Experimental Probability
Experimental probability is the probability of an event based on the results of an actual experiment or observation.
Formula
\text{P(event)} = \dfrac{\text{number of times event occurred}}{\text{total number of trials}}
Definition
Experimental probability is the probability you find by actually running an experiment many times and counting how often an event happens: (number of times event occurred) / (total number of trials). Also called empirical probability or relative frequency, it can differ from theoretical probability due to chance, especially with small sample sizes, and converges to theoretical probability as the number of trials increases. Formally, experimental probability is the empirical relative frequency $f_n(A) = (\text{count of } A \text{ in } n \text{ trials})/n$; by the strong law of large numbers, $f_n(A)$ converges almost surely to $P(A)$ as $n$ grows for i.i.d. trials, and the Berry-Esseen theorem quantifies the rate of convergence of the standardized sum to normal: $\sup|F_n(x)-\Phi(x)| \le C\rho/(\sigma^3\sqrt{n})$, where $\rho = E[|X-\mu|^3]$.
Example
You flip a coin $40$ times and get heads $22$ times: experimental $P(\text{heads}) = 22/40 = 0.55$, while the theoretical probability is $0.5$, showing the experiment came out slightly different. A spinner with $4$ equal sections comes up red $58$ times in $200$ spins: experimental $P(\text{red}) = 58/200 = 0.29$ versus theoretical $P(\text{red}) = 0.25$, the difference due to random variation with this sample size. A/B testing in technology runs $10{,}000$ trials for two website variants, with the experimental conversion rates estimating the theoretical conversion probabilities, and a z-test or chi-square test determines if the observed difference is statistically significant or due to chance.
Key Insight
Experimental probability is what actually happened, while theoretical probability is what should have happened in a perfect world; with more trials, they get closer to each other by the law of large numbers, though they are rarely exactly equal for any finite number of trials. The distinction between experimental and theoretical probability mirrors the frequentist vs. model-based statistical frameworks: experimental probability requires no model, theoretical probability requires a specified probability model, and both are essential tools in applied statistics.