Page 104 - DMTH404_STATISTICS

Statistics

Notes 2. Generalisation of Multiplicative Theorem :

If A, B and C are three events, then

P (A B C )  P ( ) ( /A .P B A ).P C ( / A  ) B ù û
é
ë
Similarly, for n events A , A , ...... A , we can write
1 2 n
.P
) ù
P (A  A  ...  A n ) P A ( ) (A 2 /A 1 ).P A 3 / (A  A 2 û
é
ë
1
1
1
2
... P A n / (A  A  ...  A n 1 û
é
)ù
ë
1
2
Further, if A , A , ...... A are independent, we have
1
2
n
1 b A g b g b g
P A  A  ...  n  P A P A .... b g.
.
P A
n
2
2
1

3. If A and B are independent, then A and B , A and B, A and B are also independent.
We can write P A B d i  P A b g b 

P A Bg (by theorem 3)

= P A a f- P A a f.P B a f = P A a f 1- P B a f = P A a f.P B d i, which shows that A and B are
independent. The other results can also be shown in a similar way.
4. The probability of occurrence of at least one of the events A , A , ...... A , is given by
1
2
n
1 d
1 b
P A  A  ....  A g   P A  A  ....  A ni .
1
n
2
2
If A , A , ...... A are independent then their compliments will also be independent, therefore,
1 2 n
the above result can be modified as
1 b
P A .... d
P A  A  ....  A g   P A d 1i d 2i P A ni .
1
.
n
2
Pair-wise and Mutual Independence
Three events A, B and C are said to be mutually independent if the following conditions are
simultaneously satisfied :
(A  ) B  P ( ) ( ), A .P B P (B C  ) P ( ) ( ), B .P C P (A C  ) P A .P C
( ) ( )
P
and P (A   C )  P A .P B .P C
B
( ) ( ) ( ) .
If the last condition is not satisfied, the events are said to be pair-wise independent.
From the above we note that mutually independent events will always be pair-wise independent
but not vice-versa.
Example 27: Among 1,000 applicants for admission to M.A. economics course in a
University, 600 were economics graduates and 400 were non-economics graduates; 30% of
economics graduate applicants and 5% of non-economics graduate applicants obtained admission.
If an applicant selected at random is found to have been given admission, what is the probability
that he/she is an economics graduate?
Solution.
Let A be the event that the applicant selected at random is an economics graduate and B be the
event that he/she is given admission.
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