Page 103 - DMTH404_STATISTICS

Unit 7: Modern Approach to Probability

For the event B/A, the sample space is the set of elements in A and out of these the number of Notes
cases favourable to B is given by n A B b g .

( n A 
 P ( /B A  ) B .
)
( )
n A
If we multiply the numerator and denominator of the above expression by n(S), we get
( n A  ) B n ( ) S P (A  B )
)
P ( /B A  ´ 
( )
( )
n A n ( ) S P A
or P (A  ) B  P ( ) ( /A .P B A ) .
The other result can also be shown in a similar way.

Notes To avoid mathematical complications, we have assumed that the elementary
events are equally likely. However, the above results will hold true even for the cases
where the elementary events are not equally likely.

(b) Multiplicative Theorem for Independent Events
If A and B are independent, the probability of their simultaneous occurrence is given by
P A Bg b g b
 b
.

P A P Bg .
Proof.
We can write A b A B  g d A Bi .


Since A B b g and A B d i are mutually exclusive, we have
P A Bg d 
P A b g b  + P A Bi (by axiom III)

d
a
= P B a f.P A/Bf+P B d i.P A/Bi
If A and B are independent, then proportion of A's in B is equal to proportion of A's in B ’s, i.e.,
d
a
P A/Bf = P A/Bi.
Thus, the above equation can be written as
a
a
P A a f = P A/Bf P B a f+P B d i = P A/Bf
Substituting this value in the formula of conditional probability theorem, we get
P A Bg b g b
 b

.
P A P Bg .
Remarks:
The addition theorem is used to find the probability of A or B i.e. P(A  B), where as multiplicative
theorem is used to find the probability of A and B i.e. P(A Ç B).
Corollaries :
1. (i) If A and B are mutually exclusive and P(A).P(B) > 0, then they cannot be independent
since P A B b g  0 .

(ii) If A and B are independent and P(A).P(B) > 0, then they cannot be mutually exclusive
since P A B b g  0 .

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