Probability Axioms
Every event has a probability. For example, P(A) means “the probability of event A happening.” A rule that assigns such probabilities to all events is called a probability law. All probability laws must satisfy three fundamental axioms, formalized by Andrey Kolmogorov in 1933.
The Three Axioms
Axiom 1: Non-negativity
For any event A:
Probability is always a non-negative number. You cannot have a “negative chance” of something happening.
Axiom 2: Normalization
The probability of the entire sample space Ω:
Some outcome must occur — the total probability across all possible outcomes equals 1 (or 100%).
Normalization means scaling all weights so the total equals 1. Imagine each outcome has a “weight” — some measure of how likely it is. These weights could sum to any number: 16, 100, even ∞. Dividing each weight by the total turns raw weights into proper probabilities.
Example: A biased dice has outcome weights 2, 3, 1, 4, 2, 4 (total = 16). Normalizing by dividing each weight by 16 gives probabilities: 2/16, 3/16, 1/16, 4/16, 2/16, 4/16. Now P(Ω) = 1.
Axiom 3: Additivity
If events A and B are mutually exclusive (disjoint, meaning A ∩ B = ∅):
In words: if two events cannot happen at the same time, the probability that either one happens equals the sum of their individual probabilities.
Example: A blood donor has blood type A with probability 0.36, and type B with probability 0.11. Since a person can only have one blood type, these are disjoint:
This extends beyond two events. For any countable sequence of pairwise disjoint events A₁, A₂, A₃, …:
Here each Aᵢ is an event, and the subscript i just tells you which one — A₁ is the first, A₂ is the second, and so on. The big union ∪ means “any of these events occurs,” and the sum Σ adds up all their probabilities. Pairwise disjoint means any two of them share no outcomes (A₁ ∩ A₂ = ∅, A₁ ∩ A₃ = ∅, A₂ ∩ A₃ = ∅, etc.).
Example: A student’s test score falls into exactly one category: below 60 (probability 0.15), 60–79 (0.45), 80–89 (0.25), or 90–100 (0.15). These are pairwise disjoint — a score can’t be in two categories at once. By additivity:
Derived Properties
From these three simple axioms, we can derive many important properties:
Complement Rule
The probability that A does not happen equals 1 minus the probability that A does happen.
Proof: A and Aᶜ are disjoint, and A ∪ Aᶜ = Ω. By Axioms 2 and 3:
Probability Bounds
Every probability is between 0 and 1.
Proof:
- Lower bound P(A) ≥ 0: this is Axiom 1 directly — all probabilities are non-negative.
- Upper bound P(A) ≤ 1: by the complement rule, P(A) = 1 − P(Aᶜ). Since Aᶜ is also an event, Axiom 1 gives P(Aᶜ) ≥ 0. Subtracting a non-negative number from 1 can only make it smaller or equal, so P(A) ≤ 1.
Inclusion-Exclusion Principle
For any two events A and B (not necessarily disjoint):
We subtract P(A ∩ B) because it was counted twice—once in P(A) and once in P(B).
Monotonicity
If A ⊆ B (every outcome in A is also in B):
In words: a larger event is at least as likely as a smaller one.
Proof: We can split B into two disjoint pieces — the outcomes in A, and the outcomes in B that are not in A (written B \ A). Since these are disjoint:
By Axiom 1, P(B \ A) ≥ 0, so P(B) ≥ P(A).
Empty Set
The impossible event has probability zero.
Proof: The empty set ∅ and the sample space Ω are disjoint (they share no outcomes), and their union is Ω itself (adding nothing changes nothing). By Axiom 3:
But Ω ∪ ∅ = Ω, so the left side is just P(Ω). This gives:
Subtracting P(Ω) from both sides: P(∅) = 0.
Example: Fair Dice
Consider rolling a fair six-sided dice. The sample space is:
Each outcome has probability 1/6. Let A = “roll an even number” = {2, 4, 6}.
Using the axioms:
This uses Axiom 3 (additivity) since {2}, {4}, {6} are disjoint.