Theoretical vs Empirical Probability

There are two ways to put a number on an event, and a trader uses both constantly.

Theoretical probability comes from a model of the mechanism. A fair coin lands heads with probability $\frac{1}{2}$ because of what "fair coin" means, not because anyone flipped it. You get the number by reasoning about structure.

Empirical probability comes from counting what actually happened. Flip a coin 10 times, see 6 heads, and your empirical estimate is $\frac{6}{10} = 0.6$. You get the number by observing frequency.

Neither is the "right" one. The theoretical number is precise but only as good as its assumptions. The empirical number makes no assumptions but is noisy, and it can only tell you about conditions that actually occurred.

The gap between them is the interesting part

In markets you rarely know the true mechanism, so you are usually reconciling a model against data. When the two disagree, exactly one of three things is happening:

• The model is wrong, which is a signal to change the model.
• The sample is too small, which is a signal to collect more data.
• The model is right and the market is mispriced, which is a signal to trade.

Distinguishing the third case from the first two is most of quantitative research. A candidate who says "the empirical number is 0.6 so the coin is biased" after 10 flips has failed to consider the second case, and that failure is exactly what interviewers probe for.

The Law of Large Numbers

Empirical estimates converge to theoretical ones as samples grow. Formally, for independent draws $X_1, X_2, \ldots$ with mean $\mu$:

This is the theoretical guarantee that data eventually reveals the truth. It is also routinely over-trusted, for two reasons worth knowing.

It says nothing about how fast. The standard error of an estimated proportion is

so at $p = 0.5$, ten flips give a standard error of about $0.16$. Observing 6 heads out of 10 is well inside one standard error of a fair coin. To pin the rate down to $\pm 1\%$ you need roughly 2,500 flips. Convergence is real but slow, and it slows as the square root, so getting ten times more precise costs a hundred times more data.

It assumes the process is stable. The law describes repeated draws from an unchanging distribution. Markets change regime, so an average taken over a period that spans a regime change converges to something that describes neither regime.

Why interviewers care

"Is this coin fair?" is a standard question, and the expected answer is not yes or no. It is a comparison: how far is the observed frequency from the model, measured in standard errors? That instinct, sizing a discrepancy against its noise rather than eyeballing it, is the same one used to decide whether a strategy's backtest is real. We develop it properly in hypothesis testing and confidence intervals.