Theoretical probability describes what a model predicts. Experimental probability describes what actually happened in a set of trials. The two should often become closer as the number of trials grows, but they do not need to match exactly in a finite experiment.
This distinction is central to probability. It separates the structure of a model from the variability of observed data and prevents students from treating random fluctuation as mathematical failure.
One-sentence answer
Theoretical probability comes from the assumed model; experimental probability comes from observed relative frequency.
The two calculations
Theoretical probability = model-based favourable probability
Experimental probability = observed successes ÷ number of trialsFor a fair coin, the theoretical probability of heads is 1/2. If 20 tosses produce 13 heads, the experimental probability is 13/20 = 0.65.
That does not prove the theory is wrong. Twenty tosses are a small random sample.
Worked example 1: short run variation
A fair die is rolled 30 times and a six appears 8 times.
Theoretical P(6) = 1/6 ≈ 0.167
Experimental P(6) = 8/30 ≈ 0.267The observed value is higher. A finite random experiment can deviate from its theoretical expectation.
Long-run stability
As an experiment is repeated many times under stable conditions, relative frequencies often settle closer to the model probability. This is the long-run behaviour behind the law of large numbers.
It does not mean the experimental probability moves smoothly toward the theoretical value after every trial. It can rise and fall while becoming more stable overall.
Expected frequency is not guaranteed frequency
If P(red) = 0.4 and a spinner is used 500 times, the expected frequency is 200 red outcomes. The actual count might be 194, 207 or another nearby number.
Expectation is a model-based centre, not a contractual result.
When persistent mismatch matters
If a supposedly fair coin is tossed 100,000 times and produces heads 80% of the time, random fluctuation becomes a less convincing explanation. We may need to question the model, apparatus, recording process or assumption of fairness.
This is a powerful scientific idea: data can be used to test whether a model remains plausible.
Worked example 2: estimate an unknown probability
A new game has no known theoretical model. In 2,000 plays, a bonus appears 318 times. An experimental estimate is:
318 / 2000 = 0.159So 0.159 is a reasonable data-based estimate of the bonus probability under the observed conditions. More trials could refine the estimate.
Simulation
Simulation uses a random process to imitate another probability situation. Computers can run thousands or millions of trials quickly, making simulation useful when exact theoretical calculation is difficult.
A simulation is only as meaningful as the model built into it. If the programmed assumptions are wrong, a very large simulation can still produce misleading results.
Relative frequency and sample size
Suppose one student gets 4 heads in 5 tosses and another gets 510 heads in 1,000 tosses. The first experimental probability is 0.8; the second is 0.51. The second is usually stronger evidence about the fair-coin probability because it is based on far more trials.
Sample size changes how much confidence we place in an experimental estimate.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| Says experiment must equal theory | Random variation misunderstood | Compare relative frequencies over increasing trials |
| Uses one tiny experiment as proof | Sample-size judgement | Discuss stability and replication |
| Calls expected frequency guaranteed | Expectation confused with outcome | Separate model centre from realised count |
| Ignores large persistent mismatch | Model checking absent | Question assumptions, fairness and data quality |
| Trusts simulation without assumptions | Model opacity | Inspect what the simulation actually represents |
Practice
- A fair coin is tossed 50 times and gives 29 heads. Find the experimental probability of heads.
- A fair die is rolled 600 times. What is the expected frequency of a 3?
- A spinner gives blue 145 times in 500 spins. Estimate P(blue).
- Explain why 7 heads in 10 tosses does not prove a coin is biased.
- Explain why 70,000 heads in 100,000 tosses of a supposedly fair coin would justify investigating the fairness assumption.
Answers
1. 29/50 = 0.58. 2. 100. 3. 145/500 = 0.29. 4. Small samples can fluctuate substantially by chance. 5. A large persistent departure from 0.5 is harder to explain as ordinary random variation and gives reason to test the model or apparatus.
Connected routes
Use Probability as a Model of Uncertainty for the conceptual base, Sample Space Diagrams for exact outcome models, and Tree Diagrams for sequential events. Return to the Mathematics Learning Hub for the wider Secondary Mathematics estate.