Page 23 - DMTH404_STATISTICS

Unit 2: Methods of Enumeration of Probability

Now look carefully at the probabilities attached to the sample points in Example 1 (i) and Notes
(ii). Did you notice that
(i) these are number’s between 0 and 1, and
(ii) the sum of the probabilities of all the sample points is one ?

This is not true of this example alone. In general, we have the following rule or axiom about the
assignment of probabilities to the points of a discrete sample space.
Axiom : Let  be a discrete samplk space containing the points  ,  , . . . ; i.e.,
1 2
 = { ,  , .... }.
1 2
To each point  of , assign a number P{}, 0  P{}  1, such that
j j j
P{ } + P{ } + ..... = 1. . . . (1)
1 2
We call P{}, the probability of .
j j
Now see if you can do the following exercise on the basis of this axiom.
If you have done El, you would have noticed that it is possible to have more than one valid
assignment of probabilities to the same sample space. If the discrete sample space is not finite,
the left side of Equation (1) should be interpreted as an infinite series. For example, suppose
 = {  ,  , ... } and
1 2
j
P{} = 1 / 2 ,  j =l , 2, .......
j
Then this assignment is valid because, 0  P{ }  1, and
j
2 3
1  1   1 
P{ } + P{ } + ... =    ...
1 2    
2  2   2 
1  1 1 
=  1    .... 
2  2 2 2 
= 1

So far we have not explained what the probability P {} assigned to the point oj signifies. We
j
have just said that they are all arbitrary numbers between 0 and 1, except for the requirement
that they add up to 1. In fact, we have not even tried to clarify the nature of the sample space
except to assert that it be a discrete sample space. Such an approach is consistent with the usual
procedure of beginning the study of a mathematical discipline with a few undefined notions and
axioms and then building a theory based on the laws of logic (Remember the axioms of
geometry?). It is for this reason that this approach to the specification of probabilities to discrete
sample spaces is called the axiomatic approach. It was introduced by the Russian mathematician
A.N. Kolmogorov in 1933. This approach is mathematically precise and is now universally
accepted. But when we try to use the mathematical theory of probability to solve some real life
problems, that we have to interpret the significance of statements like “The probability of an
event A is 0.6.”
We now define the probability of an event A for a discrete sample space.

2.1.1 Probability of an Event : Definition

Let  be a discrete sample space consisting of the points  ,  , . . . , finite or infinite in number.
1 2
Let P{ }, P{ }, . . . be the probabilities assigned to the points  ,  , . . .
1 2 1 2

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