Rules of Probability

Key Terms

Basic Axiomatic Properties

Set-Operation Rules

Additive & Inequality Rules

Multiplicative & Conditional Rules

Quick Lookup: Which Rule to Use

Welcome to the Rules of Probability.

Grounded in the three axioms of probability—non-negativity, normalization and finite additivity—this section translates those foundational principles into practical tools. You’ll begin by revisiting the basic axiomatic properties before moving on to set-operation rules (complements, differences and absorption), additive rules (addition, inclusion–exclusion and Boole’s inequality) and multiplicative rules (chain rule, law of total probability and Bayes’ theorem). Each law here is built on the axioms to ensure consistency and rigor. In upcoming chapters, you’ll see how these rules power classical (combinatorial) models, discrete and continuous distributions, conditional probability and independence, Bayesian inference, expectation and variance, limit theorems and stochastic processes. Keep this axiomatic framework in mind as your roadmap through the broader probability theory landscape.

Key Terms

Mutual Exclusiveness— events with

A \cap B = \emptyset

, simplifying the addition rule

Independent Events— events where

P(A \cap B) = P(A) \cdot P(B)

Complement of a Set—

A^c

, with

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

Union of Sets—

A \cup B

, the event in the addition rule

Intersection of Sets—

A \cap B

, the event in the multiplication rule

Conditional Probability—

P(A \mid B)

, used in the multiplication rule

See All Probability Definitions →

Basic Axiomatic Properties

	rule	formula	explanation
	Non-negativity & Bounds	$0 \le P(A) \le 1$	The probability of any event lies between 0 and 1, inclusive.
	Empty-Set Rule	$P(\varnothing) = 0$	The probability of the empty event (impossible outcome) is zero.

Set-Operation Rules

rule	formula	explanation
Complement Rule	$P(A^c) = 1 - P(A)$	The probability of the complement of A equals one minus the probability of A.
Difference Rule	$P(A \setminus B) = P(A) - P(A \cap B)$	The probability of A excluding B equals the probability of A minus the probability of A and B.
Subset Absorption	If $B \subseteq A$ , then $P(A \cap B) = P(B)$ and $P(A \cup B) = P(A)$	If B is contained in A, the intersection has B’s probability and the union has A’s.
Complement Absorption	If $A \subseteq B^c$ , then $P(A \cap B^c) = P(A)$ and $P(A \cup B^c) = P(B^c)$	When A lies entirely outside B, intersection yields A and union yields the complement of B.
Mutual Exclusivity	If $A \cap B = \varnothing$ , then $P(A \cap B) = 0$ and $P(A \cup B) = P(A) + P(B)$	Disjoint events have zero probability of occurring together, and their union is the sum of probabilities.

Additive & Inequality Rules

rule	formula	explanation
Addition Rule	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$	The probability of A or B equals the sum minus the overlap.
Inclusion–Exclusion Principle	$P\bigl(\bigcup_{i=1}^n A_i\bigr) = \sum_i P(A_i) - \sum_{i<j}P(A_i \cap A_j) + \cdots + (-1)^{n-1}P\bigl(\bigcap_{i=1}^n A_i\bigr)$	General formula for the probability of a union of n events, correcting for over-counted overlaps.
Monotonicity & Boole’s Inequality	If $A \subseteq B$ , then $P(A) \le P(B)$ ; and $P\bigl(\bigcup_i A_i\bigr) \le \sum_i P(A_i)$	Probabilities respect subset ordering, and the probability of any union is at most the sum of individual probabilities.

The addition rule on two dice

A, first die 6, is the blue bottom row: 6 outcomes. B, doubles, is the amber diagonal: 6 outcomes. Adding 6/36 + 6/36 counts the corner (6,6) twice, because it is in both. Subtracting P(A ∩ B) = 1/36 once leaves the 11 distinct outcomes of A ∪ B: P(A ∪ B) = 11/36.

For disjoint events the overlap is empty and the rule reduces to P(A) + P(B).

The additive rules, stated formally:

Complement Rule

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

Learn more about this formula: Complement Rule →

Monotonicity Rule

A \subseteq B \implies P(A) \leq P(B)

Learn more about this formula: Monotonicity Rule →

Difference Rule

P(A \setminus B) = P(A) - P(A \cap B)

Learn more about this formula: Difference Rule →

Addition Rule

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

Learn more about this formula: Addition Rule →

Inclusion-Exclusion Principle

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

Learn more about this formula: Inclusion-Exclusion Principle →

Booles Inequality

P\!\left(\bigcup_{i=1}^{n} A_i\right) \leq \sum_{i=1}^{n} P(A_i)

Learn more about this formula: Booles Inequality →

Browse all probability formulas →

Multiplicative & Conditional Rules

rule	formula	explanation
Multiplication & Chain Rules	$P(A \cap B) = P(B)\,P(A \mid B)$ ; $P\bigl(\bigcap_{i=1}^n A_i\bigr) = \prod_{i=1}^n P\bigl(A_i \mid A_1 \cap \cdots \cap A_{i-1}\bigr)$	Compute joint probabilities via conditional probabilities, extended through the chain rule.
Law of Total Probability	$P(A) = \sum_i P(A \mid B_i)\,P(B_i)$	Expresses P(A) as a weighted sum over a partition of the sample space.
Bayes’ Theorem	$P(B_j \mid A) = \dfrac{P(A \mid B_j)\,P(B_j)}{\sum_i P(A \mid B_i)\,P(B_i)}$	Relates the reverse conditional probability to the forward conditional probability and priors.

The multiplicative rules, stated formally:

Multiplication Rule

P(A \cap B) = P(A)\,P(B \mid A) = P(B)\,P(A \mid B)

Learn more about this formula: Multiplication Rule →

Chain Rule

P(A_1 \cap A_2 \cap \cdots \cap A_n) = P(A_1)\,P(A_2 \mid A_1)\,P(A_3 \mid A_1 \cap A_2) \cdots P(A_n \mid A_1 \cap \cdots \cap A_{n-1})

Learn more about this formula: Chain Rule →

Browse all probability formulas →

Quick Lookup: Which Rule to Use

The four categories above organize the rules by their mechanical type — axiomatic, set-operation, additive, and multiplicative. In practice, however, the question a reader brings to this page tends to run in the opposite direction: given a probability problem, which rule applies? The table below reorganizes the most-used rules by the kind of question they answer, providing a one-line guide from a reader's goal to the appropriate rule and its formula.

Goal	Rule	Formula
Get P(A^c) from P(A)	Complement Rule	P(A^c) = 1 − P(A)
Get P(A ∪ B) with general overlap	Addition Rule	P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Get P(A ∪ B) when A, B are disjoint	Mutual Exclusivity	P(A ∪ B) = P(A) + P(B)
Get P of a union of n events	Inclusion–Exclusion	alternating sums of pairwise, triple, … intersections
Bound P(∪ A_i) from above	Boole's Inequality	P(∪ A_i) ≤ Σ P(A_i)
Get P(A ∩ B) for general events	Multiplication Rule	P(A ∩ B) = P(B) · P(A \| B)
Get P(A) from a partition {B_i}	Law of Total Probability	P(A) = Σ_i P(A \| B_i) · P(B_i)
Reverse a conditional probability	Bayes' Theorem	P(B \| A) = P(A \| B) · P(B) / P(A)
Compare P(A) and P(B) when A ⊆ B	Monotonicity	A ⊆ B ⇒ P(A) ≤ P(B)