The Three Approaches to Probability: Classical, Relative Frequency and Axiomatic

There are three distinct, standard ways of assigning a probability to an event, and the Gujarat board Std 12 Statistics syllabus expects a student to be able to state, apply, and compare all three.

(A) Classical (Mathematical / A Priori) Approach

If a random experiment has $n$ mutually exclusive, exhaustive and equally likely outcomes, and $m$ of them are favourable to an event $A$, then:

This is called 'a priori' because the probability can be worked out before the experiment is ever performed, purely by counting. Its central limitation: it only works when the outcomes can genuinely be assumed equally likely — it says nothing about a biased coin, a loaded die, or a real-world event like 'it rains tomorrow'.

(B) Relative Frequency (Empirical / Statistical) Approach

When outcomes are NOT equally likely (or that assumption cannot be verified), probability is instead estimated by actually repeating the experiment a very large number of times and observing how often the event occurs:

where $f$ is the number of times event $A$ occurred in $n$ trials. In practice $n$ cannot literally be taken to infinity, so $f/n$ for a sufficiently large, fixed $n$ is used as a practical estimate — this is the approach behind quality-control defect rates, insurance mortality tables, and weather-forecast probabilities.

(C) Axiomatic Approach

The modern, rigorous foundation (due to Kolmogorov) defines probability not by a single formula but by three axioms that any valid probability measure $P$ on a sample space $S$ must satisfy:

1. $0 \le P(A) \le 1$ for every event $A$.
2. $P(S) = 1$ (something in the sample space is certain to happen).
3. For mutually exclusive events $A_1, A_2, A_3, \ldots$: $P(A_1\cup A_2\cup A_3\cup\cdots)=P(A_1)+P(A_2)+P(A_3)+\cdots$ …