Page 107 - DMTH404_STATISTICS

Unit 7: Modern Approach to Probability

(c) exactly two of them are able to solve the problem. Notes

(d) exactly one of them is able to solve the problem.
Solution.
Let A be the event that student A solves the problem. Similarly, we can define the events B and
C. Further, A, B and C are given to be independent.
(a) The problem is solved if at least one of them is able to solve it. This probability is given by
1 2 3 3
P (A   C ) 1 P  ( ) ( ) ( ) 1A .P B .P C   ´ ´ 
B
2 3 4 4
Here we have to find P A B b  g b B C  g b A Cg
(b)  
) 
P é ë (A  B ) (B  C ) (A  C ù û P A P B ( ) ( ) P A P C+ ( ) ( )
( ) ( ) P B P C+
( ) ( ) ( )
 2P A P B P C
1 1 1 1 1 1 1 1 1 7
 ´ + ´ + ´  2. ´ ´ 
2 3 3 4 2 4 2 3 4 24
C d
The required probability is given by P A B C  i d A B  i B Ci
d
(c)  A  
( ) ( ) ( )
( ) ( ) B +
( ) ( ) P+
 P A .P P B .P C ( ) ( ) 3A .P C  P A .P B .P C
1 1 1 1 1
 + +   .
6 12 8 8 4
The required probability is given by P A B d  C  i d A  B C  i d A  B  Ci

(d)
( ) ( )
( ) P B+
 P A ( ) P+ ( ) 2C  P ( ) ( ) 2A .P B  P B .P C
( ) ( ) ( )
 2P ( ) ( ) 3A .P C + P A .P B .P C
1 1 1 1 1 1 1 11
 + +    +  .
2 3 4 3 6 4 8 24
Note that the formulae used in (a), (b), (c) and (d) above are the modified forms of corollaries
(following theorem 4) 3, 4, 5 and 6 respectively.

Example 32: A bag contains 2 red and 1 black ball and another bag contains 2 red and 2
black balls. One ball is selected at random from each bag. Find the probability of drawing (a) at
least a red ball, (b) a black ball from the second bag given that ball from the first is red; (c) show
that the event of drawing a red ball from the first bag and the event of drawing a red ball from
the second bag are independent.
Solution.
Let A be the event of drawing a red ball from the first bag and A be the event of drawing a red
1 2
ball from the second bag. Thus, we can write:
1 b 1 d 2i
´
A
n A  g  2 2  4, n A  A  2 2  4,
´
2
1 d
2i
1 d
2i
n A  A  1 2  2, n A  A  1 2  2
´
´
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