Page 115 - DMTH404_STATISTICS

Unit 7: Modern Approach to Probability

Notes
Example 47: Computers A and B are to be marketed. A salesman who is assigned the job
of finding customers for them has 60% and 40% chances respectively of succeeding in case of
computer A and B. The two computers can be sold independently. Given that the salesman is
able to sell at least one computer, what is the probability that computer A has been sold?
Solution.
Let A be the event that the salesman is able to sell computer A and B be the event that he is able
to sell computer B. It is given that P(A) = 0.6 and P(B) = 0.4. The probability that the salesman is
able to sell at least one computer, is given by

( ) P B+
P (A  ) B  P A ( ) P (A  ) B  P A ( ) P A ( ) ( ) B
( ) P B+
.P
(note that A and B are given to be independent)
 0.6 0.4 0.6 0.4  0.76
´

+
Now the required probability, the probability that computer A is sold given that the salesman
is able to sell at least one computer, is given by
0.60
P ( /A A  B )   0.789
0.76

Example 48: Two men M and M and three women W , W and W , in a big industrial
1 2 1 2 3
firm, are trying for promotion to a single post which falls vacant. Those of the same sex have
equal probabilities of getting promotion but each man is twice as likely to get the promotion as
any women.

(a) Find the probability that a woman gets the promotion.
(b) If M and W are husband and wife, find the probability that one of them gets the promotion.
2 2
Solution.
Let p be the probability that a woman gets the promotion, therefore 2p will be the probability
that a man gets the promotion. Thus, we can write, P(M ) = P(M ) = 2p and P(W ) = P(W ) = P(W )
1 2 1 2 3
= p, where P(M ) denotes the probability that i th man gets the promotion (i = 1, 2) and P(W)
i j
denotes the probability that j th woman gets the promotion.
Since the post is to be given only to one of the five persons, the events M , M , W , W and W are
1 2 1 2 3
mutually exclusive and exhaustive.
 P (M  M  W  W  W 3 ) P (M 1 ) P+ (M 2 ) P W+ ( ) P W+ ( ) P+ ( ) 1W 
1
2
3
1
2
2
1
1
+
+
+
Þ 2p + 2p p p p  1 or p 
7
(a) The probability that a woman gets the promotion
1 b 3
P W  W  g b g b g b g 
W 
P W +
P W +
P W
2
3
3
1
2
7
(b) The probability that M or W gets the promotion
2
2
2 b 3
W 
P M  g b g b g 
P W
P M +
2
2
2
7
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