Probability theory enters psychology in multiple ways: as a normative standard for rational belief, as a descriptive model of stochastic behavior, and as a mathematical tool for statistical inference. Kolmogorov's axioms (1933) provide the mathematical foundation, while psychological research has explored when and how human probability judgments deviate from these norms.
The Kolmogorov Axioms
2. Normalization: P(Ω) = 1 (certainty)
3. Additivity: P(A ∪ B) = P(A) + P(B) for disjoint A, B
Subjective Probability
De Finetti and Ramsey independently developed frameworks for subjective probability — probability as degree of belief rather than frequency. The Dutch book argument shows that if an agent's beliefs violate the probability axioms, a set of bets can be constructed that guarantee the agent loses money. This provides a compelling normative argument for calibrated beliefs, though empirical research consistently shows that human probability judgments violate additivity, are subject to base rate neglect, and exhibit the conjunction fallacy documented by Tversky and Kahneman.
Bayesian approaches in psychology use the probability axioms as the foundation for models of perception, learning, and decision making, treating the brain as an approximate Bayesian inference engine that updates beliefs in light of evidence.