Statistics Foundations
Experimental vs. Theoretical Probability
Comparing observed frequencies to a theoretical probability model.
Prerequisites
- Understanding Probability
What Actually Happens vs. What Should Happen
Flip a fair coin 10 times. Before reading on, predict: will you get exactly 5 heads? Most people sense that's not guaranteed — so what does the coin's true 1/2 probability of heads actually promise, if not exactly half heads in every short run?
Definition — Experimental and Theoretical Probability
In a small number of trials, experimental probability often drifts noticeably from the theoretical value — 10 coin flips might reasonably land 7 heads and 3 tails. But as the number of trials grows very large, the experimental probability tends to settle in closer and closer to the theoretical probability. The short-run wobble isn't a contradiction of the theory; it's exactly what genuine randomness looks like over a small sample.
Worked Example — Comparing Experimental Results to a Theoretical Prediction
Worked Example — Predicting an Outcome from Experimental Data
Probability Simulator
Experiment
Run a trial to see the outcome here.
Simulation
Histogram — experimental vs theoretical
Running probability — Law of Large Numbers
Statistics
Mean
— experimental
3.5 theoretical
Variance
— experimental
2.917 theoretical
Standard deviation
— experimental
1.708 theoretical
Trials
0 experimental
— theoretical
Frequency table
| Outcome | Count | Experimental | Theoretical |
|---|---|---|---|
| 1 | 0 | — | 0.167 |
| 2 | 0 | — | 0.167 |
| 3 | 0 | — | 0.167 |
| 4 | 0 | — | 0.167 |
| 5 | 0 | — | 0.167 |
| 6 | 0 | — | 0.167 |
Central Limit Theorem
Draws 300 independent samples of 5trials each from the current experiment, averages each sample, and histograms the resulting sample means — the distribution should look progressively more bell-shaped (normal) as the sample size grows, even though the experiment’s own distribution usually isn’t.
Tip
Common Mistakes
Treating a small mismatch between experimental and theoretical probability as evidence that the theoretical probability must be wrong.
Short-run experimental results normally differ somewhat from theoretical probability — a mismatch in a small number of trials doesn't disprove the theoretical value, especially as more trials tend to close the gap.
Confusing which probability a question is asking for, calculating a theoretical probability when the question specifically asks about collected experimental data, or vice versa.
Check whether the question describes reasoning about equally likely outcomes (theoretical) or describes results actually observed from trials (experimental) before choosing which calculation to use.
Key Takeaways
- Theoretical probability comes from reasoning about equally likely outcomes; experimental probability comes from actual collected trial data.
- Experimental probability typically varies from theoretical probability in a small number of trials, but tends to approach it as the number of trials grows.
- Experimental data can be used to predict outcomes in future trials by applying the observed rate.
Summary
Comparing experimental results to theoretical predictions completes Grade 7's probability foundations. This closes out every remaining Grade 7 unit — Algebra Foundations was already complete, and ratios, the number system, geometry, and statistics now join it, finishing Grade 7 entirely. Grade 8 continues into linear functions, systems of equations, and the deeper algebraic reasoning that leads directly into Algebra I.
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