Probability Spaces and Events - Key Properties

The axioms of probability lead to numerous fundamental properties that form the toolkit for probability calculations. These properties are used constantly in both theoretical and applied work.

Basic Properties of Probability

From Kolmogorov's axioms, we can derive several essential properties:

Monotonicity: If $A \subseteq B$, then $P(A) \leq P(B)$

Complement Rule: For any event $A$, we have $P(A^c) = 1 - P(A)$

Empty Set: The impossible event has probability $P(\emptyset) = 0$

Inclusion-Exclusion Principle: For any two events $A$ and $B$: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

This generalizes to $n$ events as: $P\left(\bigcup_{i=1}^n A_i\right) = \sum_{i=1}^n P(A_i) - \sum_{i<j} P(A_i \cap A_j) + \sum_{i<j<k} P(A_i \cap A_j \cap A_k) - \cdots$

Properties of Conditional Probability

Conditional probability inherits structure from the original probability measure:

Theorem (Conditional Probability as Probability Measure): For fixed $B$ with $P(B) > 0$, the function $P(\cdot|B): \mathcal{F} \to [0,1]$ defined by $P(A|B) = P(A \cap B)/P(B)$ satisfies all axioms of probability.

Multiplication Rule: For events $A_1, \ldots, A_n$ with $P(A_1 \cap \cdots \cap A_{n-1}) > 0$: $P(A_1 \cap \cdots \cap A_n) = P(A_1) \cdot P(A_2|A_1) \cdot P(A_3|A_1 \cap A_2) \cdots P(A_n|A_1 \cap \cdots \cap A_{n-1})$

Law of Total Probability

Law of Total Probability: If $\{B_i\}$ is a partition of $\Omega$, then for any event $A$: $P(A) = \sum_{i=1}^n P(A|B_i) P(B_i)$

This powerful result allows us to compute probabilities by conditioning on different scenarios.