3. Axiomatic Approach to probability

Axiomatic Approach to probability:

Axiomatic probability is a unifying probability theory in Mathematics. The axiomatic approach to probability sets down a set of axioms that apply to all of the approaches of probability which includes frequentist probability and classical probability. These rules are generally based on Kolmogorov's Three Axioms.

Axiomatic approach is another way of describing probability of an event. In this approach some axioms or rules are depicted to assign probabilities.

Let S be the sample space of a random experiment.

The probability P is a real valued function whose domain is the power set of S and range is the interval [0,1] satisfying the following axioms

1. For any event E, P (E) ³ 0
2.  (ii) P (S) = 1
3. If E and F are mutually exclusive events, then P(E È F) = P(E) + P(F).

P (E È f) = P (E) + P (f)

 P(E) = P(E) + P (f)

i.e. P (f) = 0.

Let S be a sample space containing outcomes w₁, w₂, ..., w_n .

i.e., S = {w₁, w₂, ..., w_n}

It follows from the axiomatic definition of probability that

(i) 0 £ P (w_i) £ 1 for each w_i Î S

(ii) P (w₁) + P (w₂) + ... + P (w_n) = 1

(iii) For any event A, P(A) = å P(w_i ), w_i Î A.

Example: A coin tossing

S={H,T}

P(H)= ½  ,P(T) = ½

i.e., each number is neither less than zero nor greater than 1

and P(H) + P(T) = ½ + ½  = 1

In fact, we can assign the numbers p and (1 – p) to both the outcomes such that

0 £ p £ 1 and P(H) + P(T) = p + (1 – p) = 1

This assignment, too, satisfies both conditions of the axiomatic approach of probability.

Hence, we can say that there are many ways (rather infinite) to assign probabilities to outcomes of an experiment.