Page 108 - DMTH404_STATISTICS

Statistics

Also, n S b g b A g d A i d A i d
Notes  n A  2 + n A  2 + n A  2 + n A  A i  12
1
1
1
2
1
Writing the given information in the form of a nine-square table, we get
A A Total
2 2
A 1 4 4 8
A 2 2 4
1
Total 6 6 12
(a) The probability of drawing at least a red ball is given by

( n A  A 2 ) 2 5
1
P (A  A 2 ) 1   1  
1
n ( ) S 12 6
d
We have to find P A /A i
(b) 2 1
( n A  A 2 ) 4 1
1
P ( A 2 /A 1 )   
( )
n A 1 8 2
A and A will be independent if P A  g b g b g
1 b
.
(c) A  P A P A 2
2
1
1 2
( n A  A ) 4 1
Now P (A  A )  1 2  
1
2
n ( ) S 12 3
( ) ( )
n A n A 8 6 1
P A 1 .P A 2 1 . 2  ´ 
( ) ( ) 
n ( ) S n ( ) S 12 12 3
Hence, A and A are independent.
1 2
Example 33: An urn contains 3 red and 2 white balls. 2 balls are drawn at random. Find
the probability that either both of them are red or both are white.
Solution.
Let A be the event that both the balls are red and B be the event that both the balls are white.
Thus, we can write

n ( ) S  5 C  10, n A 3 C  3, n ( ) B  2 C  1, also n (A  B ) 0
( ) 
2
2
2
n A ( ) 3 1 2
+
( ) n B+
 The required probability is (A  ) B   
P
n ( ) S 10 5
Example 34: A bag contains 10 red and 8 black balls. Two balls are drawn at random.
Find the probability that (a) both of them are red, (b) one is red and the other is black.
Solution.
Let A be the event that both the balls are red and B be the event that one is red and the other is
black.
18
Two balls can be drawn from 18 balls in C equally likely ways.
2
18!
 n ( ) S  18 C   153
2
2!16!

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