Q.Explain the classical, relative-frequency and axiomatic approaches to probability. What is the main limitation of the classical approach?

Each of the three approaches assigns probability differently — by counting, by observing, or by axioms — and the classical approach's counting method only works under one specific condition.

Classical (a priori) approach: $P(A)=\dfrac{m}{n}$, computed by counting favourable outcomes ($m$) out of total possible outcomes ($n$), assuming all $n$ outcomes are equally likely — computable BEFORE the experiment is ever performed.

Relative-frequency (empirical) approach: $P(A)=\lim_{n\to\infty}\dfrac{f}{n}$, estimated by actually repeating the experiment a large number of times and observing how often $A$ occurs — used precisely when outcomes cannot be assumed equally likely.

Axiomatic approach: defines probability as any function $P$ satisfying three rules — $0\le P(A)\le1$; $P(S)=1$; and additivity, $P(A_1\cup A_2\cup\cdots)=P(A_1)+P(A_2)+\cdots$ for mutually exclusive events — without reference to counting or repeated trials at all; every formula used elsewhere in the chapter can be derived from these three axioms.

Main limitation of the classical approach: it presumes the outcomes are equally likely, which is often untrue or unverifiable in practice — e.g. it cannot assign a meaningful probability to 'this specific patient recovers' or 'a biased coin shows heads', because there is no valid basis for treating the outcomes as equally likely in either case.

A common answer wrongly claims the classical approach 'always works' — it fails whenever equal likelihood cannot genuinely be assumed, which is exactly why the relative-frequency approach exists as an alternative.