2.2 Estimating Probabilities Using Simulation

What is a simulation?

A simulation is a way to model a chance process by mimicking it repeatedly using random outcomes (like random digits, coins, or spinners) to estimate probabilities. Since it's often impractical to observe an event many times in real life, simulation lets us approximate the long-run relative frequency of an outcome without needing formulas.

Steps to design a simulation

1) State the question and identify the component to simulate. 2) Assign random digits/numbers to represent outcomes, matching their true probabilities. 3) Run many trials, recording the outcome of each. 4) Stop and analyze: compute the relative frequency of the outcome of interest across all trials — this is your probability estimate.

Law of Large Numbers

The Law of Large Numbers states that as the number of trials in a simulation increases, the simulated relative frequency gets closer to the true probability. More trials produce more reliable and precise estimates, reducing the effect of random chance variation seen in small samples.

Key terms

Simulation

A technique that models a chance process by repeating random trials to estimate probabilities.

Steps of a simulation

State the question, Assign random digits to outcomes, Run many trials, and analyze the relative frequency (Stop and analyze).

Law of Large Numbers

As the number of trials increases, the simulated relative frequency approaches the true probability.

Relative frequency

The proportion of trials in a simulation in which a specific outcome occurred; used to estimate probability.

Why use simulation?

To estimate probabilities of events that are hard or impossible to calculate exactly or observe repeatedly in real life.

Try an AP-style question

Using random digits 0–9 to simulate an event with probability 0.3, which assignment works?

1. Digits 0–9 = success
2. Digits 0–2 = success; 3–9 = failure
3. Digit 3 = success; others = failure
4. Digits 0–3 = success; 4–9 = failure