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7.4 Definitions of Probability

We shall now consider two definitions of probability :

(1) Mathematical or a priori or classical.

(2) Statistical or empirical.

1. Mathematical (or A Priori or Classic) Definition

If there are ‘n’ exhaustive, mutually exclusive and equally likely cases and m of them are favorable to an event A, the probability of A happening is defined as the ratio m/n

Expressed as a formula :-

This definition is due to ‘Laplace.’ Thus probability is a concept which measures numerically the degree of certainty or uncertainty of the occurrence of an event.

For example, the probability of randomly drawing taking from a well-shuffled deck of cards is 4/52. Since 4 is the number of favorable outcomes (i.e. 4 kings of diamond, spade, club and heart) and 52 is the number of total outcomes (the number of cards in a deck).


If A is any event of sample space having probability P, then clearly, P is a positive number (expressed as a fraction or usually as a decimal) not greater than unity. 0 £ P £ 1 i.e. 0 (no chance or for impossible event) to a high of 1 (certainty). Since the number of cases not favorable to A are (n - m), the probability q that event A will not happen is, q = or q = 1 - m/n or q = 1 - p.

Now note that the probability q is nothing but the probability of the complementary event A i.e.

Thus p () = 1 - p or p () = 1 - p ()

so that p (A) + p () = 1 i.e. p + q = 1

Relative Frequency Definition

The classical definition of probability has a disadvantage i.e. the words ‘equally likely’ are vague. In fact, since these words seem to be synonymous with "equally probable". This definition is circular as it is defining (in terms) of itself. Therefore, the estimated or empirical probability of an event is taken as the relative frequency of the occurrence of the event when the number of observations is very large.


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"Verdana"7. 1 Introduction
7.2 Trial
7.3 Sample Space
7. 4 Definition of Probability
7. 5 The Laws of Probability
7. 6 Conditional Probability
7. 7 Theoretical Distribution
7. 8 Binomial Distribution
7. 9 Normal Distribution
Chapter 8

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