Probability describes uncertainty before an experiment; experimental probability summarises what happened after repeated trials. A simulation lets us repeat a random process many times, compare observed frequencies with theoretical expectations, and investigate how sample size changes the stability of results.
This Secondary 2 Mathematics Learning Guide develops simulation, randomness, fair trial design, experimental probability, expected frequency, sample size, relative frequency and long-run behaviour. It also explains the limits: a large simulation can provide strong evidence about a probability model, but it cannot turn a random event into a guaranteed short-run outcome.
Secondary Mathematics Hub: S1–S4 Capability Map · Secondary 2 Learning Guide, Batch 8, Guide 4. Companion guides cover spreadsheet Mathematics, graphing technology, and computational thinking and debugging.
Course boundary. This guide strengthens the existing probability, sample spaces and combined events guide. It focuses specifically on repeated experiments and technology-assisted simulation. The examples are mathematical models of random processes, not claims that every real-world event is equally likely or independent.
Navigate: random experiments · theoretical probability · experimental probability · simulation design · sample size · expected frequency · bias and independence · practice and answers · teaching and transfer.
1. A random experiment has uncertain individual outcomes but describable possibilities
Tossing a fair coin, rolling a fair die and selecting one well-mixed card from a set are standard random experiments. Before a trial, we may know the possible outcomes and their probabilities without knowing which individual outcome will occur.
Random does not mean patternless in every long run. It means the next individual outcome is not determined by the probability model in advance.
Worked example 1: fair die
A fair six-sided die has sample space {1,2,3,4,5,6}. Probability of rolling an even number is 3/6 = 1/2. The next roll is still uncertain; the fraction describes the model over possible outcomes.
2. Theoretical probability comes from a mathematical model of the outcomes
When outcomes are equally likely, probability = favourable outcomes / total outcomes. This relationship depends on the equally-likely assumption. If a spinner has unequal sectors, counting sectors alone is insufficient.
Worked example 2: coloured counters
A bag contains 4 red, 3 blue and 3 green counters. One counter is chosen at random. P(red) = 4/10 = 0.4. P(not red) = 6/10 = 0.6.
The model assumes each counter is equally likely to be selected, which requires a random selection method rather than choosing by sight.
3. Experimental probability is a relative frequency
If an event occurs f times in n trials, experimental probability = f/n. This value describes observed results. It may differ from theoretical probability, especially when n is small.
Worked example 3: coin experiment
A coin is tossed 40 times and heads appears 23 times. Experimental probability of heads = 23/40 = 0.575. This differs from the theoretical fair-coin value 0.5.
The difference does not by itself prove the coin unfair. Random variation can produce deviations in finite samples.
Worked example 4: compare two sample sizes
Experiment A: 7 heads in 10 tosses gives 0.7. Experiment B: 527 heads in 1000 tosses gives 0.527. The larger sample is closer to 0.5 here, although a larger sample is not guaranteed to be closer on every single run.
The important long-run idea is increasing stability, not guaranteed monotonic improvement after each extra trial.
4. A simulation maps random numbers to outcomes
A simulation replaces a physical random experiment with a mathematically equivalent random mechanism. For a fair coin, two equally likely random outcomes are enough. For a fair die, six equally likely outcomes are required.
The mapping must preserve the probabilities of the original experiment. A simulation that assigns three random digits to heads and seven to tails does not model a fair coin.
Worked simulation 1: one fair coin
Use random integers 0 and 1 with equal probability. Map 0 → tails and 1 → heads. Repeat many times. Count heads and divide by number of trials.
The computational procedure is simple: generate, map, count, divide. The Mathematics lies in the fair mapping and the interpretation of the relative frequency.
Worked simulation 2: event with probability 0.3
Use random digits 0–9, each equally likely. Map 0,1,2 to success and 3–9 to failure. Then simulated P(success) = 3/10 = 0.3.
If digits are not generated equally likely, the simulation no longer matches the intended probability.
Worked simulation 3: two fair coins
Generate two independent fair coin outcomes. The equally likely ordered outcomes are HH, HT, TH, TT. Probability of exactly one head is 2/4 = 1/2.
A simulation can repeat this pair thousands of times and estimate the same event frequency.
5. Sample size affects variability in experimental probability
With few trials, relative frequency can move dramatically after one new outcome. With many trials, one additional outcome changes the proportion by a much smaller amount.
Worked example 5: sensitivity after one extra success
Suppose an event has occurred 3 times in 5 trials: relative frequency 0.6. One more success changes it to 4/6 ≈ 0.667, a noticeable shift.
Suppose instead it has occurred 600 times in 1000 trials: 0.6. One more success gives 601/1001 ≈ 0.6004, a very small shift.
This explains part of the increasing stability seen in larger samples.
Larger samples do not remove randomness
A run of 1000 fair coin tosses need not contain exactly 500 heads. The theoretical probability predicts a long-run proportion around 0.5, not an exact requirement for every finite sample.
6. Expected frequency connects probability to a planned number of trials
If an event has probability p and there are n trials under the model, expected frequency is np. This is a long-run expectation, not a promise that the event will occur exactly np times.
Worked example 6: expected red outcomes
If P(red) = 0.3 over 200 independent trials, expected red frequency = 200 × 0.3 = 60.
Observed frequency might be 54, 63 or another nearby value. The expectation is the model’s average target across many repetitions of the entire 200-trial experiment.
Worked example 7: reverse from data
An event occurs 84 times in 300 trials. Experimental probability = 84/300 = 0.28. If the same model is used for another 500 trials, a rough expected count based on this estimate is 500 × 0.28 = 140.
This uses experimental probability as an estimate of an unknown underlying probability. The estimate itself carries sampling uncertainty.
7. A simulation is only as fair as its random mechanism and assumptions
A biased physical spinner, non-random selection method or unequal random-number mapping can invalidate the simulation. Technology does not guarantee fairness merely because a “random” command is used.
The simulation designer must ask whether the mechanism matches the intended probabilities and whether repeated trials are being treated as independent when that assumption is required.
Worked example 8: sampling without replacement
A bag contains 2 red and 3 blue counters. Two counters are drawn without replacement. The second draw is not independent of the first because the bag composition changes.
A simulation that resets the bag after the first draw would model sampling with replacement instead and therefore answer a different question.
Worked example 9: fair mapping check
To simulate a fair six-sided die using random digits 0–9, mapping 0→1, 1→2, …, 5→6 and discarding 6–9 is fair if the digits are equally likely and rejected digits are regenerated. Mapping 0–1 to face 1 and one digit each to other faces would not be fair.
The mapping probabilities must match the target probabilities.
8. Long-run behaviour is about stabilising proportion, not deterministic alternation
After five heads in a row, a fair coin is not “due” to produce tails on the next toss. Under independence, the next toss remains 1/2 heads and 1/2 tails.
Long-run balance emerges across many trials through aggregate behaviour, not because the random process keeps a memory and compensates immediately for short runs.
Worked example 10: gambler’s-fallacy contrast
A fair coin has produced HHHHH. P(heads next) is still 1/2 under the independent fair-coin model. The sequence history may look unusual, but it does not alter the next-trial probability.
9. Simulation can compare theoretical and observed distributions
For two fair dice, sums are not equally likely. Sum 7 can occur in six ordered ways, while sum 2 occurs only as (1,1). A simulation of many two-die rolls should reflect this uneven distribution.
Worked simulation 4: two-die sums
Theoretical P(sum 7) = 6/36 = 1/6. Over 6000 simulated rolls, an expected count is about 1000, but the observed value will normally differ somewhat.
Comparing the whole frequency table with the theoretical distribution is more informative than checking one sum only.
10. Common simulation and experimental-probability errors
- Experimental probability treated as exact theoretical probability: repair model-versus-data distinction.
- Small sample overinterpreted: repair variability awareness.
- Large sample assumed to match theory exactly: repair long-run interpretation.
- Unequal random mapping used for a fair event: repair simulation design.
- Without-replacement experiment simulated with replacement: repair state update.
- Expected frequency treated as guaranteed count: repair expectation meaning.
- Previous independent outcomes assumed to change next probability: repair independence.
- Technology output trusted without checking mechanism: repair random-generator assumptions.
11. Practice: design the simulation before running it
Questions 1–6. 1. A fair die is rolled. Find P(number greater than 4). 2. In 50 rolls, numbers greater than 4 occur 19 times. Find experimental probability. 3. Does this prove the die unfair? 4. Design a random-digit simulation for an event with probability 0.4. 5. What sample size usually gives a more stable relative frequency: 20 or 2000? 6. Does “more stable” mean guaranteed closer on every run?
Questions 7–12. 7. P(success) = 0.35 over 400 trials. Find expected success frequency. 8. Observed success frequency is 151. Find experimental probability. 9. Compare it with 0.35. 10. A fair coin has produced six tails in a row. Find P(tails next). 11. Explain why. 12. A bag sample is drawn without replacement. Why are successive draws generally dependent?
Questions 13–18. 13. Two fair dice are rolled. Find P(sum 2). 14. Find P(sum 7). 15. In 3600 simulations, about how many sum-7 outcomes are expected? 16. Why should observed count not be required to equal this exactly? 17. Describe one unfair way to simulate a fair coin using digits 0–9. 18. Describe one fair way.
Explained answers 1–6
1. Outcomes 5 or 6: 2/6 = 1/3. 2. 19/50 = 0.38. 3. No; finite random variation can produce differences. 4. Map four equally likely digits, such as 0–3, to success and 4–9 to failure. 5. 2000. 6. No. A larger sample tends to reduce variability but one particular run can still be farther from theory than a smaller run.
Explained answers 7–12
7. 400 × 0.35 = 140. 8. 151/400 = 0.3775. 9. It is 0.0275 above the theoretical model value. 10. 1/2. 11. Under independence, previous tosses do not change the next probability. 12. The first draw changes the composition of the bag.
Explained answers 13–18
13. 1/36. 14. 6/36 = 1/6. 15. 3600 × 1/6 = 600. 16. Expected frequency is a long-run average, not a guaranteed finite count. 17. Example: digits 0–6 → heads and 7–9 → tails. 18. Example: 0–4 → heads and 5–9 → tails.
12. Teaching sequence: predict → simulate → compare → explain
Begin with theoretical probability and ask the learner to predict what a small sample might look like. Run 10 or 20 trials. Then increase to hundreds or thousands using technology. Record relative frequency after several checkpoints.
Ask what becomes more stable, what remains random and what assumptions the simulation is making. Then deliberately introduce a biased mapping or dependence error and ask the learner to diagnose why the results no longer match the intended model.
Questions parents and tutors can ask
What is the theoretical probability? How is the simulation mapping random numbers to outcomes? Is the mapping fair? Are trials independent? What happens to relative frequency as the number of trials grows? Does expected frequency mean guaranteed frequency?
13. The transfer test: can the learner separate randomness from poor modelling?
If simulation results differ from theory, there are at least two possibilities: ordinary random variation or a flawed model/simulation. The learner should inspect sample size, mapping, independence and data-generation assumptions before declaring the theory wrong.
Define the random experiment. Build a fair mapping. Repeat the process. Track relative frequency. Compare with theory. Interpret larger samples as more stable, not deterministic. Check independence and model assumptions before trusting the simulation.
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