Axiomatic Approach to Probability

The Axiomatic Approach to Probability

In earlier sections, you learned about random experiments, sample spaces, and events. But how do we actually assign a number — a probability — to an event? The classical approach you may have seen before works only when all outcomes are equally likely. The axiomatic approach is far more general: it lays down a small set of rules (axioms) that any valid probability assignment must satisfy, and then builds everything else from those rules.

Think of it like the rules of a game. The axioms don't tell you which numbers to assign to outcomes — they only tell you what constraints those numbers must obey. Within those constraints, many different assignments are possible.

The Three Axioms of Probability

Let $S$ be the sample space of a random experiment. Probability $P$ is a real-valued function whose domain is the power set of $S$ (that is, the set of all possible events, including $S$ itself and the empty set $\phi$) and whose range is the interval $[0, 1]$. The function $P$ must satisfy three axioms:

Axiom I says probabilities are never negative. Axiom II says the sample space — the certain event — has probability 1. Axiom III is the addition rule for disjoint events: if two events cannot happen together, the probability that at least one of them occurs is simply the sum of their individual probabilities.

A Crucial Consequence: Probability of the Empty Set

From Axiom III, we can immediately derive that $P(\phi) = 0$. Here is the reasoning.

Take any event $E$. Notice that $E$ and $\phi$ are mutually exclusive — they have no outcomes in common because $\phi$ contains nothing. So by Axiom III:

But $E \cup \phi = E$ (union with the empty set leaves $E$ unchanged). Therefore:

Subtracting $P(E)$ from both sides gives $P(\phi) = 0$.

This makes intuitive sense: the empty event — an event that can never occur — must have probability zero.

Assigning Probabilities to Individual Outcomes

Now consider a finite sample space $S = \{\omega_1, \omega_2, \ldots, \omega_n\}$, where each $\omega_i$ is a single outcome. The axioms lead to three important conditions for how we assign probabilities to these individual outcomes:

Condition 1 follows from Axiom I (non-negativity) and the fact that probabilities cannot exceed 1 (since the total probability of all outcomes is 1, no single outcome can exceed that total). Condition 2 follows from Axiom II and Axiom III: since the outcomes are mutually exclusive (only one can occur in a single trial), the probability of their union — which is $S$ itself — must equal the sum of their individual probabilities, and that sum must be 1. Condition 3 is the general rule: the probability of any event is the sum of the probabilities of the individual outcomes that make up that event.

Why This Approach Allows Many Assignments

The axiomatic approach does not force a unique probability assignment. Consider a coin toss. One valid assignment is:

Both conditions are satisfied: each probability is between 0 and 1, and their sum is 1. But another valid assignment is:

This also satisfies both conditions. In fact, for any number $p$ such that $0 \leq p \leq 1$, the assignment $P(H) = p$ and $P(T) = 1-p$ is valid. There are infinitely many possible assignments.

The choice of which assignment to use depends on the real-world situation. For a fair coin, we choose $p = \frac{1}{2}$. For a biased coin, we might choose a different $p$. The axioms only tell us what is allowed — they do not tell us what is correct for a given experiment.

Checking Validity of Probability Assignments

To check whether a given assignment of probabilities to outcomes is valid, you must verify two things:

1. Each individual probability $P(\omega_i)$ lies in the interval $[0, 1]$.
2. The sum of all probabilities equals exactly 1.

If either condition fails, the assignment is invalid.

Let us work through the examples from the textbook. Consider a sample space $S = \{\omega_1, \omega_2, \omega_3, \omega_4, \omega_5, \omega_6\}$ with the following proposed assignments: