Q.A letter is taken at random from the letters of the word 'ASSISTANT' and another letter is taken at random from the letters of the word 'STATISTICS'. The probability that the selected letters are the same is

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: $P(S)=1$. …