Q.A probability function assigns $P(1) = 0.1$, $P(2) = 0.2$, $P(3) = 0.3$, $P(4) = 0.4$ to the outcomes of $S = \{1, 2, 3, 4\}$. Verify that this satisfies the axioms of probability, and find the probability of the event 'the outcome is even'.

Classical (a priori) definition. When a random experiment's outcomes are all equally likely, the classical or a priori probability of an event $A$ is

This is the everyday 'favourable over total' rule, and it silently needs two things: the outcomes must be equally likely, and there must be finitely many of them. Neither the coin-till-first-head experiment (infinite outcomes) nor a biased die (unequal chances) can be handled by this definition alone -- which is why the axiomatic approach was developed.

Axiomatic approach (Kolmogorov, 1933). Let $S$ be a finite sample space, $\mathcal P(S)$ the class of all events, and $P$ a real-valued function on $\mathcal P(S)$. $P(A)$ is a probability function exactly when it obeys three axioms:

• $[P_1]$ Non-negativity: $P(A)\ge 0$ for every event $A$.
• $[P_2]$ Additivity: for mutually exclusive $A,B$: $P(A\cup B)=P(A)+P(B)$ (and more generally, for mutually exclusive $A_1,\ldots,A_n$: $P(A_1\cup\cdots\cup A_n)=P(A_1)+\cdots+P(A_n)$).
• $[P_3]$ Normalisation: $P(S)=1$. …