ↗