Probability and Its Interpretations

What is probability?

Probability seems intuitively clear — but upon closer examination raises profound philosophical questions. There are three main interpretations, each with different implications.

Frequentist interpretation (Fisher, von Mises): the probability of an event is the limit of its frequency as the experiment is repeated infinitely. This works for casinos and insurance. But what does the "probability" mean that Napoleon would have won Waterloo with a different disposition? Such an "experiment" cannot be repeated.

Subjective (Bayesian) interpretation (de Finetti, Savage): probability is a measure of the degree of confidence of a rational agent. This allows for talk about unique events. But different agents can have different "subjective probabilities"—and that's normal: they should converge as evidence accumulates.

Objective Bayesian interpretation (Keynes, Jeffreys): probability is objectively determined by the available evidence. This is an attempt to avoid subjectivity while preserving the Bayesian formalism.

Conditional probability and intuition errors

$P(A|B)$ — the probability of A given that B has occurred. This is a fundamental concept, whose intuition is systematically violated.

"Base rate": If a rare disease (1 in 10,000) has a test with 99% accuracy, and your test is positive—what is the probability that you are sick? Intuitive answer: 99%. Correct: about 1%. Because out of 10,000 people, one is sick (1 true positive), but 100 healthy people will have a false positive result (1% of 9,999 ≈ 100). Of 101 positive results, only one person is actually ill.

This is not an abstraction: courts, media, and physicians systematically ignore the base rate. "DNA matches—so guilty" without accounting for the a priori probability of guilt.

Question for reflection: What "base rates" do you ignore when making business decisions? How often do your forecasts take into account the base statistics of the success of similar projects?