Page 96 - DMTH404_STATISTICS

Statistics

Notes Further, if there are m elementary events in an event A, we have,

in
of
1 1 1 m n ( ), . .,A i e number elements A
( )  + + ...... + ( timesm ) 
P A
in
n n n n n ( ), . .,S i e number elements S
of
We note that the above expression is similar to the formula obtained under classical
definition.
2. Using Statistical Definition
Using this definition, the assignment of probabilities to various elementary events of a
sample space can be done by repeating an experiment a large number of times or by using
the past records.
3. Subjective Assignment
The assignment of probabilities on the basis of the statistical and the classical definitions
is objective. Contrary to this, it is also possible to have subjective assignment of
probabilities. Under the subjective assignment, the probabilities to various elementary
events are assigned on the basis of the expectations or the degree of belief of the statistician.
These probabilities, also known as personal probabilities, are very useful in the analysis
of various business and economic problems where it is neither possible to repeat the
experiment nor the outcomes are equally likely.
It is obvious from the above that the Modern Definition of probability is a general one
which includes the classical and the statistical definitions as its particular cases. Besides
this, it provides a set of mathematical rules that are useful for further mathematical
treatment of the subject of probability.

7.2 Theorems on Probability

Theorem 1.

0
P ( )   , where  is a null set.
Proof.
For a sample space S of an experiment, we can write S    S .
Taking probability of both sides, we have P S g b
b

P Sg .
Since S and  are mutually exclusive, using axiom III, we can write
P(S) + P(f ) = P(S). Hence, P f b g =0.
Theorem 2.

P A d i =1- P A a f, where A is compliment of A.
Proof.
Let A be any event in the sample space S. We can write
P Sg
A  A  S or Pd A  i b
A 
Since A and A are mutually exclusive, we can write

P A a f+P A d i = P S a f =1. Hence, P A d i =1- P A a f.

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