Probability Axioms
Probability theory is built on measure-theoretic axioms: a sample space, events, and a probability measure satisfying countable additivity. These foundations support all stochastic modeling.
Key Points
- Kolmogorov's axioms define probability as a measure on a sigma-algebra.
- Conditional probability and independence are derived from the axioms.
- The law of total probability and Bayes' theorem follow from these rules.
Formulas
Axioms
$$P(A) \ge 0, \quad P(\Omega) = 1, \quad P(\bigcup_{i} A_i) = \sum_i P(A_i) \text{ for disjoint } A_i$$
Conditional probability
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
Bayes' theorem
$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$
Code Example
# Estimate P(A|B) by counting
import numpy as np
samples = np.random.rand(10000)
A = samples < 0.3
B = samples < 0.5
print(A[B].mean()) # ~ 0.6