Axiomatic Probability

The axiomatic approach to probability starts from a solid foundation—clear, logical rules that define how probabilities behave. With just a few principles, we can build a whole world of mathematical reasoning about chance! 🧠

Learning Objectives

Learning objectives of the Axiomatic Probability section.

• Probabilities
• Conditional Probabilities
• Independence

Summary Table

Summary of the Axiomatic Probability section.

Concept	Description	Example
Sample Space (\(S\))	Set of all possible outcomes of an experiment	\(S = \{Heads, Tails\}\)
Event (\(A\))	A subset of the sample space	\(A = \{1, 3, 5\}\) when rolling a die
Outcome	A single result from the sample space	\(Heads \in \{Heads, Tails\}\)
Axiom 1: Nonnegativity	Probability of any event is ≥ 0	\(P(A) \geq 0\)
Axiom 2: Normalization	Probability of the sample space is 1	\(P(S) = 1\)
Axiom 3: Additivity	For disjoint events, the probability of their union is the sum of parts	\(P(A \cup B) = P(A) + P(B)\) if \(A \cap B = \emptyset\)
Conditional Probability	Probability of \(A\) given \(B\) has occurred	\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
Product Rule	Probability of intersection using conditional probability	\(P(A \cap B) = P(A|B)P(B)\)
Total Probability Theorem	Compute probability over partitions of the sample space	\(P(B) = \sum_i P(B|A_i)P(A_i)\)
Bayes’ Theorem	Reverse conditional probability using prior and likelihood	\(P(S|W) = \frac{P(W|S)P(S)}{P(W|S)P(S) + P(W|NS)P(NS)}\)
Independent Events	Events that do not affect each other’s probabilities	\(P(A \cap B) = P(A)P(B)\) if \(A \perp B\)
Conditioning and Independence	Independence may break down when conditioning on a third event	\(A \perp B\) might not imply \(A \perp B | C\)