Axiomatic Probability

Once outcomes and events are in place, we attach a number — the probability — to each event, measuring how likely it is. The axiomatic approach, due to the Russian mathematician A. N. Kolmogorov, fixes a few simple rules that any sensible probability must obey, and everything else in the chapter follows from them.

For a sample space $S$ and any event $E$, the axioms of probability are:

In words: (1) every probability lies between $0$ and $1$, (2) the sure event has probability $1$, and (3) for mutually exclusive events the probabilities simply add. From these the impossible event follows immediately: $P(\varnothing) = 0$.

A clean way to assign probabilities is to give each elementary event $\{\omega_i\}$ a number $p_i$ with $0 \le p_i \le 1$ and $p_1 + p_2 + \dots + p_n = 1$; the probability of any event is then the sum of the $p_i$ of the outcomes it contains. This single requirement — that the probabilities of all outcomes add to $1$ — is the practical heart of the axioms.

When all outcomes are equally likely — a fair coin, a fair die, a well-shuffled pack — each $p_i = \dfrac{1}{n(S)}$, and the probability of an event reduces to a ratio of counts. This is the classical (Laplace) definition you will use most:

Notice that this formula only applies when the outcomes are equally likely. A loaded die or a bent coin breaks the assumption, and then you must use observed (empirical) frequencies instead.

A constant companion of any event is its complement $\overline{A}$, the event "$A$ does not occur". Since $A$ and $\overline{A}$ are mutually exclusive and together exhaust $S$, axiom (3) with $A \cup \overline{A} = S$ gives:

This complement rule is often the fastest route to an answer: when "at least one" appears in a question, computing $1 - P(\text{none})$ is usually far shorter than counting the favourable cases directly.

Two further consequences of the axioms are worth stating. For any two events (not necessarily disjoint) the probability of "$A$ or $B$" is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$, the subtraction removing the doubly-counted overlap (this is the addition theorem, developed fully in the next topic). And monotonicity: if $A \subseteq B$ then $P(A) \le P(B)$ — a larger event is at least as likely.

Odds. Examiners sometimes phrase likelihood as odds. If an event has $m$ favourable and $k$ unfavourable equally likely outcomes, the odds in favour are $m : k$ and the odds against are $k : m$. Converting back, $P(A) = \dfrac{m}{m+k}$. For instance, odds in favour of $3 : 2$ mean $P(A) = \dfrac{3}{5}$.

Deeper Insight — why axioms beat a single "definition": Earlier you may have met probability as nothing more than favourable over total. That classical formula is genuinely useful, but it quietly assumes the outcomes are equally likely — an assumption that fails for a weighted coin, an unfair die, or a continuous quantity like a person's exact height. The axiomatic framework sidesteps this by not committing to how the numbers arise; it only demands that whatever numbers we assign are non-negative, sum to $1$ over the whole sample space, and add up over disjoint events. The familiar $\tfrac{n(A)}{n(S)}$ is then just one way to satisfy the axioms — the way that happens to work when symmetry makes every outcome equally likely. Because the complement rule, the addition theorem, monotonicity, and every later result are all derived from these axioms, learning to reason from them — rather than memorising isolated formulas — is what makes harder problems tractable.