Foundations of Probability

Axiomatic Probability

This is a prototype build and does not reflect actual lessons.

To build statistical intuition, we avoid informal heuristics and begin with the foundational Kolmogorov axioms.

The Probability Space $(\Omega, \mathcal{F}, P)$

A probability space consists of three structural elements:

1. Sample Space ($\Omega$): The set of all possible outcomes.
2. Event Space ($\mathcal{F}$): A $\sigma$-algebra of subsets of $\Omega$.
3. Probability Measure ($P$): A function mapping events in $\mathcal{F}$ to real values in $[0, 1]$.

# Sample event simulation in Python
import random

def coin_toss_experiment(trials=1000):
    heads = sum(1 for _ in range(trials) if random.random() > 0.5)
    return heads / trials

print(f"Empirical probability: {coin_toss_experiment()}")