Step 2: Chance, Random Experiments & Empirical Probability
In empirical science and everyday observation, probability begins with experimental trials.
1. Essential Terminology
- Random Experiment: An experiment whose individual outcome cannot be predicted with certainty in advance, but whose set of all possible outcomes is known.
- Trial: A single performance of a random experiment (e.g. flipping a coin once, rolling a die once).
- Event ($E$): A specific collection of outcomes of an experiment (e.g. getting an odd number on a die).
2. Empirical (Experimental) Probability Formula
If a trial is repeated $n$ times under identical conditions, and event $E$ occurs $m$ times:
$$P(E) = \frac{m}{n} = \frac{\text{Number of trials in which event } E \text{ occurred}}{\text{Total number of trials performed}}$$
3. Historical Coin-Toss Experiments
| Experimenter | Number of Tosses ($n$) | Number of Heads ($m$) | Empirical Probability ($m/n$) |
|---|---|---|---|
| Comte de Buffon (18th Century) | 4,040 | 2,048 | 0.5069 |
| J.E. Kerrich (1940s) | 10,000 | 5,067 | 0.5067 |
| Karl Pearson (1900) | 24,000 | 12,012 | 0.5005 |
[!NOTE] As the number of trials increases without bound, the empirical probability converges to the theoretical probability of $0.5$. This convergence is the Law of Large Numbers.