Page 112 - DMTH404_STATISTICS

Statistics

Notes 5 1 5 5 5 1 5 5 5 5 5 1
Thus, (B winsP )  ´ + ´ ´ ´ + ´ ´ ´ ´ ´ + ......
6 6 6 6 6 6 6 6 6 6 6 6

5 é æ 5ö 2 æ 5ö 4 ù 5 1 5
 ê 1+ ç ÷ + ç ÷ + ......  ´ 2 
ú
36 ë è 6 ø è 6 ø û 36 æ 5ö 11
1 ç ÷
è 6 ø

Example 40: A bag contains 5 red and 3 black balls and second bag contains 4 red and 5
black balls.
(a) If one ball is selected at random from each bag, what is the probability that both of them
are of same colour?
(b) If a bag is selected at random and two balls are drawn from it, what is the probability that
they are of (i) same colour, (ii) different colours?
Solution.

é Probability that ball ù é Probability that balls ù
+
(a) a a fRequired Probability  ê ú ê ú
ë from both bags are red û ë from both bags are black û
5 4 3 5 35
 ´ + ´ 
8 9 8 9 72

(b) Let A be the event that first bag is drawn so that A denotes the event that second bag is
drawn. Since the two events are equally likely, mutually exclusive and exhaustive, we
1
have P A a f = P A d i = .
2
(i) Let R be the event that two drawn balls are red and B be the event that they are black.
The required probability is given by
)
 P A /A ) P+ ( /B A )] ( ) ( /P A P R A+ é ) ( /P B A ù
+
( ) ( [P R
ë û
5
3
1 é 5 C + C ù 1 é 4 C + C ù 1 10 3 ù 1 6 10 ù 229
+
+
é
é
 ê 2 2 ú + ê 2 2 ú  + 
2 ë 8 C 2 û 2 ë 9 C 2 û 2 ë ê 28 û ú 2 ë ê 36 û ú 504
(ii) Let C denote the event that the drawn balls are of different colours. The required
probability is given by
P C P ( ) ( /A P C A ) ( ) ( /P A P C A+ )
( ) 
1 5 3ù 1 4 5ù 1 15 20 ù 275
é ´
é ´
é
 ê ú + ê ú  ê + ú 
2 ë 8 C 2 û 2 ë 9 C 2 û 2 28 36û 504
ë
Example 41: There are two urns U and U . U contains 9 white and 4 red balls and U
1 2 1 2
contains 3 white and 6 red balls. Two balls are transferred from U to U and then a ball is drawn
1 2
from U . What is the probability that it is a white ball?
2
Solution.
Let A be the event that the two transferred balls are white, B be the event that they are red and
C be the event that one is white and the other is red. Further, let W be the event that a white ball

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