Theoretical vs Empirical Probability

Idealized models versus probabilities estimated from observed data.

Theoretical probability is what we compute from an idealized model. A fair coin has $P(\text{heads}) = 1/2$ by symmetry; a fair die has $P(\text{six}) = 1/6$ . We never had to flip the coin to know that.

Empirical probability is what we estimate from data. Flip a real coin 1,000 times, count the heads, and divide by 1,000. If the result is, say, $0.498$ , that's our empirical estimate.

The Law of Large Numbers ties them together. As the number of trials grows, the empirical proportion converges to the theoretical value, provided the model is correct:

\hat{P}_n(A) = \frac{\text{number of times A occurred}}{n} \to P(A) \quad \text{as } n \to \infty

In quant finance, both views show up daily. Black–Scholes is theoretical: it derives an option price from a model. Implied volatility is empirical: it backs the volatility number out of observed market prices. The gap between the two is where many trading edges live.