Probability

Definition

A probability is a number attached to an event, between 0 and 1, measuring the share it occupies among all possible outcomes.

The set of those outcomes forms the sample space, written $\Omega$Ω. An event is a subset of it: "the die shows an even number" is the subset $\{2, 4, 6\}${2,4,6}.

The three axioms

The whole calculus of probability rests on three statements, and on nothing else:

The third extends to countable unions, which suffices to handle infinite sample spaces.

What follows from them

These three rules are not an arbitrary starting point: they generate the usual identities.

$P(\bar{A}) = 1 - P(A) \qquad P(\emptyset) = 0 \qquad P(A \cup B) = P(A) + P(B) - P(A \cap B)$P(Aˉ)=1−P(A)P(∅)=0P(A∪B)=P(A)+P(B)−P(A∩B)

The last one corrects double counting: adding $P(A)$P(A) and $P(B)$P(B) counts twice whatever belongs to both, so it must be subtracted once. Additivity alone applies only to disjoint events.

Conditioning, and independence

Knowing that $B$B occurred changes the share occupied by $A$A. The sample space is then restricted to $B$B:

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0$P(A∣B)=P(B)P(A∩B)​,P(B)>0

Two events are independent when this restriction changes nothing: $P(A \mid B) = P(A)$P(A∣B)=P(A), which is equivalent to $P(A \cap B) = P(A)\,P(B)$P(A∩B)=P(A)P(B).

Independence is a numerical property, not an intuition. Two events can have no causal link and still be dependent, and the converse occurs too.

What the axioms do not say

The formalism fixes the rules of calculation and says nothing about their meaning. Two readings coexist: the frequentist, where a probability is the limit of a frequency over repetitions; the Bayesian, where it measures a revisable degree of belief.

Both obey the same three axioms and produce the same numbers. Their disagreement bears on interpretation, not on the calculation.

Summary

Three axioms found the whole calculus of probability, and none of them says what a probability means.