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Introduction

Probability theory rests on three simple axioms proposed by Andrey Kolmogorov in 1933. These axioms provide the mathematical foundation for all probabilistic reasoning, from coin flips to quantum mechanics.

Learning Objectives:

Understand Kolmogorov's axioms of probability
Derive fundamental properties from axioms
Apply complement rule, union bound, and inclusion-exclusion
Verify properties with Python simulations

Kolmogorov's Axioms

A probability function $P$ assigns a real number to each event in a sample space $\Omega$ , satisfying three axioms:

Axiom 1 (Non-negativity): For any event $A$ , $P(A) \geq 0$

Axiom 2 (Normalization): The probability of the entire sample space is 1, $P(\Omega) = 1$

Axiom 3 (Additivity): For mutually exclusive events $A_1, A_2, A_3, ...$ , $P(A_1 \cup A_2 \cup A_3 \cup \cdots) = P(A_1) + P(A_2) + P(A_3) + \cdots$

Note: "Mutually exclusive" means $A_i \cap A_j = \emptyset$ for $i \neq j$ .

Why These Axioms?

Axiom 1: Probabilities can't be negative (you can't have "negative chance")
Axiom 2: Something must happen (certainty = 100%)
Axiom 3: If events can't occur together, their probabilities add

From just these three axioms, we can derive all properties of probability!

Derived Properties

Property 1: Probability of Empty Set

$P(\emptyset) = 0$

Proof: $\Omega$ and $\emptyset$ are disjoint, and $\Omega \cup \emptyset = \Omega$ . By Axiom 3: $P(\Omega) = P(\Omega \cup \emptyset) = P(\Omega) + P(\emptyset)$ $1 = 1 + P(\emptyset)$ $P(\emptyset) = 0$

Property 2: Complement Rule

$P(A^c) = 1 - P(A)$

Proof: $A$ and $A^c$ are disjoint and $A \cup A^c = \Omega$ : $P(\Omega) = P(A \cup A^c) = P(A) + P(A^c)$ $1 = P(A) + P(A^c)$ $P(A^c) = 1 - P(A)$

python

import random

def simulate_complement_rule(n_trials=10000):
    """Verify complement rule: P(A) + P(A^c) = 1"""
    # Event A: die roll is even
    omega = [1, 2, 3, 4, 5, 6]
    A = {2, 4, 6}
    A_complement = {1, 3, 5}

    count_A = 0
    count_A_complement = 0

    for _ in range(n_trials):
        roll = random.choice(omega)
        if roll in A:
            count_A += 1
        if roll in A_complement:
            count_A_complement += 1

    p_A = count_A / n_trials
    p_A_complement = count_A_complement / n_trials

    print(f"Simulated P(A) = {p_A:.4f}")
    print(f"Simulated P(A^c) = {p_A_complement:.4f}")
    print(f"Sum: P(A) + P(A^c) = {p_A + p_A_complement:.4f}")
    print(f"\nTheoretical: P(A) = 3/6 = 0.5000")
    print(f"Theoretical: P(A^c) = 3/6 = 0.5000")

simulate_complement_rule()

Addition Rule

Property: Addition Rule (for two events)

For any events $A$ and $B$ :

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Why subtract? Without it, we'd count $A \cap B$ twice!

python

import random

def simulate_addition_rule(n_trials=10000):
    """Verify: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)"""
    omega = [1, 2, 3, 4, 5, 6]
    A = {2, 4, 6}  # even
    B = {2, 3, 5}  # prime

    count_A, count_B, count_union, count_inter = 0, 0, 0, 0

    for _ in range(n_trials):
        roll = random.choice(omega)
        if roll in A: count_A += 1
        if roll in B: count_B += 1
        if roll in A or roll in B: count_union += 1
        if roll in A and roll in B: count_inter += 1

    p_A = count_A / n_trials
    p_B = count_B / n_trials
    p_union = count_union / n_trials
    p_inter = count_inter / n_trials

    print(f"P(A ∪ B) simulated: {p_union:.4f}")
    print(f"P(A) + P(B) - P(A ∩ B): {p_A + p_B - p_inter:.4f}")

simulate_addition_rule()

Key Takeaways

Three axioms define all of probability
Complement rule: $P(A^c) = 1 - P(A)$
Addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Bounds: $0 \leq P(A) \leq 1$ for all events

Next Lesson: Apply axioms to finite sample spaces with counting!

Property	Formula
Complement	$P(A^c) = 1 - P(A)$
Addition	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Bounds	$0 \leq P(A) \leq 1$

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