Three Views of Probability — Section 1: Foundations of Probability

Three different philosophical interpretations of probability all give us the same math but different intuitions about what a probability *is*.

The classical view says probabilities come from symmetry. A fair coin has $P(\text{heads}) = 1/2$ because the two outcomes are equally likely a priori. A fair die has $P(\text{six}) = 1/6$ for the same reason. This works beautifully when the system has obvious symmetries.

The frequentist view defines probability as the long-run frequency of an event. $P(\text{heads})$ is the limit of the fraction of heads as the number of flips approaches infinity:

P(A) = \lim_{n \to \infty} \frac{\text{number of times A occurred in n trials}}{n}

The Bayesian view treats probability as a degree of belief. $P(\text{heads})$ measures how strongly you believe the next flip will land heads, given everything you currently know. New evidence updates that belief.

Frequentist methods (confidence intervals, p-values, hypothesis tests) dominate classical statistics. Bayesian methods (priors, posteriors, MCMC) dominate modern machine learning and any problem where you have to make decisions under uncertainty with limited data. Quants borrow from both depending on the situation.

Three different philosophical interpretations of probability all give us the same math but different intuitions about what a probability *is*.

The frequentist view defines probability as the long-run frequency of an event. $P(\text{heads})$ is the limit of the fraction of heads as the number of flips approaches infinity:

P(A) = \lim_{n \to \infty} \frac{\text{number of times A occurred in n trials}}{n}