Detailed Notes: Probabiliity | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


Probability 
Probability is a fascinating field of mathematics that quantifies uncertainty and chance. 
It provides us with powerful tools to analyze and predict outcomes in situations where 
randomness plays a role. From weather forecasts to game strategies, probability theory 
helps us make sense of the unpredictable aspects of our world. 
At its core, probability deals with the likelihood of events occurring. It allows us to move 
beyond simple yes-or-no predictions and instead assign numerical values to the chances 
of different outcomes. This nuanced approach to uncertainty opens up a wealth of 
applications across various disciplines, including science, finance, and decision-making. 
Understanding probability is increasingly important in our data-driven society. It forms 
the foundation for statistical analysis, risk assessment, and many machine learning 
algorithms. By grasping the principles of probability, we can better interpret 
information, make informed decisions, and understand the complex systems that 
surround us. Whether you're a student, professional, or simply curious about the world, 
exploring probability offers valuable insights into the nature of chance and uncertainty. 
There are various phenomena in nature, leading to an outcome, which cannot be 
predicted apriori e.g. in tossing of a coin, a head or a tail may result. Probability theory 
aims at measuring the uncertainties of such outcomes. 
(I) Important terminology: 
(i) Random experiment: 
It is a process which results in an outcome which is one of the various possible outcomes 
that are known to us before hand e.g. throwing of a die is a random experiment as it 
leads to fall of one of the outcome from {1,2,3,4,5,6}. Similarly taking a card from a pack 
of 52 cards is also a random experiment. 
(ii) Sample space: 
It is the set of all possible outcomes of a random experiment e.g. {?? , ?? } is the sample 
space associated with tossing of a coin. 
In set notation it can be interpreted as the universal set. 
 
Problem 1: Write the sample space of the experiment 'A coin is tossed and a die is 
thrown'. 
Solution:   The sample space ?? = {?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6, ?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6}. 
Page 2


Probability 
Probability is a fascinating field of mathematics that quantifies uncertainty and chance. 
It provides us with powerful tools to analyze and predict outcomes in situations where 
randomness plays a role. From weather forecasts to game strategies, probability theory 
helps us make sense of the unpredictable aspects of our world. 
At its core, probability deals with the likelihood of events occurring. It allows us to move 
beyond simple yes-or-no predictions and instead assign numerical values to the chances 
of different outcomes. This nuanced approach to uncertainty opens up a wealth of 
applications across various disciplines, including science, finance, and decision-making. 
Understanding probability is increasingly important in our data-driven society. It forms 
the foundation for statistical analysis, risk assessment, and many machine learning 
algorithms. By grasping the principles of probability, we can better interpret 
information, make informed decisions, and understand the complex systems that 
surround us. Whether you're a student, professional, or simply curious about the world, 
exploring probability offers valuable insights into the nature of chance and uncertainty. 
There are various phenomena in nature, leading to an outcome, which cannot be 
predicted apriori e.g. in tossing of a coin, a head or a tail may result. Probability theory 
aims at measuring the uncertainties of such outcomes. 
(I) Important terminology: 
(i) Random experiment: 
It is a process which results in an outcome which is one of the various possible outcomes 
that are known to us before hand e.g. throwing of a die is a random experiment as it 
leads to fall of one of the outcome from {1,2,3,4,5,6}. Similarly taking a card from a pack 
of 52 cards is also a random experiment. 
(ii) Sample space: 
It is the set of all possible outcomes of a random experiment e.g. {?? , ?? } is the sample 
space associated with tossing of a coin. 
In set notation it can be interpreted as the universal set. 
 
Problem 1: Write the sample space of the experiment 'A coin is tossed and a die is 
thrown'. 
Solution:   The sample space ?? = {?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6, ?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6}. 
Problem 2: Write the sample space of the experiment 'A coin is tossed, if it shows head a 
coin tossed again else a die is thrown. 
Solution:   The sample space ?? = {???? , ???? , ?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6} 
Problem 3: Find the sample space associated with the experiment of rolling a pair of dice 
(plural of die) once. Also find the number of elements of the sample space. 
Solution: Let one die be blue and the other be green. Suppose ' 1 ' appears on blue die 
and ' 2 ' appears on green die. We denote this outcome by an ordered pair (1,2). 
Similarly, if ' 3 ' appears on blue die and ' 5 ' appears on green die, we denote this 
outcome by (3,5) and so on. Thus, each outcome can be denoted by an ordered pair 
(?? , ?? ), where ?? is the number appeared on the first die (blue die) and y appeared on the 
second die (green die). Thus, the sample space is given by 
?? = {(?? , ?? )?? is the number on blue die and ?? is the number on green die } 
We now list all the possible outcomes (figure) 
 
 1 2 3 4 5 6 
1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) 
2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) 
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) 
 
Figure 
Number of elements (outcomes) of the above sample space is 6 × 6 i.e., 36 
 
Page 3


Probability 
Probability is a fascinating field of mathematics that quantifies uncertainty and chance. 
It provides us with powerful tools to analyze and predict outcomes in situations where 
randomness plays a role. From weather forecasts to game strategies, probability theory 
helps us make sense of the unpredictable aspects of our world. 
At its core, probability deals with the likelihood of events occurring. It allows us to move 
beyond simple yes-or-no predictions and instead assign numerical values to the chances 
of different outcomes. This nuanced approach to uncertainty opens up a wealth of 
applications across various disciplines, including science, finance, and decision-making. 
Understanding probability is increasingly important in our data-driven society. It forms 
the foundation for statistical analysis, risk assessment, and many machine learning 
algorithms. By grasping the principles of probability, we can better interpret 
information, make informed decisions, and understand the complex systems that 
surround us. Whether you're a student, professional, or simply curious about the world, 
exploring probability offers valuable insights into the nature of chance and uncertainty. 
There are various phenomena in nature, leading to an outcome, which cannot be 
predicted apriori e.g. in tossing of a coin, a head or a tail may result. Probability theory 
aims at measuring the uncertainties of such outcomes. 
(I) Important terminology: 
(i) Random experiment: 
It is a process which results in an outcome which is one of the various possible outcomes 
that are known to us before hand e.g. throwing of a die is a random experiment as it 
leads to fall of one of the outcome from {1,2,3,4,5,6}. Similarly taking a card from a pack 
of 52 cards is also a random experiment. 
(ii) Sample space: 
It is the set of all possible outcomes of a random experiment e.g. {?? , ?? } is the sample 
space associated with tossing of a coin. 
In set notation it can be interpreted as the universal set. 
 
Problem 1: Write the sample space of the experiment 'A coin is tossed and a die is 
thrown'. 
Solution:   The sample space ?? = {?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6, ?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6}. 
Problem 2: Write the sample space of the experiment 'A coin is tossed, if it shows head a 
coin tossed again else a die is thrown. 
Solution:   The sample space ?? = {???? , ???? , ?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6} 
Problem 3: Find the sample space associated with the experiment of rolling a pair of dice 
(plural of die) once. Also find the number of elements of the sample space. 
Solution: Let one die be blue and the other be green. Suppose ' 1 ' appears on blue die 
and ' 2 ' appears on green die. We denote this outcome by an ordered pair (1,2). 
Similarly, if ' 3 ' appears on blue die and ' 5 ' appears on green die, we denote this 
outcome by (3,5) and so on. Thus, each outcome can be denoted by an ordered pair 
(?? , ?? ), where ?? is the number appeared on the first die (blue die) and y appeared on the 
second die (green die). Thus, the sample space is given by 
?? = {(?? , ?? )?? is the number on blue die and ?? is the number on green die } 
We now list all the possible outcomes (figure) 
 
 1 2 3 4 5 6 
1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) 
2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) 
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) 
 
Figure 
Number of elements (outcomes) of the above sample space is 6 × 6 i.e., 36 
 
It is subset of sample space. e.g. getting a head in tossing a coin or getting a prime 
number in throwing a die. In general if a sample space consists ' ?? ' elements, then a 
maximum of 2
?? events can be associated with it. 
 
(iii) Complement of event: 
The complement of an event 'A' with respect to a sample space ?? is the set of all elements 
of ' ?? ' which are not in A. It is usually denoted by ?? '
, or ?? ?? . 
(iv) Simple event: 
If an event covers only one point of sample space, then it is called a simple event e.g. 
getting a head followed by a tail in throwing of a coin 2 times is a simple event. 
(v) Compound event: 
When two or more than two events occur simultaneously, the event is said to be a 
compound event. Symbolically ?? n ?? or ???? represent the occurrence of both ?? &?? 
simultaneously. 
Note: "A B" or A + B represent the occurrence of either ?? or ?? . 
Problem 4: Write down all the events of the experiment 'tossing of a coin'. 
Solution:  ?? = {?? , ?? } 
the events are ?? , {?? }, {?? }, {?? , ?? } 
Problem 5: A die is thrown. Let ?? be the event ' an odd number turns up' and ?? be the 
event 'a number  ?????????????????? ???? 3 ?????????? ???? '. ?????????? ?? h?? ????????????   (?? ) ?? ???? ??  (?? ) ?? ?????? ??  
Solution:  ?? = {1,3,5}, ?? = {3,6} 
?  ?? ???? ?? = ?? ? ?? = {1,3,5,6}  ?? ?????? ?? = ?? n ?? = {3}  
 (vii) Equally likely events: 
If events have same chance of occurrence, then they are said to be equally likely. e.g 
(i) In a single toss of a fair coin, the events {?? } and {?? } are equally likely. 
(ii) In a single throw of an unbiased die the events {1}, {2}, {3} and {4}, are equally likely. 
Page 4


Probability 
Probability is a fascinating field of mathematics that quantifies uncertainty and chance. 
It provides us with powerful tools to analyze and predict outcomes in situations where 
randomness plays a role. From weather forecasts to game strategies, probability theory 
helps us make sense of the unpredictable aspects of our world. 
At its core, probability deals with the likelihood of events occurring. It allows us to move 
beyond simple yes-or-no predictions and instead assign numerical values to the chances 
of different outcomes. This nuanced approach to uncertainty opens up a wealth of 
applications across various disciplines, including science, finance, and decision-making. 
Understanding probability is increasingly important in our data-driven society. It forms 
the foundation for statistical analysis, risk assessment, and many machine learning 
algorithms. By grasping the principles of probability, we can better interpret 
information, make informed decisions, and understand the complex systems that 
surround us. Whether you're a student, professional, or simply curious about the world, 
exploring probability offers valuable insights into the nature of chance and uncertainty. 
There are various phenomena in nature, leading to an outcome, which cannot be 
predicted apriori e.g. in tossing of a coin, a head or a tail may result. Probability theory 
aims at measuring the uncertainties of such outcomes. 
(I) Important terminology: 
(i) Random experiment: 
It is a process which results in an outcome which is one of the various possible outcomes 
that are known to us before hand e.g. throwing of a die is a random experiment as it 
leads to fall of one of the outcome from {1,2,3,4,5,6}. Similarly taking a card from a pack 
of 52 cards is also a random experiment. 
(ii) Sample space: 
It is the set of all possible outcomes of a random experiment e.g. {?? , ?? } is the sample 
space associated with tossing of a coin. 
In set notation it can be interpreted as the universal set. 
 
Problem 1: Write the sample space of the experiment 'A coin is tossed and a die is 
thrown'. 
Solution:   The sample space ?? = {?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6, ?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6}. 
Problem 2: Write the sample space of the experiment 'A coin is tossed, if it shows head a 
coin tossed again else a die is thrown. 
Solution:   The sample space ?? = {???? , ???? , ?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6} 
Problem 3: Find the sample space associated with the experiment of rolling a pair of dice 
(plural of die) once. Also find the number of elements of the sample space. 
Solution: Let one die be blue and the other be green. Suppose ' 1 ' appears on blue die 
and ' 2 ' appears on green die. We denote this outcome by an ordered pair (1,2). 
Similarly, if ' 3 ' appears on blue die and ' 5 ' appears on green die, we denote this 
outcome by (3,5) and so on. Thus, each outcome can be denoted by an ordered pair 
(?? , ?? ), where ?? is the number appeared on the first die (blue die) and y appeared on the 
second die (green die). Thus, the sample space is given by 
?? = {(?? , ?? )?? is the number on blue die and ?? is the number on green die } 
We now list all the possible outcomes (figure) 
 
 1 2 3 4 5 6 
1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) 
2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) 
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) 
 
Figure 
Number of elements (outcomes) of the above sample space is 6 × 6 i.e., 36 
 
It is subset of sample space. e.g. getting a head in tossing a coin or getting a prime 
number in throwing a die. In general if a sample space consists ' ?? ' elements, then a 
maximum of 2
?? events can be associated with it. 
 
(iii) Complement of event: 
The complement of an event 'A' with respect to a sample space ?? is the set of all elements 
of ' ?? ' which are not in A. It is usually denoted by ?? '
, or ?? ?? . 
(iv) Simple event: 
If an event covers only one point of sample space, then it is called a simple event e.g. 
getting a head followed by a tail in throwing of a coin 2 times is a simple event. 
(v) Compound event: 
When two or more than two events occur simultaneously, the event is said to be a 
compound event. Symbolically ?? n ?? or ???? represent the occurrence of both ?? &?? 
simultaneously. 
Note: "A B" or A + B represent the occurrence of either ?? or ?? . 
Problem 4: Write down all the events of the experiment 'tossing of a coin'. 
Solution:  ?? = {?? , ?? } 
the events are ?? , {?? }, {?? }, {?? , ?? } 
Problem 5: A die is thrown. Let ?? be the event ' an odd number turns up' and ?? be the 
event 'a number  ?????????????????? ???? 3 ?????????? ???? '. ?????????? ?? h?? ????????????   (?? ) ?? ???? ??  (?? ) ?? ?????? ??  
Solution:  ?? = {1,3,5}, ?? = {3,6} 
?  ?? ???? ?? = ?? ? ?? = {1,3,5,6}  ?? ?????? ?? = ?? n ?? = {3}  
 (vii) Equally likely events: 
If events have same chance of occurrence, then they are said to be equally likely. e.g 
(i) In a single toss of a fair coin, the events {?? } and {?? } are equally likely. 
(ii) In a single throw of an unbiased die the events {1}, {2}, {3} and {4}, are equally likely. 
(iii) In tossing a biased coin the events {?? } and {?? } are not equally likely. 
(viii) Mutually exclusive / disjoint / incompatible events: 
Two events are said to be mutually exclusive if occurrence of one of them rejects the 
possibility of occurrence of the other i.e. both cannot occur simultaneously. 
In the vein diagram the events ?? and ?? are mutually exclusive. Mathematically, we write 
?? n ?? = ?? 
Events ?? 1
, ?? 2
, ?? 3
, … . . ?? ?? are said to be mutually exclusive events iff ?? ?? n ?? ?? = ?? ??? , ?? ?
{1,2, … , ?? } where ?? ? ?? 
 
Note: If ?? ?? n ?? ?? = ?? ??? , ?? ? {1,2, … , ?? } where ?? ? ?? , then ?? 1
n ?? 2
n ?? 3
… .n ?? ?? = ?? but 
converse need not to be true. 
Problem 6: In a single toss of a coin find whether the events {?? }, {?? } are mutually 
exclusive or not. 
Solution:   Since {?? } n {?? } = ?? , 
?  the events are mutually exclusive. 
Problem 7: In a single throw of a die, find whether the events {1,2}, {2,3} are mutually 
exclusive or not. 
Solution: Since {1,2} n {2,3} = {2} ? ?? ?  the events are not mutually exclusive. 
 (vi) Exhaustive system of events: 
If each outcome of an experiment is associated with at least one of the events 
?? 1
, ?? 2
, ?? 3
, … … . . ?? ?? , then collectively the events are said to be exhaustive. Mathematically 
we write 
?? 1
? ?? 2
? ?? 3
… … … ?? ?? = ?? . (Sample space) 
Problem 8: In throwing of a die, let A be the event 'even number turns up', B be the event 
'an odd prime turns up' and ?? be the event 'a numbers less than 4 turns up'. Find 
whether the events ?? , ?? and ?? form an exhaustive system or not. 
Solution:  ?? = {2,4,6}, ?? = {3,5} and ?? = {1,2,3}. 
Page 5


Probability 
Probability is a fascinating field of mathematics that quantifies uncertainty and chance. 
It provides us with powerful tools to analyze and predict outcomes in situations where 
randomness plays a role. From weather forecasts to game strategies, probability theory 
helps us make sense of the unpredictable aspects of our world. 
At its core, probability deals with the likelihood of events occurring. It allows us to move 
beyond simple yes-or-no predictions and instead assign numerical values to the chances 
of different outcomes. This nuanced approach to uncertainty opens up a wealth of 
applications across various disciplines, including science, finance, and decision-making. 
Understanding probability is increasingly important in our data-driven society. It forms 
the foundation for statistical analysis, risk assessment, and many machine learning 
algorithms. By grasping the principles of probability, we can better interpret 
information, make informed decisions, and understand the complex systems that 
surround us. Whether you're a student, professional, or simply curious about the world, 
exploring probability offers valuable insights into the nature of chance and uncertainty. 
There are various phenomena in nature, leading to an outcome, which cannot be 
predicted apriori e.g. in tossing of a coin, a head or a tail may result. Probability theory 
aims at measuring the uncertainties of such outcomes. 
(I) Important terminology: 
(i) Random experiment: 
It is a process which results in an outcome which is one of the various possible outcomes 
that are known to us before hand e.g. throwing of a die is a random experiment as it 
leads to fall of one of the outcome from {1,2,3,4,5,6}. Similarly taking a card from a pack 
of 52 cards is also a random experiment. 
(ii) Sample space: 
It is the set of all possible outcomes of a random experiment e.g. {?? , ?? } is the sample 
space associated with tossing of a coin. 
In set notation it can be interpreted as the universal set. 
 
Problem 1: Write the sample space of the experiment 'A coin is tossed and a die is 
thrown'. 
Solution:   The sample space ?? = {?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6, ?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6}. 
Problem 2: Write the sample space of the experiment 'A coin is tossed, if it shows head a 
coin tossed again else a die is thrown. 
Solution:   The sample space ?? = {???? , ???? , ?? 1, ?? 2, ?? 3, ?? 4, ?? 5, ?? 6} 
Problem 3: Find the sample space associated with the experiment of rolling a pair of dice 
(plural of die) once. Also find the number of elements of the sample space. 
Solution: Let one die be blue and the other be green. Suppose ' 1 ' appears on blue die 
and ' 2 ' appears on green die. We denote this outcome by an ordered pair (1,2). 
Similarly, if ' 3 ' appears on blue die and ' 5 ' appears on green die, we denote this 
outcome by (3,5) and so on. Thus, each outcome can be denoted by an ordered pair 
(?? , ?? ), where ?? is the number appeared on the first die (blue die) and y appeared on the 
second die (green die). Thus, the sample space is given by 
?? = {(?? , ?? )?? is the number on blue die and ?? is the number on green die } 
We now list all the possible outcomes (figure) 
 
 1 2 3 4 5 6 
1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) 
2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) 
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) 
 
Figure 
Number of elements (outcomes) of the above sample space is 6 × 6 i.e., 36 
 
It is subset of sample space. e.g. getting a head in tossing a coin or getting a prime 
number in throwing a die. In general if a sample space consists ' ?? ' elements, then a 
maximum of 2
?? events can be associated with it. 
 
(iii) Complement of event: 
The complement of an event 'A' with respect to a sample space ?? is the set of all elements 
of ' ?? ' which are not in A. It is usually denoted by ?? '
, or ?? ?? . 
(iv) Simple event: 
If an event covers only one point of sample space, then it is called a simple event e.g. 
getting a head followed by a tail in throwing of a coin 2 times is a simple event. 
(v) Compound event: 
When two or more than two events occur simultaneously, the event is said to be a 
compound event. Symbolically ?? n ?? or ???? represent the occurrence of both ?? &?? 
simultaneously. 
Note: "A B" or A + B represent the occurrence of either ?? or ?? . 
Problem 4: Write down all the events of the experiment 'tossing of a coin'. 
Solution:  ?? = {?? , ?? } 
the events are ?? , {?? }, {?? }, {?? , ?? } 
Problem 5: A die is thrown. Let ?? be the event ' an odd number turns up' and ?? be the 
event 'a number  ?????????????????? ???? 3 ?????????? ???? '. ?????????? ?? h?? ????????????   (?? ) ?? ???? ??  (?? ) ?? ?????? ??  
Solution:  ?? = {1,3,5}, ?? = {3,6} 
?  ?? ???? ?? = ?? ? ?? = {1,3,5,6}  ?? ?????? ?? = ?? n ?? = {3}  
 (vii) Equally likely events: 
If events have same chance of occurrence, then they are said to be equally likely. e.g 
(i) In a single toss of a fair coin, the events {?? } and {?? } are equally likely. 
(ii) In a single throw of an unbiased die the events {1}, {2}, {3} and {4}, are equally likely. 
(iii) In tossing a biased coin the events {?? } and {?? } are not equally likely. 
(viii) Mutually exclusive / disjoint / incompatible events: 
Two events are said to be mutually exclusive if occurrence of one of them rejects the 
possibility of occurrence of the other i.e. both cannot occur simultaneously. 
In the vein diagram the events ?? and ?? are mutually exclusive. Mathematically, we write 
?? n ?? = ?? 
Events ?? 1
, ?? 2
, ?? 3
, … . . ?? ?? are said to be mutually exclusive events iff ?? ?? n ?? ?? = ?? ??? , ?? ?
{1,2, … , ?? } where ?? ? ?? 
 
Note: If ?? ?? n ?? ?? = ?? ??? , ?? ? {1,2, … , ?? } where ?? ? ?? , then ?? 1
n ?? 2
n ?? 3
… .n ?? ?? = ?? but 
converse need not to be true. 
Problem 6: In a single toss of a coin find whether the events {?? }, {?? } are mutually 
exclusive or not. 
Solution:   Since {?? } n {?? } = ?? , 
?  the events are mutually exclusive. 
Problem 7: In a single throw of a die, find whether the events {1,2}, {2,3} are mutually 
exclusive or not. 
Solution: Since {1,2} n {2,3} = {2} ? ?? ?  the events are not mutually exclusive. 
 (vi) Exhaustive system of events: 
If each outcome of an experiment is associated with at least one of the events 
?? 1
, ?? 2
, ?? 3
, … … . . ?? ?? , then collectively the events are said to be exhaustive. Mathematically 
we write 
?? 1
? ?? 2
? ?? 3
… … … ?? ?? = ?? . (Sample space) 
Problem 8: In throwing of a die, let A be the event 'even number turns up', B be the event 
'an odd prime turns up' and ?? be the event 'a numbers less than 4 turns up'. Find 
whether the events ?? , ?? and ?? form an exhaustive system or not. 
Solution:  ?? = {2,4,6}, ?? = {3,5} and ?? = {1,2,3}. 
Clearly ?? ? ?? ? ?? = {1,2,3,4,5,6} = ?? . Hence the system of events is exhaustive. 
Problem 9: Three coins are tossed. Describe 
(i) two events ?? and ?? which are mutually exclusive 
(ii) three events ?? , ?? and ?? which are mutually exclusive and exhaustive. 
(iii) two events ?? and ?? which are not mutually exclusive. 
(iv) two events ?? and ?? which are mutually exclusive but not exhaustive. 
(v) three events ?? , ?? and ?? which are mutually exclusive but not exhaustive. 
Ans. (i) A: "getting at least two heads" 
B: "getting at least two tails" 
(ii) A: "getting at most one heads" 
B: "getting exactly two heads" 
C: "getting exactly three heads" 
(iii) A: "getting at most two tails" 
B: "getting exactly two heads" 
(iv) A: "getting exactly one head" 
B: "getting exactly two heads" 
(v) A: "getting exactly one tail" 
B: "getting exactly two tails" 
C: "getting exactly three tails" 
[Note: There may be other cases also] 
 (II) Classical (a priori) definition of 
probability: 
If an experiment results in a total of (?? + ?? ) outcomes which are equally likely and if ' ?? 
' outcomes are favorable to an event 'A' while ' ?? ' are unfavorable, then the probability of 
occurrence of the event 'A', denoted by ?? (?? ), is defined by 
?? ?? +?? =
 ???????????? ???? ???????????????????? ???????????????? 
 ?????????? ???????????? ???? ???????????????? 
 
i.e.  ?? (?? ) =
?? ?? +?? .