Page 99 - DMTH404_STATISTICS

Unit 7: Modern Approach to Probability

6. The probability of occurrence of exactly one of the three events can be written as Notes
 d
P A B  i A  B C  i d A  B  Ci = P(at least one of the three events occur) -
C d

P(at least two of the three events occur).
b
 P A b g b + P Cg 2 b   P B Cg 2 b  + P A B Cg .
+ P Bg b

b  
 P A Bg 3
 P A Cg 3
Example 23: In a group of 1,000 persons, there are 650 who can speak Hindi, 400 can
speak English and 150 can speak both Hindi and English. If a person is selected at random, what
is the probability that he speaks (i) Hindi only, (ii) English only, (iii) only one of the two
languages, (iv) at least one of the two languages?
Solution.
Let A denote the event that a person selected at random speaks Hindi and B denotes the event
that he speaks English.
Thus, we have n(A) = 650, n(B) = 400, n A B b g  150 and n(S) = 1000, where n(A), n(B), etc.

denote the number of persons belonging to the respective event.
(i) The probability that a person selected at random speaks Hindi only, is given by

( )
n A ( n A  ) B 650 150 1
P (A  B )     
n ( ) S n ( ) S 1000 1000 2
(ii) The probability that a person selected at random speaks English only, is given by
n A Bg

P A  d Bi  n B b g b n S b g  1000  1000  1
150
400
n S b g

4
(iii) The probability that a person selected at random speaks only one of the languages, is
given by
 b
 d
P A B  i d A  Bi  P A b g b P A Bg (see corollary 2)
P Bg  2
+
n A b g + n B b g b  650 + 400 300 3
 n A Bg
2

  
n S b g 1000 4
(iv) The probability that a person selected at random speaks at least one of the languages, is
given by
 b 650 + 400 150 9

P A Bg  
1000 10
Alternative Method
The above probabilities can easily be computed by the following nine-square table :
B Total
B
A 150 500 650
A 250 100 350
Total 400 600 1000

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