Experimental Probability
Statistics & ProbabilityExperimental 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.
Example
You flip a coin $40$ times and get heads $22$ times. Experimental $P(\text{heads}) = 22/40 = 0.55$. The theoretical probability is $0.5$, but your experiment came out slightly different.
Key Insight
Experimental probability is what actually happened. Theoretical probability is what should have happened in a perfect world. With more trials, they get closer to each other.
Definition
Experimental probability = (number of times event occurred) / (total number of trials). It is also called empirical probability or relative frequency. It can differ from theoretical probability due to chance, especially with small sample sizes. As the number of trials increases, experimental probability converges to theoretical probability.
Example
A spinner has $4$ equal sections. After $200$ spins, red comes up $58$ times. Experimental $P(\text{red}) = 58/200 = 0.29$. Theoretical $P(\text{red}) = 1/4 = 0.25$. The difference ($0.04$) is due to random variation with this sample size.
Key Insight
The law of large numbers guarantees that experimental probability approaches theoretical probability as trials increase. But they are rarely exactly equal for any finite number of trials.
Definition
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. 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
A/B testing in technology: an experiment runs $10{,}000$ trials for two website variants. The experimental conversion rates (experimental probabilities) estimate the theoretical conversion probabilities. A z-test or chi-square test determines if the observed difference is statistically significant or due to chance.
Key Insight
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. Both are essential tools in applied statistics.