JEE Mathematics / Lakshmikanta Satapathy / Probability / Algebra of Events / Mutually exclusive and Exhaustive Events

1. Physics Helpline
L K Satapathy
Probability Theory 4

2. Physics Helpline
L K Satapathy
Algebra of Events : Two or more events can be combined in a manner
similar to the combination of two or more sets
(Union , Intersection , difference , complement etc.)
Complementary Event: For every event A , there is another event A,
which is the complement of event A (also called ‘not A’)
Consider the experiment of tossing three coins.
 S = { HHH , HHT , HTH , HTT , THH , THT , TTH , TTT }
Consider the event ‘ only one tail occurs ’  A = { HHT , HTH , THH }
For every outcome of the experiment which is not in A , we say that
A has not occurred. Which means that ‘not A’ ( or A) has occurred
 A = { HHH , HTT , THT , TTH , TTT }
Or A = {  :   S and  A } = S – A
Probability Theory 4

3. Physics Helpline
L K Satapathy
If A , B are two events associated with a sample space S ,
then AB is the event ‘A or B’ [ either A or B or both ]
Event A or B :
Consider the events A = an even number occurs  A = { 2 , 4 , 6 }
and B = a multiple of 3 occurs  B = { 3 , 6 }
Example : The experiment of throwing a die.  S = { 1 , 2 , 3 , 4 , 5 , 6 }
 The event A or B = A  B = {  : A or B}
The event ‘an even number or a multiple of 3 occurs’ = the event A or B
 The event ‘A or B’ = A  B = { 2 , 3 , 4 , 6 }
 When the outcome of the throw is either 2 , 3 , 4 or 6 ,
we say that the event A or B has occurred
Probability Theory 4

4. Physics Helpline
L K Satapathy
If A , B are two events associated with a sample space S ,
then A  B is the event ‘A and B’
Event A and B :
Consider the events A = an odd number occurs  A = { 1 , 3 , 5 }
and B = a prime number occurs  B = { 2 , 3 , 5 }
Example : The experiment of throwing a die.  S = { 1 , 2 , 3 , 4 , 5 , 6 }
The event A and B = A  B = {  : A and B }
The event ‘an odd prime number occurs’ = the event A and B
 The event ‘A and B’ = A  B = { 3 , 5 }
 When the outcome of the throw is either 3 or 5 ,
we say that the event ‘A and B’ has occurred
Probability Theory 4

5. Physics Helpline
L K Satapathy
If A , B are two events associated with a sample space S ,
then A – B = A  B is the event ‘A but not B’
Event A but not B :
Consider the events A = an odd number occurs  A = { 1 , 3 , 5 }
and B = a prime number occurs  B = { 2 , 3 , 5 }
Example : The experiment of throwing a die.  S = { 1 , 2 , 3 , 4 , 5 , 6 }
The event A but not B = A  B = {  : A and B }
The event ‘an odd number which is not prime occurs’ = A but not B
 The event ‘ A but not B ’ = A  B = A – B = { 1 }
 When the outcome of the throw is 1 ,
we say that the event ‘A but not B’ has occurred
Probability Theory 4

6. Physics Helpline
L K Satapathy
Two events A and B of a random experiment are said to be mutually
exclusive , if the occurrence of any one of them excludes the
occurrence of the other (both cannot occur simultaneously) .
Mutually exclusive events :
Consider the events A = an odd number occurs  A = { 1 , 3 , 5 }
B = an even number occurs  B = { 2 , 4 , 6 }
Example : The experiment of throwing a die.  S = { 1 , 2 , 3 , 4 , 5 , 6 }
 A  B =   A and B are mutually exclusive events
and C = a number less than 4 occurs  C = { 1 , 2 , 3 }
A  C = { 1 , 3 }    A and C are not mutually exclusive events
B  C = { 2 }    B and C are not mutually exclusive events
 A  B = 
Probability Theory 4

7. Physics Helpline
L K Satapathy
A number of events of a sample space are said to be exhaustive if
their union produces the sample space.
Exhaustive Events :
Consider n events of a sample space S .1 2, , . . . , nE E E
1 2
1
. . .
n
n i
i
E E E E S

     They are said to be exhaustive if
Consider the experiment of throwing a die.  S = { 1 , 2 , 3 , 4 , 5 , 6 }
Consider the events A = an odd number occurs  A = { 1 , 3 , 5 }
B = a prime number occurs  B = { 2 , 3 , 5 }
C = a number greater than 3 occurs  C = { 4 , 5 , 6 }
We observe that A  B  C = { 1 , 2 , 3 , 4 , 5 , 6 } = S
 A , B and C are exhaustive events . Hence at least one of
them will occur when the experiment is performed
Probability Theory 4

8. Physics Helpline
L K Satapathy
Mutually Exclusive and Exhaustive Events :
Consider n events of a sample space S .1 2, , . . . , nE E E
1 2
1
( ) . . .
n
n i
i
ii E E E E S

     
They are said to be mutually exclusive and exhaustive if
( ) , , ( )i ji E E i j i j   
The collection of elementary events associated with a random
experiment form a system of mutually exclusive and exhaustive events.
For S = {1,2,3,4,5,6} , elementary events are {1}, {2}, {3}, {4}, {5} and {6} ,
which are mutually exclusive and exhaustive.
In the experiment of throwing a die , the events A = an odd number
occurs and B = an even number occurs , are mutually exclusive and
exhaustive. Here S = { 1,2,3,4,5,6 }
Since A = { 1,3,5 } and B = { 2,4,6 } A  B = S and A  B = 
(pair wise disjoint)
Probability Theory 4

9. Physics Helpline
L K Satapathy
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