Pretend experiments
A simulation models a real situation using random tools. It’s useful when a real experiment would be too slow, expensive or dangerous, or when the maths is complicated.
Tools
| Tool | Outcomes |
|---|---|
| Coin | 2 (½ each) |
| Die | 6 (⅙ each) |
| Spinner | any you design |
| Cards | 52, or a chosen subset |
| Random number generator | any range, any number of trials |
Spreadsheets (RAND, RANDBETWEEN) and programs can run thousands of trials in seconds.
Steps of a simulation
- Define the problem. What probability do you want?
- Choose a model. Match random outcomes to real outcomes.
- Define a trial. What does one run look like?
- Run many trials and record results.
- Calculate the experimental probability.
- Interpret and compare with theory if possible.
Example: cereal prizes
A cereal box contains one of 4 equally likely toys. How many boxes, on average, to collect all 4?
- Model: roll a 4-sided die (or random numbers 1–4); each number is a toy.
- Trial: keep rolling until all 4 numbers appear; count rolls.
- Run 50 trials. Suppose the average is 8.4 boxes.
The theoretical answer is about 8.3, so the simulation is close.
Example: free throws
A player makes 70% of free throws. Estimate P(making at least 2 of 3).
- Model: random digits 0–9; 0–6 = make, 7–9 = miss.
- Trial: generate 3 digits; count makes.
- After 100 trials, suppose 79 had at least 2 makes → P ≈ 0.79.
(The theoretical value is 0.784.)
The more trials, the better
Simulations follow the law of large numbers: more trials → closer to the true probability. That’s why computer simulations often run thousands or millions of trials. Using huge numbers of random trials to estimate answers is called the Monte Carlo method, named after the casino city in Monaco.
Where simulations are used
- Weather forecasting
- Traffic planning
- Medical trials
- Video games and sports analytics
- Insurance risk
Worked examples
Example 1: How could you simulate a 25% chance? A spinner with 4 equal parts (1 part = success) or random numbers 1–4 (1 = success).
Example 2: 30 trials; the event happened 9 times. Experimental probability? 9/30 = 0.3.
Example 3: Simulate whether a family with 3 children has all girls. Flip 3 coins per trial (H = girl).
Common mistakes
- A model that doesn’t match the real probabilities.
- Too few trials to trust the result.
- Treating the simulation result as exact.
Try it yourself
- How could you simulate a 1/6 chance?
- Design a model for a 40% chance of rain using digits 0–9.
- In 200 trials, an event happened 46 times. Estimate its probability.
- Why might two classes get different simulation results?
Answers: 1) roll a die; one number = success 2) 0–3 = rain, 4–9 = no rain 3) 0.23 4) random variation, especially with few trials
Quick recap
- A simulation uses random tools to model real events.
- Define the model and the trial, then run many trials.
- More trials give better estimates (Monte Carlo method).