What is a simulation in probability?

Imitating a chance process with random numbers, many times, to estimate how likely an event is.

What is the difference between experimental and theoretical probability?

Theoretical comes from reasoning about equally likely outcomes; experimental comes from the relative frequency in actual or simulated trials.

How many trials should a simulation use?

As many as practical: hundreds or thousands. The estimate's typical error shrinks like 1 ÷ √(trials).

Estimating Probability Using Simulation

A simulation imitates a chance process with random numbers so we can estimate a probability that is hard to calculate. Plan it: state the question, give each outcome a set of random digits that matches its probability, define one trial and what counts as the event, then repeat many trials. The relative frequency (event count ÷ trials) is the estimate, and it gets closer to the true probability as the number of trials grows (the law of large numbers).

Why simulate?

The theoretical probability of an event comes from reasoning: a fair die has 6 equally likely faces, so P(6) = 1/6. The experimental (empirical) probability comes from doing it: roll many times and find the relative frequency = (number of times the event happened) ÷ (number of trials).

Some questions are hard to work out by theory, like "How many cereal boxes must I buy to collect all 6 toys?" A simulation imitates the random process using random numbers (from a calculator, a computer or a random digit table) so you can estimate the answer quickly and cheaply.

Designing a simulation: step by step

Question: state what you want to estimate, e.g. P(at least one 6 in 4 rolls).
Model: give each outcome random digits in the right proportion. For a 30% chance, use digits 0–2 = "yes" and 3–9 = "no". For a die, use 1–6 and skip 0, 7, 8, 9. For 2-digit chances like 0.45, use 00–44 = yes, 45–99 = no.
Trial: say exactly what one repetition is (e.g. read 4 usable digits) and what counts as the event. Say whether repeats are allowed (with replacement) or not.
Run: do many trials and record results.
Answer: estimate = event count ÷ number of trials, written in context.

Each trial must be independent and use the same rules, otherwise the estimate is biased.

The law of large numbers

With few trials the relative frequency can be far from the truth: 3 sixes in 10 rolls gives 0.3, almost double 1/6. As the number of trials grows, the relative frequency settles closer and closer to the true probability. This is the law of large numbers.

It works only in the long run. It does NOT mean that after many non-sixes a six is "due". The die has no memory; each roll is still 1/6. That wrong idea is called the gambler's fallacy.

A rough rule: the typical error of an estimate shrinks like 1 ÷ √(number of trials). So 4 times as many trials halves the typical error.

Worked simulation and Try it

Example: A basketball player makes 70% of free throws. Estimate P(she makes all 3 shots).

Model: digits 0–6 = make, 7–9 = miss.
Trial: read 3 digits; event = all three are 0–6.
Digits 512 | 873 | 046 | 390 | 621 → trials 1, 3 and 5 are all makes, trial 2 has 8 and 7, trial 4 has 9.
Estimate from 5 trials: 3/5 = 0.6. With many more trials it settles near 0.7³ = 0.343, which shows why 5 trials are not enough.

Try it: Toss a coin 20 times and record the running proportion of heads after every toss. Draw it on a graph. Then press "Run 1000" in 3D free play and compare the wobbly start with the flat end.

Key formulas and definitions

Relative frequency = event count ÷ number of trials
Theoretical probability = favourable outcomes ÷ total equally likely outcomes
Digits for probability p: give a fraction p of all digits to "yes"
P(at least one) = 1 − P(none); e.g. 1 − (5/6)⁴ ≈ 0.518
Typical error ≈ 1 ÷ √n (shrinks as trials n grow)

Worked examples

1. A spinner lands on red with probability 0.4. Assign random digits to model one spin.

Use one digit: 0–3 = red (4 of 10 digits = 0.4), 4–9 = not red.

2. In 50 simulated trials, the event happened 13 times. What is the estimate?

Relative frequency = 13 ÷ 50 = 0.26.

3. A family has 3 children; each child is equally likely a girl or boy. Plan a simulation for P(at least 2 girls).

Model: even digit = girl, odd = boy. Trial: read 3 digits. Event: at least 2 even digits. Run 100+ trials. Answer: count ÷ trials (theory: 4/8 = 0.5).

4. Using the digits 3 8 1 7 2 9 5 0 4 6 for the family model (even = girl), how many trials have at least 2 girls in the first three triples?

381 → 8 is even: 1 girl (no). 729 → 2: 1 girl (no). 504 → 0, 4: 2 girls (yes). 1 of 3 trials, estimate 1/3, too few trials to trust.

5. A quiz has 5 true/false questions. Design a simulation for P(guessing at least 4 right).

Model: digit 0–4 = right, 5–9 = wrong. Trial: 5 digits. Event: at least 4 are 0–4. Run many trials; theory gives 6/32 ≈ 0.19 for comparison.

6. A simulation of 400 trials gives 0.52 for "at least one 6 in 4 rolls". Is this reasonable?

Theory: 1 − (5/6)⁴ = 1 − 625/1296 ≈ 0.518. Typical error ≈ 1/√400 = 0.05, and 0.52 is within 0.002 of the truth, so yes.

Common mistakes

Using digits in the wrong proportion, e.g. 0–4 for a 40% chance (that is 5 digits = 50%).
Forgetting to skip unused digits (like 0, 7, 8, 9 for a die) or to state the rule for repeats.
Trusting an estimate from very few trials. Ten trials can be far off; use hundreds.
Believing in the gambler's fallacy: past results do not change the next independent trial.

Practice quiz

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Estimating Probability Using Simulation

🎬 Step-by-step story

🤔 Common doubts, cleared

If we know P(6) = 1/6, why simulate a die at all?

Why skip digits 0, 7, 8 and 9?

Why are the columns not exactly equal?

Why does the blue line wobble at the start?

Why is "at least one 6 in 4 rolls" not 4 × 1/6?

Will 1000 trials always give 0.167 exactly?