NCERT Textbook: Probability | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


vWhere a mathematical reasoning can be had, it is as great a folly to
make use of any other, as to grope for a thing in the dark, when
you have a candle in your hand. – JOHN ARBUTHNOT v
14.1  Event
We have studied about random experiment and sample space associated with an
experiment. The sample space serves as an universal set for all questions concerned
with the experiment.
Consider the experiment of tossing a coin two times. An associated sample space
is S = {HH, HT, TH, TT}.
Now suppose that we are interested in those outcomes which correspond to the
occurrence of exactly one head. We find that HT and TH are the only elements of S
corresponding to the occurrence of this happening (event). These two elements form
the set  E = { HT, TH}
We know that the set E is a subset of the sample space S . Similarly, we find the
following correspondence between events and subsets of S.
Description of events Corresponding subset of ‘S’
Number of tails is exactly 2 A = {TT}
Number of tails is atleast one B = {HT, TH, TT}
Number of heads is atmost one C = {HT, TH, TT}
Second toss is not head D = { HT, TT}
Number of tails is atmost two S  = {HH, HT, TH, TT}
Number of tails is more than two f
The above discussion suggests that a subset of sample space is associated with
an event and an event is associated with a subset of sample space. In the light of this
we define an event as follows.
Definition  Any subset E of a sample space S is called an event.
14 Chapter
PROBABILITY
2024-25
Page 2


vWhere a mathematical reasoning can be had, it is as great a folly to
make use of any other, as to grope for a thing in the dark, when
you have a candle in your hand. – JOHN ARBUTHNOT v
14.1  Event
We have studied about random experiment and sample space associated with an
experiment. The sample space serves as an universal set for all questions concerned
with the experiment.
Consider the experiment of tossing a coin two times. An associated sample space
is S = {HH, HT, TH, TT}.
Now suppose that we are interested in those outcomes which correspond to the
occurrence of exactly one head. We find that HT and TH are the only elements of S
corresponding to the occurrence of this happening (event). These two elements form
the set  E = { HT, TH}
We know that the set E is a subset of the sample space S . Similarly, we find the
following correspondence between events and subsets of S.
Description of events Corresponding subset of ‘S’
Number of tails is exactly 2 A = {TT}
Number of tails is atleast one B = {HT, TH, TT}
Number of heads is atmost one C = {HT, TH, TT}
Second toss is not head D = { HT, TT}
Number of tails is atmost two S  = {HH, HT, TH, TT}
Number of tails is more than two f
The above discussion suggests that a subset of sample space is associated with
an event and an event is associated with a subset of sample space. In the light of this
we define an event as follows.
Definition  Any subset E of a sample space S is called an event.
14 Chapter
PROBABILITY
2024-25
290 MATHEMATICS
14.1.1  Occurrence of an event  Consider the experiment of throwing a die. Let E
denotes the event “ a number  less than 4 appears”. If actually ‘1’ had appeared on the
die then we say that event E has occurred. As a matter of fact if outcomes are 2 or 3,
we say that event E has occurred
Thus, the event E of a sample space S is said to have occurred if the outcome
? of the experiment is such that ? ? E. If the outcome ? is such that ? ? E, we say
that the event E has not occurred.
14.1.2  Types of events  Events can be classified into various types on the basis  of the
elements they have.
1. Impossible and Sure Events  The empty set f and the sample space S describe
events. In fact f is called an impossible event and S, i.e., the whole sample space is
called the sure event.
To understand these let us consider the experiment of rolling a die. The associated
sample space is
S = {1, 2, 3, 4, 5, 6}
Let E be the event “ the number appears on the die is a multiple of 7”. Can you
write the subset associated with the event E?
Clearly no outcome satisfies the condition given in the event, i.e., no element of
the sample space ensures the occurrence of the event E. Thus, we say that the empty
set only correspond to the event E. In other words we can say that it is impossible to
have a multiple of 7 on the upper face of the die. Thus, the event E = f is an impossible
event.
Now let us take up another event F “the number turns up is odd or even”. Clearly
F = {1, 2, 3, 4, 5, 6,} = S, i.e., all outcomes of the experiment ensure the occurrence of
the event F. Thus, the event F = S is a sure event.
2. Simple Event  If an event E has only one sample point of a sample space, it is
called a simple (or elementary) event.
In a sample space containing n distinct elements, there are exactly n simple
events.
For example in the experiment of tossing two coins, a sample space is
S={HH, HT, TH, TT}
There are four simple events corresponding to this sample space. These are
E
1
= {HH}, E
2
={HT}, E
3
= { TH} and E
4
={TT}.
2024-25
Page 3


vWhere a mathematical reasoning can be had, it is as great a folly to
make use of any other, as to grope for a thing in the dark, when
you have a candle in your hand. – JOHN ARBUTHNOT v
14.1  Event
We have studied about random experiment and sample space associated with an
experiment. The sample space serves as an universal set for all questions concerned
with the experiment.
Consider the experiment of tossing a coin two times. An associated sample space
is S = {HH, HT, TH, TT}.
Now suppose that we are interested in those outcomes which correspond to the
occurrence of exactly one head. We find that HT and TH are the only elements of S
corresponding to the occurrence of this happening (event). These two elements form
the set  E = { HT, TH}
We know that the set E is a subset of the sample space S . Similarly, we find the
following correspondence between events and subsets of S.
Description of events Corresponding subset of ‘S’
Number of tails is exactly 2 A = {TT}
Number of tails is atleast one B = {HT, TH, TT}
Number of heads is atmost one C = {HT, TH, TT}
Second toss is not head D = { HT, TT}
Number of tails is atmost two S  = {HH, HT, TH, TT}
Number of tails is more than two f
The above discussion suggests that a subset of sample space is associated with
an event and an event is associated with a subset of sample space. In the light of this
we define an event as follows.
Definition  Any subset E of a sample space S is called an event.
14 Chapter
PROBABILITY
2024-25
290 MATHEMATICS
14.1.1  Occurrence of an event  Consider the experiment of throwing a die. Let E
denotes the event “ a number  less than 4 appears”. If actually ‘1’ had appeared on the
die then we say that event E has occurred. As a matter of fact if outcomes are 2 or 3,
we say that event E has occurred
Thus, the event E of a sample space S is said to have occurred if the outcome
? of the experiment is such that ? ? E. If the outcome ? is such that ? ? E, we say
that the event E has not occurred.
14.1.2  Types of events  Events can be classified into various types on the basis  of the
elements they have.
1. Impossible and Sure Events  The empty set f and the sample space S describe
events. In fact f is called an impossible event and S, i.e., the whole sample space is
called the sure event.
To understand these let us consider the experiment of rolling a die. The associated
sample space is
S = {1, 2, 3, 4, 5, 6}
Let E be the event “ the number appears on the die is a multiple of 7”. Can you
write the subset associated with the event E?
Clearly no outcome satisfies the condition given in the event, i.e., no element of
the sample space ensures the occurrence of the event E. Thus, we say that the empty
set only correspond to the event E. In other words we can say that it is impossible to
have a multiple of 7 on the upper face of the die. Thus, the event E = f is an impossible
event.
Now let us take up another event F “the number turns up is odd or even”. Clearly
F = {1, 2, 3, 4, 5, 6,} = S, i.e., all outcomes of the experiment ensure the occurrence of
the event F. Thus, the event F = S is a sure event.
2. Simple Event  If an event E has only one sample point of a sample space, it is
called a simple (or elementary) event.
In a sample space containing n distinct elements, there are exactly n simple
events.
For example in the experiment of tossing two coins, a sample space is
S={HH, HT, TH, TT}
There are four simple events corresponding to this sample space. These are
E
1
= {HH}, E
2
={HT}, E
3
= { TH} and E
4
={TT}.
2024-25
  PROBABILITY            291
3. Compound Event  If an event has more than one sample point, it is called  a
Compound event.
For example, in the experiment of “tossing a coin thrice” the events
E: ‘Exactly one head appeared’
F: ‘Atleast one head appeared’
G: ‘Atmost one head appeared’ etc.
are all compound events. The subsets of S associated with these events are
E={HTT,THT,TTH}
F={HTT,THT, TTH, HHT, HTH, THH, HHH}
G= {TTT, THT, HTT, TTH}
Each of the above subsets contain more than one sample point, hence they are all
compound events.
14.1.3  Algebra of events  In the Chapter on Sets, we have studied about different
ways of combining two or more sets, viz, union, intersection, difference, complement
of a set etc. Like-wise we can combine two or more events by using the analogous set
notations.
Let A, B, C be events associated with an experiment whose sample space is S.
1. Complementary Event For every event A, there corresponds another event
A' called the complementary event to A. It is also called the event ‘not A ’.
For example, take the experiment ‘of tossing three coins’. An associated sample
space is
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Let  A={HTH, HHT, THH} be the event ‘only one tail appears’
Clearly for the outcome HTT, the event A has not occurred. But we may say that
the event ‘not A’ has occurred. Thus, with every outcome which is not in A, we say
that ‘not A’ occurs.
Thus the complementary event ‘not A’ to the event A is
A' = {HHH, HTT, THT, TTH, TTT}
or A' = {? : ? ? S and ? ?A} = S – A.
2. The Event ‘A or B’ Recall that union of two sets A and B denoted by A ? B
contains all those elements which are either in A  or in B or in both.
When the sets A and B are two events associated with a sample space, then
‘A ? B’ is the event ‘either A or B or both’. This event ‘A ? B’ is also called ‘A or B’.
Therefore Event ‘A or B’ = A ? B
= {? : ? ? A or ? ? B}
2024-25
Page 4


vWhere a mathematical reasoning can be had, it is as great a folly to
make use of any other, as to grope for a thing in the dark, when
you have a candle in your hand. – JOHN ARBUTHNOT v
14.1  Event
We have studied about random experiment and sample space associated with an
experiment. The sample space serves as an universal set for all questions concerned
with the experiment.
Consider the experiment of tossing a coin two times. An associated sample space
is S = {HH, HT, TH, TT}.
Now suppose that we are interested in those outcomes which correspond to the
occurrence of exactly one head. We find that HT and TH are the only elements of S
corresponding to the occurrence of this happening (event). These two elements form
the set  E = { HT, TH}
We know that the set E is a subset of the sample space S . Similarly, we find the
following correspondence between events and subsets of S.
Description of events Corresponding subset of ‘S’
Number of tails is exactly 2 A = {TT}
Number of tails is atleast one B = {HT, TH, TT}
Number of heads is atmost one C = {HT, TH, TT}
Second toss is not head D = { HT, TT}
Number of tails is atmost two S  = {HH, HT, TH, TT}
Number of tails is more than two f
The above discussion suggests that a subset of sample space is associated with
an event and an event is associated with a subset of sample space. In the light of this
we define an event as follows.
Definition  Any subset E of a sample space S is called an event.
14 Chapter
PROBABILITY
2024-25
290 MATHEMATICS
14.1.1  Occurrence of an event  Consider the experiment of throwing a die. Let E
denotes the event “ a number  less than 4 appears”. If actually ‘1’ had appeared on the
die then we say that event E has occurred. As a matter of fact if outcomes are 2 or 3,
we say that event E has occurred
Thus, the event E of a sample space S is said to have occurred if the outcome
? of the experiment is such that ? ? E. If the outcome ? is such that ? ? E, we say
that the event E has not occurred.
14.1.2  Types of events  Events can be classified into various types on the basis  of the
elements they have.
1. Impossible and Sure Events  The empty set f and the sample space S describe
events. In fact f is called an impossible event and S, i.e., the whole sample space is
called the sure event.
To understand these let us consider the experiment of rolling a die. The associated
sample space is
S = {1, 2, 3, 4, 5, 6}
Let E be the event “ the number appears on the die is a multiple of 7”. Can you
write the subset associated with the event E?
Clearly no outcome satisfies the condition given in the event, i.e., no element of
the sample space ensures the occurrence of the event E. Thus, we say that the empty
set only correspond to the event E. In other words we can say that it is impossible to
have a multiple of 7 on the upper face of the die. Thus, the event E = f is an impossible
event.
Now let us take up another event F “the number turns up is odd or even”. Clearly
F = {1, 2, 3, 4, 5, 6,} = S, i.e., all outcomes of the experiment ensure the occurrence of
the event F. Thus, the event F = S is a sure event.
2. Simple Event  If an event E has only one sample point of a sample space, it is
called a simple (or elementary) event.
In a sample space containing n distinct elements, there are exactly n simple
events.
For example in the experiment of tossing two coins, a sample space is
S={HH, HT, TH, TT}
There are four simple events corresponding to this sample space. These are
E
1
= {HH}, E
2
={HT}, E
3
= { TH} and E
4
={TT}.
2024-25
  PROBABILITY            291
3. Compound Event  If an event has more than one sample point, it is called  a
Compound event.
For example, in the experiment of “tossing a coin thrice” the events
E: ‘Exactly one head appeared’
F: ‘Atleast one head appeared’
G: ‘Atmost one head appeared’ etc.
are all compound events. The subsets of S associated with these events are
E={HTT,THT,TTH}
F={HTT,THT, TTH, HHT, HTH, THH, HHH}
G= {TTT, THT, HTT, TTH}
Each of the above subsets contain more than one sample point, hence they are all
compound events.
14.1.3  Algebra of events  In the Chapter on Sets, we have studied about different
ways of combining two or more sets, viz, union, intersection, difference, complement
of a set etc. Like-wise we can combine two or more events by using the analogous set
notations.
Let A, B, C be events associated with an experiment whose sample space is S.
1. Complementary Event For every event A, there corresponds another event
A' called the complementary event to A. It is also called the event ‘not A ’.
For example, take the experiment ‘of tossing three coins’. An associated sample
space is
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Let  A={HTH, HHT, THH} be the event ‘only one tail appears’
Clearly for the outcome HTT, the event A has not occurred. But we may say that
the event ‘not A’ has occurred. Thus, with every outcome which is not in A, we say
that ‘not A’ occurs.
Thus the complementary event ‘not A’ to the event A is
A' = {HHH, HTT, THT, TTH, TTT}
or A' = {? : ? ? S and ? ?A} = S – A.
2. The Event ‘A or B’ Recall that union of two sets A and B denoted by A ? B
contains all those elements which are either in A  or in B or in both.
When the sets A and B are two events associated with a sample space, then
‘A ? B’ is the event ‘either A or B or both’. This event ‘A ? B’ is also called ‘A or B’.
Therefore Event ‘A or B’ = A ? B
= {? : ? ? A or ? ? B}
2024-25
292 MATHEMATICS
3. The Event ‘A and B’ We know that intersection of two sets A n B is the set of
those elements which are common to both A and B. i.e., which belong to both
‘A and B’.
If A and B are two events, then the set A n B denotes the event ‘A and B’.
Thus,      A n B = {? : ? ? A and ? ? B}
For example, in the experiment of ‘throwing a die twice’ Let A be the event
‘score on the first throw is six’ and B is the event ‘sum of two scores is atleast 11’ then
A = {(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}, and  B = {(5,6), (6,5), (6,6)}
so A n B = {(6,5), (6,6)}
Note that the set A n B = {(6,5), (6,6)} may represent the event ‘the score on the first
throw  is six and the sum of the scores is atleast 11’.
4. The Event ‘A but not B’ We know that A–B is the set of all those elements
which are in A but not in B. Therefore, the set A–B may denote the event ‘A but not
B’.We know that
       A – B = A n B´
Example 1 Consider the experiment of rolling a die. Let A be the event ‘getting a
prime number’, B be the event ‘getting an odd number’. Write the sets representing
the events (i) Aor B (ii) A and B (iii)  A but not B (iv) ‘not A’.
Solution Here S = {1, 2, 3, 4, 5, 6}, A = {2, 3, 5} and B = {1, 3, 5}
Obviously
(i) ‘A or B’ = A ? B = {1, 2, 3, 5}
(ii) ‘A and B’ = A n B = {3,5}
(iii) ‘A but not B’ =  A – B = {2}
(iv) ‘not  A’ = A' = {1,4,6}
14.1.4 Mutually exclusive events In the experiment of rolling a die, a sample space
is S = {1, 2, 3, 4, 5, 6}. Consider events, A ‘an odd number appears’ and B ‘an even
number appears’
Clearly the event A excludes the event B and vice versa. In other words, there is
no outcome which ensures the occurrence of events A and B simultaneously. Here
A = {1, 3, 5} and B = {2, 4, 6}
Clearly A n B = f,  i.e., A and B are disjoint sets.
In general, two events A and B are called mutually exclusive events if the
occurrence of any one of them excludes the occurrence of the other event, i.e., if they
can not occur simultaneously. In this case the sets A and B are disjoint.
2024-25
Page 5


vWhere a mathematical reasoning can be had, it is as great a folly to
make use of any other, as to grope for a thing in the dark, when
you have a candle in your hand. – JOHN ARBUTHNOT v
14.1  Event
We have studied about random experiment and sample space associated with an
experiment. The sample space serves as an universal set for all questions concerned
with the experiment.
Consider the experiment of tossing a coin two times. An associated sample space
is S = {HH, HT, TH, TT}.
Now suppose that we are interested in those outcomes which correspond to the
occurrence of exactly one head. We find that HT and TH are the only elements of S
corresponding to the occurrence of this happening (event). These two elements form
the set  E = { HT, TH}
We know that the set E is a subset of the sample space S . Similarly, we find the
following correspondence between events and subsets of S.
Description of events Corresponding subset of ‘S’
Number of tails is exactly 2 A = {TT}
Number of tails is atleast one B = {HT, TH, TT}
Number of heads is atmost one C = {HT, TH, TT}
Second toss is not head D = { HT, TT}
Number of tails is atmost two S  = {HH, HT, TH, TT}
Number of tails is more than two f
The above discussion suggests that a subset of sample space is associated with
an event and an event is associated with a subset of sample space. In the light of this
we define an event as follows.
Definition  Any subset E of a sample space S is called an event.
14 Chapter
PROBABILITY
2024-25
290 MATHEMATICS
14.1.1  Occurrence of an event  Consider the experiment of throwing a die. Let E
denotes the event “ a number  less than 4 appears”. If actually ‘1’ had appeared on the
die then we say that event E has occurred. As a matter of fact if outcomes are 2 or 3,
we say that event E has occurred
Thus, the event E of a sample space S is said to have occurred if the outcome
? of the experiment is such that ? ? E. If the outcome ? is such that ? ? E, we say
that the event E has not occurred.
14.1.2  Types of events  Events can be classified into various types on the basis  of the
elements they have.
1. Impossible and Sure Events  The empty set f and the sample space S describe
events. In fact f is called an impossible event and S, i.e., the whole sample space is
called the sure event.
To understand these let us consider the experiment of rolling a die. The associated
sample space is
S = {1, 2, 3, 4, 5, 6}
Let E be the event “ the number appears on the die is a multiple of 7”. Can you
write the subset associated with the event E?
Clearly no outcome satisfies the condition given in the event, i.e., no element of
the sample space ensures the occurrence of the event E. Thus, we say that the empty
set only correspond to the event E. In other words we can say that it is impossible to
have a multiple of 7 on the upper face of the die. Thus, the event E = f is an impossible
event.
Now let us take up another event F “the number turns up is odd or even”. Clearly
F = {1, 2, 3, 4, 5, 6,} = S, i.e., all outcomes of the experiment ensure the occurrence of
the event F. Thus, the event F = S is a sure event.
2. Simple Event  If an event E has only one sample point of a sample space, it is
called a simple (or elementary) event.
In a sample space containing n distinct elements, there are exactly n simple
events.
For example in the experiment of tossing two coins, a sample space is
S={HH, HT, TH, TT}
There are four simple events corresponding to this sample space. These are
E
1
= {HH}, E
2
={HT}, E
3
= { TH} and E
4
={TT}.
2024-25
  PROBABILITY            291
3. Compound Event  If an event has more than one sample point, it is called  a
Compound event.
For example, in the experiment of “tossing a coin thrice” the events
E: ‘Exactly one head appeared’
F: ‘Atleast one head appeared’
G: ‘Atmost one head appeared’ etc.
are all compound events. The subsets of S associated with these events are
E={HTT,THT,TTH}
F={HTT,THT, TTH, HHT, HTH, THH, HHH}
G= {TTT, THT, HTT, TTH}
Each of the above subsets contain more than one sample point, hence they are all
compound events.
14.1.3  Algebra of events  In the Chapter on Sets, we have studied about different
ways of combining two or more sets, viz, union, intersection, difference, complement
of a set etc. Like-wise we can combine two or more events by using the analogous set
notations.
Let A, B, C be events associated with an experiment whose sample space is S.
1. Complementary Event For every event A, there corresponds another event
A' called the complementary event to A. It is also called the event ‘not A ’.
For example, take the experiment ‘of tossing three coins’. An associated sample
space is
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
Let  A={HTH, HHT, THH} be the event ‘only one tail appears’
Clearly for the outcome HTT, the event A has not occurred. But we may say that
the event ‘not A’ has occurred. Thus, with every outcome which is not in A, we say
that ‘not A’ occurs.
Thus the complementary event ‘not A’ to the event A is
A' = {HHH, HTT, THT, TTH, TTT}
or A' = {? : ? ? S and ? ?A} = S – A.
2. The Event ‘A or B’ Recall that union of two sets A and B denoted by A ? B
contains all those elements which are either in A  or in B or in both.
When the sets A and B are two events associated with a sample space, then
‘A ? B’ is the event ‘either A or B or both’. This event ‘A ? B’ is also called ‘A or B’.
Therefore Event ‘A or B’ = A ? B
= {? : ? ? A or ? ? B}
2024-25
292 MATHEMATICS
3. The Event ‘A and B’ We know that intersection of two sets A n B is the set of
those elements which are common to both A and B. i.e., which belong to both
‘A and B’.
If A and B are two events, then the set A n B denotes the event ‘A and B’.
Thus,      A n B = {? : ? ? A and ? ? B}
For example, in the experiment of ‘throwing a die twice’ Let A be the event
‘score on the first throw is six’ and B is the event ‘sum of two scores is atleast 11’ then
A = {(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}, and  B = {(5,6), (6,5), (6,6)}
so A n B = {(6,5), (6,6)}
Note that the set A n B = {(6,5), (6,6)} may represent the event ‘the score on the first
throw  is six and the sum of the scores is atleast 11’.
4. The Event ‘A but not B’ We know that A–B is the set of all those elements
which are in A but not in B. Therefore, the set A–B may denote the event ‘A but not
B’.We know that
       A – B = A n B´
Example 1 Consider the experiment of rolling a die. Let A be the event ‘getting a
prime number’, B be the event ‘getting an odd number’. Write the sets representing
the events (i) Aor B (ii) A and B (iii)  A but not B (iv) ‘not A’.
Solution Here S = {1, 2, 3, 4, 5, 6}, A = {2, 3, 5} and B = {1, 3, 5}
Obviously
(i) ‘A or B’ = A ? B = {1, 2, 3, 5}
(ii) ‘A and B’ = A n B = {3,5}
(iii) ‘A but not B’ =  A – B = {2}
(iv) ‘not  A’ = A' = {1,4,6}
14.1.4 Mutually exclusive events In the experiment of rolling a die, a sample space
is S = {1, 2, 3, 4, 5, 6}. Consider events, A ‘an odd number appears’ and B ‘an even
number appears’
Clearly the event A excludes the event B and vice versa. In other words, there is
no outcome which ensures the occurrence of events A and B simultaneously. Here
A = {1, 3, 5} and B = {2, 4, 6}
Clearly A n B = f,  i.e., A and B are disjoint sets.
In general, two events A and B are called mutually exclusive events if the
occurrence of any one of them excludes the occurrence of the other event, i.e., if they
can not occur simultaneously. In this case the sets A and B are disjoint.
2024-25
  PROBABILITY            293
Again in the experiment of rolling a die, consider the events A ‘an odd number
appears’ and event B ‘a number less than 4 appears’
Obviously A = {1, 3, 5} and B = {1, 2, 3}
Now 3 ? A  as well as 3 ? B
Therefore,  A and B are not mutually exclusive events.
Remark Simple events of a sample space are always mutually exclusive.
14.1.5 Exhaustive events Consider the experiment of throwing a die. We have
S = {1, 2, 3, 4, 5, 6}. Let us define the following events
A: ‘a number less than 4 appears’,
B: ‘a number greater than 2 but less than 5 appears’
and C: ‘a number greater than 4 appears’.
Then A = {1, 2, 3}, B = {3,4} and C = {5, 6}. We observe that
A ? B ? C = {1, 2, 3} ? {3, 4} ? {5, 6} = S.
Such events A, B and C are called exhaustive events. In general, if E
1
, E
2
, ..., E
n
 are n
events of a sample space S and if
1 2 3
1
E E E E E S
n
n i
i
...
=
? ? ? ? = ? =
then E
1
, E
2
, ...., E
n
 are called exhaustive events.In other words, events E
1
, E
2
, ..., E
n
are said to be exhaustive if atleast one of them necessarily occurs whenever the
experiment is performed.
Further, if E
i
 n E
j
 = f for i ? j  i.e., events E
i
 and E
j 
are pairwise disjoint and
S E
1
= ?
=
i
n
i
, then events E
1
, E
2
, ..., E
n 
are called mutually exclusive and exhaustive
events.
We now consider some examples.
Example 2 Two dice are thrown and the sum of the numbers which come up on the
dice is noted. Let us consider the following events associated with this experiment
A: ‘the sum is even’.
B: ‘the sum is a multiple of 3’.
C: ‘the sum is less than 4’.
D: ‘the sum is greater than 11’.
Which pairs of these events are mutually exclusive?
2024-25