Page 111 - DMTH404_STATISTICS

Unit 7: Modern Approach to Probability

Notes
7 2 7 2 53
 (A  ) B  +  ´ 
P
16 5 16 5 80
Alternative Method :

9 3 53
(A  B ) 1  ´ 
P
16 5 80
Example 38: If A and B are two events such that P A B b g  , find P(B), P A B b g ,
1
3
P(A/B), P(B/A), P A  d Bi , P A  d Bi and P B d i. Also examine whether the events A and B are
: (a) Equally likely, (b) Exhaustive, (c) Mutually exclusive, and (d) Independent.
Solution.

The probabilities of various events are obtained as follows:
P B b g d  Bi b  1 + 1  1
+
P A Bg 

P A
6 3 2
 b 2 1 1 5
P A Bg  +  
3 2 3 6
/ b P A Bg 1 2 2
b 
P A Bg  P B b g  3 ´ 1  3
b 
/ b P A Bg 1 3 1
P B Ag  P A b g  3 ´ 2  2

P Ai b
P A  d Bi d + P Bg d Bi  1 + 1  1  2
P A 


3 2 6 3
P A  d Bi   P A Bg   5  1
 b
1
1
6 6
1 1
P B d i =1- P B a f =1- =
2 2
(a) Since P(A)  P(B), A and B are not equally likely events.
Since P A B b
(b) g  1, A and B are not exhaustive events.
Since P A B b
(c) g  0 , A and B are not mutually exclusive.
b g

(d) Since P A P B b g b 
P A Bg , A and B are independent events.
Example 39: Two players A and B toss an unbiased die alternatively. He who first
throws a six wins the game. If A begins, what is the probability that B wins the game?
Solution.

Let A and B be the respective events that A and B throw a six in i th toss, i = 1, 2, .... . B
i i
will win the game if any one of the following mutually exclusive events occur:
A B or A B A B 2 or A B A B A B 3 , etc.
1 1
1 1
1 1
3
2
2
2
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