Page 116 - DMTH404_STATISTICS

Statistics

Notes
Example 49: An unbiased die is thrown 8 times. What is the probability of getting a six
in at least one of the throws?
Solution.

1
P A
Let A be the event that a six is obtained in the ith throw (i = 1, 2, ...... 8). Therefore, ( ) 
i i .
6
The event that a six is obtained in at least one of the throws is represented by (A  A  ....  A 8 ) .
1
2
Thus, we have
P (A  A  ....  A 8 ) 1 P A  ( 1  A  ....  A 8 )
1
2
2
Since A , A , ...... A are independent, we can write
1 2 8
8
æ
5ö
P (A  A  ....  A ) 1 P  ( ) ( ) . .... A .P A P ( ) 1A   ç ÷ .
2
8
1
2
8
1
è 6 ø
Example 50: Two students X and Y are very weak students of mathematics and their
chances of solving a problem correctly are 0.11 and 0.14 respectively. If the probability of their
making a common mistake is 0.081 and they get the same answer, what is the chance that their
answer is correct?
Solution.
Let A be the event that both the students get a correct answer, B be the event that both get
incorrect answer by making a common mistake and C be the event that both get the same
answer. Thus, we have
P (A C  ) P ( gets correct answer .X ) ( gets correct answerP Y )

´
 0.11 0.14  0.0154 (note that the two events are independent)
Similarly,
P (B C  ) P ( gets incorrect answerX ) ´ P ( gets incorrect answerY )

P ( and make a common mistakeX Y )
´
 (1 0.11 )(1 0.14 ) 0.081 0.062´ 
) , since A C b g and B C b
( )
Further, C  (A C ) (B C  ) or P C  P (A C ) P+ (B C g are
mutually exclusive. Thus, we have

P ( ) 0.0154 0.0620C  +  0.0774

We have to find the probability that the answers of both the students are correct given that they
are same, i.e.,

P (A  C ) 0.0154
)
P ( /A C    0.199
P C 0.0774
( )

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