BTT Mathematics / Primary Mathematics Learning Hub / Approximation & Uncertainty

Primary Mathematics: Simulation, Experimental Probability and Random Trials | Worked Learning Guide

A simulation replaces a difficult, slow or unavailable chance process with a simpler random model that preserves the probabilities that matter.

The existing BTT Probability, Chance and Fairness guide develops equally likely outcomes and basic probability reasoning. This guide adds a different layer: repeated random trials, experimental probability, simulation design, relative frequency and the question of how much evidence repeated trials provide.

Flip a fair coin ten times and you may not get exactly five heads. Flip it one thousand times and the proportion of heads will often be closer to one-half, though no finite set of trials is guaranteed to be exact. That gap between theoretical probability and observed frequency is the heart of experimental probability.

This is an upper-Primary enrichment and transition bridge. Formal simulation and long-run probability language is introduced as preparation for later mathematics, not as a universal Primary requirement.

Experiment · Relative frequency · Design simulations · Repeated trials · Theory versus experiment · Limits · 24 questions · Worked solutions

1. A random trial has an uncertain individual outcome

Examples:

• flip a fair coin,
• roll a fair die,
• spin a fair spinner,
• randomly draw one card from a well-shuffled set.

We can describe possible outcomes and probabilities even though we do not know the next result with certainty.

Worked example: coin

Possible outcomes: H,T.

Theoretical probability of H for a fair coin = 1/2.

One actual flip gives either0 or1 head, not “half a head”. Probability describes chance across possible outcomes and repeated trials.

2. An experiment is a sequence of observed trials

Ten coin flips:

H,T,H,H,T,T,H,T,H,H.

Heads count=6.

Tails count=4.

The result6 heads is an observed frequency, not a change in the coin’s theoretical fairness.

3. Relative frequency converts count into proportion

Heads in10 flips =6.

Relative frequency=6/10=0.6=60%.

Experimental probability is often represented by this observed proportion.

Worked example

A spinner lands blue38 times in100 spins.

Experimental probability of blue=38/100=0.38.

4. Theoretical and experimental probability can differ

A fair coin has theoretical probability0.5 for heads.

Observed10-flip frequency0.6 differs by0.1.

This does not by itself prove the coin is unfair. Random variation is expected in finite samples.

5. Short runs can look very uneven

Four fair-coin flips can all be heads. Probability of that exact four-head sequence is(1/2)^4=1/16.

Unusual does not mean impossible.

Small samples are more volatile.

6. Larger numbers of trials often stabilise relative frequency

Imagine these fair-coin experiments:

Trials	Heads	Relative frequency
10	7	0.70
100	54	0.54
1000	508	0.508

The proportion moves closer to0.5 in this example.

Do not turn this into a guarantee that every longer run must be closer than every shorter run. The long-run tendency is statistical, not deterministic step-by-step convergence.

7. Running relative frequency shows how evidence changes

After each batch of trials, calculate cumulative successes divided by cumulative trials.

This gives a graph or table showing how the estimate changes as evidence accumulates.

8. Resetting versus accumulating

Ten separate experiments of20 trials each are not the same record as one cumulative200-trial experiment, although the total outcomes can be combined when the model and conditions are identical.

Keep track of whether proportions are batch-specific or cumulative.

9. A simulation should preserve relevant probabilities

Suppose a bus is late on about1 out of5 comparable days.

A simulation could use random digits0–9 and define:

• late = digits0 or1,
• on time = digits2–9.

Two of ten equally likely digits represent probability2/10=1/5.

10. A fair spinner can simulate unequal probabilities

To simulate probability3/4 of success, divide a spinner into4 equal sectors and mark3 success,1 failure.

The physical spinner remains fair at the sector level while labels create unequal event probabilities.

11. A die can simulate several models

Fair six-sided die:

• success on1,2 → probability2/6=1/3,
• success on1,2,3 →1/2,
• success on1–5 →5/6.

The mapping from die faces to events must match the intended probability.

12. A bad simulation changes the probability

To simulate1/3 probability, learner marks success on die faces1,2,3. That produces3/6=1/2, not1/3.

The random device may be fair while the event coding is wrong.

13. Simulate multi-stage events carefully

Two coin flips can simulate four equally likely ordered outcomes:

HH, HT, TH, TT.

Event “exactly one head” has outcomes HT and TH → probability2/4=1/2.

14. Simulation can estimate difficult probabilities

If exact counting is complex, many simulated trials can provide an empirical estimate.

But the estimate’s quality depends on:

• valid random model,
• enough trials for the purpose,
• accurate recording,
• correct interpretation.

15. Simulation can support planning without predicting one exact future

If a process succeeds about70% of the time, simulation can explore how often different totals might occur across repeated attempts.

It does not tell us the exact order of future successes and failures.

16. Compare experimental results with theory

A fair die is rolled60 times. Expected long-run share for each face≈1/6, so about10 occurrences per face is a reasonable benchmark.

Observed counts may differ from10.

The comparison asks whether the variation seems plausible, not whether each count equals10 exactly.

17. Expected count is not guaranteed count

Probability1/4 over100 trials gives expected count25.

Actual result might be21,27 or another value.

Expected value is a long-run centre, not a promise for one experiment.

18. Repeated experiments themselves vary

Five groups each flip a coin50 times. Their head counts may be21,26,24,29,25.

Variation across experiments is itself data.

19. Experimental evidence can suggest unfairness but needs care

A coin gives90 heads in100 flips. This is highly unusual for a fair coin.

The result provides evidence worth investigating, but good practice also checks procedure, recording and whether the coin/spinning method was unbiased.

At this level, avoid pretending one threshold automatically “proves” unfairness.

20. Random-looking output is not automatically random

A deterministic pattern such as H,T,H,T,H,T has50% heads but no randomness.

Matching proportions alone does not prove a process is random.

21. Computer simulation depends on the algorithm

A computer can generate large numbers of pseudo-random trials rapidly.

But if the event mapping or code is wrong, a million trials produce a precise answer to the wrong model.

Automation strengthens valid models; it does not validate assumptions automatically.

22. Simulation cannot fix poor input probabilities

If a model assumes “rain probability=20%” but the estimate itself is wrong for the location/time, simulation outputs inherit that weakness.

Model quality precedes trial count.

23. Common simulation errors

• Confusing expected count with guaranteed count.
• Assuming small experimental deviations prove unfairness.
• Designing event mappings with the wrong probability.
• Comparing batch and cumulative relative frequencies as if identical.
• Believing longer runs must improve monotonically.
• Treating a matching proportion as proof of randomness.
• Using many simulated trials to hide a bad model.
• Claiming exact prediction from a probabilistic simulation.

24. A simulation-design protocol

1. Event: What outcome are we modelling?
2. Probability: What probability should the model preserve?
3. Device: Which random mechanism can represent it?
4. Mapping: Which device outcomes mean success/failure or categories?
5. Trials: How many repetitions are useful?
6. Record: Count and relative frequency.
7. Compare: How does experimental frequency compare with theory?
8. Limit: What can the simulation not predict or prove?

25. Practice: 24 original questions

Questions 1–8: Experimental probability

1. Ten coin flips give6 heads. Find relative frequency.
2. A spinner gives blue38 times in100 spins. Find experimental probability.
3. A fair coin gives7 heads in10 flips. Does this prove unfairness?
4. Why can four fair-coin flips all be heads?
5. Head counts are54 in100 flips. Find relative frequency.
6. Heads508 in1000 flips. Find relative frequency.
7. Explain why a1000-trial result need not be closer to0.5 than every possible100-trial result.
8. Five groups run50 flips each. What kind of variability should be expected across groups?

Questions 9–16: Design simulations

9. Use random digits0–9 to simulate probability1/5.
10. Design a four-sector spinner for probability3/4 success.
11. Use a fair die to simulate probability1/3.
12. A learner uses die faces1,2,3 to simulate1/3. Critique.
13. Two coin flips: list all ordered outcomes.
14. Find probability of exactly one head in two flips.
15. Design a simple simulation for event probability2/5.
16. Explain why simulation mappings must preserve probability.

Questions 17–24: Interpret and critique

17. A fair die is rolled60 times. What benchmark count per face is expected?
18. Probability1/4 over100 trials gives what expected count?
19. Does expected25 mean exactly25 must occur? Explain.
20. A coin gives90 heads in100 flips. What is a responsible conclusion?
21. Sequence H,T,H,T,H,T has50% heads. Why does this not prove randomness?
22. Why can one million trials fail if simulation code uses wrong event mapping?
23. Explain why simulation cannot predict exact future order.
24. Create a random-trial simulation, state its intended probability and one limitation.

26. Worked solutions

1. 0.6 or60%. 2. 0.38 or38%. 3. No; finite random samples vary. 4. HHHH is one possible sequence with probability1/16.

5. 0.54. 6. 0.508. 7. Long-run closeness is a tendency, not a monotonic guarantee for every realised sample. 8. Head counts should vary around the expected centre rather than all being identical.

9. Success on digits0,1; failure2–9. 10. Three equal sectors success, one failure. 11. Success faces1,2; failure3–6. 12. Three of six faces gives1/2, not1/3.

13. HH,HT,TH,TT. 14. 2/4=1/2. 15. Example random digits0–9: success0,1,2,3; failure4–9. 16. Otherwise experimental results estimate a different event from the intended one.

17. About10. 18. 25. 19. No; actual random count varies. 20. The result is strong evidence worth investigating, while procedure and model should also be checked.

21. The sequence is deterministic despite equal counts. 22. More trials increase precision around the model being simulated, not the correctness of the model. 23. Simulation generates possible patterns/distributions, not a guaranteed future sequence. 24. Answers vary; valid simulations need a correct mapping and stated limit.

27. Simulation laboratory: estimate a two-stage event

Event: roll a fair die twice and succeed if the sum is at least10.

An exact method can enumerate36 ordered outcomes. A simulation can instead roll two dice repeatedly and record success.

Suppose1000 simulated trials produce169 successes. Experimental probability=0.169.

The exact probability is6/36=1/6≈0.167. The simulation estimate is close but not identical, illustrating random sampling variation.

28. Parent and tutor guide

Use coins, dice and spinners before digital simulation. Learners should understand the event mapping before software accelerates the trials.

Ask learners to predict an approximate proportion before running many trials and explain differences afterward.

29. Mastery receipt

• I distinguish theoretical and experimental probability.
• I calculate relative frequency.
• I expect finite random samples to vary.
• I design simulations that preserve event probabilities.
• I use repeated trials to estimate chance.
• I understand expected counts are not guaranteed counts.
• I recognise cycles or patterns are not automatically random.
• I know simulation quality depends on model and mapping quality.

Sources and scope

OECD PISA 2022 Mathematics Framework identifies computer simulation as a mathematical-literacy focus for exploring systems and variables, including experimental probability contexts.

For Singapore Primary scope, use the MOE Primary Mathematics Syllabus P1–P6, updated October 2025. This page is an enrichment/transition bridge.

Continue the Approximation & Uncertainty collection

• Geometric Approximation, Irregular Shapes and Estimating Area & Volume
• Measurement Precision, Error, Tolerance and Accuracy
• Conditional Decision Making, Two-Way Tables and Trade-Offs
• BTT Primary Mathematics Learning Hub

The Quiet Return

Simulation does not remove uncertainty. It lets us study uncertainty by making it repeatable.