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Unit 7: Modern Approach to Probability

7.1.1 Definition of Probability (Modern Approach) Notes

Let S be a sample space of an experiment and A be any event of this sample space. The probability
of A, denoted by P(A), is defined as a real value set function which associates a real value
corresponding to a subset A of the sample space S. In order that P(A) denotes a probability
function, the following rules, popularly known as axioms or postulates of probability, must be
satisfied.
Axiom I : For any event A in sample space S, we have 0  P(A)  1.
Axiom II : P(S) = 1.
Axiom III : If A , A , ...... A are k mutually exclusive events (i.e., A  A   , where  denotes
1 2 k i i j j

a null set) of the sample space S, then
k
( )
P (A  A 2 ......  A k )  å P A i
1
i 1
The first axiom implies that the probability of an event is a non-negative number less than or
equal to unity. The second axiom implies that the probability of an event that is certain to occur
must be equal to unity. Axiom III gives a basic rule of addition of probabilities when events are
mutually exclusive.
The above axioms provide a set of basic rules that can be used to find the probability of any
event of a sample space.

Probability of an Event

Let there be a sample space consisting of n elements, i.e., S = {e , e , ...... e }. Since the elementary
1 2 n
n
( ) .
events e , e , ...... e are mutually exclusive, we have, according to axiom III, ( ) S  å P e
P
1 2 n i
i 1
Similarly, if A = {e , e , ...... e } is any subset of S consisting of m elements, where m £ n, then
1 2 m
m
( ) . Thus, the probability of a sample space or an event is equal to the sum of
P A P e i
( )  å
i 1
probabilities of its elementary events.
It is obvious from the above that the probability of an event can be determined if the probabilities
of elementary events, belonging to it, are known.
The Assignment of Probabilities to various Elementary Events

The assignment of probabilities to various elementary events of a sample space can be done in
any one of the following three ways :
1. Using Classical Definition

We know that various elementary events of a random experiment, under the classical
definition, are equally likely and, therefore, can be assigned equal probabilities. Thus, if
there are n elementary events in the sample space of an experiment and in view of the fact

n 1
that ( ) S  å P e i = 1 (from axiom II), we can assign a probability equal to to every
P
( )
i 1 n
1
elementary event or, using symbols, we can write ( )  for i = 1, 2, .... n.
P e
i
n

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