Probability - Notes, Module 1, Lecture 1-6, Probability and Distributions JEE Notes | EduRev

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JEE : Probability - Notes, Module 1, Lecture 1-6, Probability and Distributions JEE Notes | EduRev

 Page 1


 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   1
   
 
MODULE 1 
PROBABILITY 
LECTURES 1-6 
Topics  
1.1 INTRODUCTION  
1.1.1 Classical Method 
1.1.2 Relative Frequency Method 
1.2 AXIOMATIC APPROACH TO PROBABILITY AND  
         PROPERTIES OF PROBABILITY MEASURE 
1.2.1 Inclusion-Exclusion Formula 
1.2.1.1 Boole’s Inequality 
1.2.1.2 Bonferroni’s Inequality 
1.2.2 Equally Likely Probability Models 
1.3 CONDITIONAL PROBABILITY AND 
         INDEPENDENCE OF EVENTS        
1.3.1 Theorem of Total Probability 
1.3.2 Bayes’ Theorem 
 
1.4 CONTINUITY OF PROBABILITY MEASURES 
 
MODULE 1 
PROBABILITY 
LECTURE 1 
Topics 
1.1 INTRODUCTION  
1.1.1 Classical Method 
1.1.2 Relative Frequency Method  
 
Page 2


 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   1
   
 
MODULE 1 
PROBABILITY 
LECTURES 1-6 
Topics  
1.1 INTRODUCTION  
1.1.1 Classical Method 
1.1.2 Relative Frequency Method 
1.2 AXIOMATIC APPROACH TO PROBABILITY AND  
         PROPERTIES OF PROBABILITY MEASURE 
1.2.1 Inclusion-Exclusion Formula 
1.2.1.1 Boole’s Inequality 
1.2.1.2 Bonferroni’s Inequality 
1.2.2 Equally Likely Probability Models 
1.3 CONDITIONAL PROBABILITY AND 
         INDEPENDENCE OF EVENTS        
1.3.1 Theorem of Total Probability 
1.3.2 Bayes’ Theorem 
 
1.4 CONTINUITY OF PROBABILITY MEASURES 
 
MODULE 1 
PROBABILITY 
LECTURE 1 
Topics 
1.1 INTRODUCTION  
1.1.1 Classical Method 
1.1.2 Relative Frequency Method  
 
 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   2
   
 
1.1 INTRODUCTION  
In our daily life we come across many processes whose nature cannot be predicted in 
advance. Such processes are referred to as random processes.  The only way to derive 
information about random processes is to conduct experiments. Each such experiment 
results in an outcome which cannot be predicted beforehand. In fact even if the 
experiment is repeated under identical conditions, due to presence of factors which are 
beyond control, outcomes of the experiment may vary from trial to trial. However we 
may know in advance that each outcome of the experiment will result in one of the 
several given possibilities. For example, in the cast of a die under a fixed environment the 
outcome (number of dots on the upper face of the die) cannot be predicted in advance and 
it varies from trial to trial. However we know in advance that the outcome has to be 
among one of the numbers 1, 2,… , 6. Probability theory deals with the modeling and 
study of random processes. The field of Statistics is closely related to probability theory 
and it deals with drawing inferences from the data pertaining to random processes. 
Definition 1.1 
(i) A random experiment is an experiment in which: 
(a) the set of all possible outcomes of the experiment is known in advance; 
(b) the outcome of a particular performance (trial) of the experiment cannot be 
predicted in advance; 
(c) the experiment can be repeated under identical conditions. 
(ii) The collection of all possible outcomes of a random experiment is called the 
sample space. A sample space will usually be denoted by ?? . _    
Example 1.1 
(i) In the random experiment of casting a die one may take the sample space as 
?? = 1, 2, 3, 4, 5, 6 , where ?? ? ?? indicates that the experiment results in ??   ?? =
1,…,6) dots on the upper face of die. 
(ii) In the random experiment of simultaneously flipping a coin and casting a die one 
may take the sample space as 
 
?? = ?? ,?? × 1, 2,… , 6 =  ?? ,?? ): ?? ? ?? ,?? ,?? ? 1, 2,… , 6   
where  ?? ,?? )  ?? ,?? )  indicates that the flip of the coin resulted in head (tail) on the 
upper face and the cast of the die resulted in ??  ?? = 1, 2,… , 6) dots on the upper 
face.  
(iii) Consider an experiment where a coin is tossed repeatedly until a head is observed. 
In this case the sample space may be taken as ?? = 1, 2,… (or 
Page 3


 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   1
   
 
MODULE 1 
PROBABILITY 
LECTURES 1-6 
Topics  
1.1 INTRODUCTION  
1.1.1 Classical Method 
1.1.2 Relative Frequency Method 
1.2 AXIOMATIC APPROACH TO PROBABILITY AND  
         PROPERTIES OF PROBABILITY MEASURE 
1.2.1 Inclusion-Exclusion Formula 
1.2.1.1 Boole’s Inequality 
1.2.1.2 Bonferroni’s Inequality 
1.2.2 Equally Likely Probability Models 
1.3 CONDITIONAL PROBABILITY AND 
         INDEPENDENCE OF EVENTS        
1.3.1 Theorem of Total Probability 
1.3.2 Bayes’ Theorem 
 
1.4 CONTINUITY OF PROBABILITY MEASURES 
 
MODULE 1 
PROBABILITY 
LECTURE 1 
Topics 
1.1 INTRODUCTION  
1.1.1 Classical Method 
1.1.2 Relative Frequency Method  
 
 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   2
   
 
1.1 INTRODUCTION  
In our daily life we come across many processes whose nature cannot be predicted in 
advance. Such processes are referred to as random processes.  The only way to derive 
information about random processes is to conduct experiments. Each such experiment 
results in an outcome which cannot be predicted beforehand. In fact even if the 
experiment is repeated under identical conditions, due to presence of factors which are 
beyond control, outcomes of the experiment may vary from trial to trial. However we 
may know in advance that each outcome of the experiment will result in one of the 
several given possibilities. For example, in the cast of a die under a fixed environment the 
outcome (number of dots on the upper face of the die) cannot be predicted in advance and 
it varies from trial to trial. However we know in advance that the outcome has to be 
among one of the numbers 1, 2,… , 6. Probability theory deals with the modeling and 
study of random processes. The field of Statistics is closely related to probability theory 
and it deals with drawing inferences from the data pertaining to random processes. 
Definition 1.1 
(i) A random experiment is an experiment in which: 
(a) the set of all possible outcomes of the experiment is known in advance; 
(b) the outcome of a particular performance (trial) of the experiment cannot be 
predicted in advance; 
(c) the experiment can be repeated under identical conditions. 
(ii) The collection of all possible outcomes of a random experiment is called the 
sample space. A sample space will usually be denoted by ?? . _    
Example 1.1 
(i) In the random experiment of casting a die one may take the sample space as 
?? = 1, 2, 3, 4, 5, 6 , where ?? ? ?? indicates that the experiment results in ??   ?? =
1,…,6) dots on the upper face of die. 
(ii) In the random experiment of simultaneously flipping a coin and casting a die one 
may take the sample space as 
 
?? = ?? ,?? × 1, 2,… , 6 =  ?? ,?? ): ?? ? ?? ,?? ,?? ? 1, 2,… , 6   
where  ?? ,?? )  ?? ,?? )  indicates that the flip of the coin resulted in head (tail) on the 
upper face and the cast of the die resulted in ??  ?? = 1, 2,… , 6) dots on the upper 
face.  
(iii) Consider an experiment where a coin is tossed repeatedly until a head is observed. 
In this case the sample space may be taken as ?? = 1, 2,… (or 
 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   3
   
 
?? = {T, TH, TTH,… }), where ?? ? ?? (or TT? TH ? ?? with  ?? - 1) Ts and one H) 
indicates that the experiment terminates on the ?? -th trial with first ?? - 1 trials 
resulting in tails on the upper face and the ?? -th trial resulting in the head on the 
upper face. 
(iv)  In the random experiment of measuring lifetimes (in hours) of a particular brand 
of batteries manufactured by a company one may take ?? = 0,70,000 , where we 
have assumed that no battery lasts for more than 70,000 hours. _ 
Definition 1.2  
(i) Let ?? be the sample space of a random experiment and let ?? ? ?? . If the outcome of 
the random experiment is a member of the set ?? we say that the event ?? has occurred. 
(ii)  Two events ?? 1
and ?? 2
are said to be mutually exclusive if they cannot occur 
simultaneously, i.e., if ?? 1
n?? 2
= ?? , the empty set. _ 
In a random experiment some events may be more likely to occur than the others. For 
example, in the cast of a fair die (a die that is not biased towards any particular 
outcome), the occurrence of an odd number of dots on the upper face is more likely 
than the occurrence of 2 or 4 dots on the upper face. Thus it may be desirable to 
quantify the likelihoods of occurrences of various events. Probability of an event is a 
numerical measure of chance with which that event occurs. To assign probabilities to 
various events associated with a random experiment one may assign a real number 
?? ?? ) ? 0,1  to each event ?? with the interpretation that there is a  100 ×?? ?? ) % 
chance that the event ?? will occur and a  100 × 1-?? ?? )  % chance that the 
event ?? will not occur. For example if the probability of an event is 0.25 it would 
mean that there is a 25% chance that the event will occur and that there is a 75% 
chance that the event will not occur. Note that, for any such assignment of 
possibilities to be meaningful, one must have ?? ?? ) = 1. Now we will discuss two 
methods of assigning probabilities. 
1.1.1 Classical Method 
This method of assigning probabilities is used for random experiments which result in 
a finite number of equally likely outcomes. Let ?? = ?? 1
,… ,?? ??  be a finite sample 
space with ??  ? N) possible outcomes; here N denotes the set of natural numbers. For 
?? ? ?? , let  ??  denote the number of elements in ?? . An outcome ?? ? ?? is said to be 
favorable to an event ?? if ?? ? ?? . In the classical method of assigning probabilities, 
the probability of an event ?? is given by 
    ?? ?? ) =
number of outcomes favorable to ?? total number of outcomes
=
 ?? 
 ?? 
=
 ?? 
?? . 
Page 4


 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   1
   
 
MODULE 1 
PROBABILITY 
LECTURES 1-6 
Topics  
1.1 INTRODUCTION  
1.1.1 Classical Method 
1.1.2 Relative Frequency Method 
1.2 AXIOMATIC APPROACH TO PROBABILITY AND  
         PROPERTIES OF PROBABILITY MEASURE 
1.2.1 Inclusion-Exclusion Formula 
1.2.1.1 Boole’s Inequality 
1.2.1.2 Bonferroni’s Inequality 
1.2.2 Equally Likely Probability Models 
1.3 CONDITIONAL PROBABILITY AND 
         INDEPENDENCE OF EVENTS        
1.3.1 Theorem of Total Probability 
1.3.2 Bayes’ Theorem 
 
1.4 CONTINUITY OF PROBABILITY MEASURES 
 
MODULE 1 
PROBABILITY 
LECTURE 1 
Topics 
1.1 INTRODUCTION  
1.1.1 Classical Method 
1.1.2 Relative Frequency Method  
 
 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   2
   
 
1.1 INTRODUCTION  
In our daily life we come across many processes whose nature cannot be predicted in 
advance. Such processes are referred to as random processes.  The only way to derive 
information about random processes is to conduct experiments. Each such experiment 
results in an outcome which cannot be predicted beforehand. In fact even if the 
experiment is repeated under identical conditions, due to presence of factors which are 
beyond control, outcomes of the experiment may vary from trial to trial. However we 
may know in advance that each outcome of the experiment will result in one of the 
several given possibilities. For example, in the cast of a die under a fixed environment the 
outcome (number of dots on the upper face of the die) cannot be predicted in advance and 
it varies from trial to trial. However we know in advance that the outcome has to be 
among one of the numbers 1, 2,… , 6. Probability theory deals with the modeling and 
study of random processes. The field of Statistics is closely related to probability theory 
and it deals with drawing inferences from the data pertaining to random processes. 
Definition 1.1 
(i) A random experiment is an experiment in which: 
(a) the set of all possible outcomes of the experiment is known in advance; 
(b) the outcome of a particular performance (trial) of the experiment cannot be 
predicted in advance; 
(c) the experiment can be repeated under identical conditions. 
(ii) The collection of all possible outcomes of a random experiment is called the 
sample space. A sample space will usually be denoted by ?? . _    
Example 1.1 
(i) In the random experiment of casting a die one may take the sample space as 
?? = 1, 2, 3, 4, 5, 6 , where ?? ? ?? indicates that the experiment results in ??   ?? =
1,…,6) dots on the upper face of die. 
(ii) In the random experiment of simultaneously flipping a coin and casting a die one 
may take the sample space as 
 
?? = ?? ,?? × 1, 2,… , 6 =  ?? ,?? ): ?? ? ?? ,?? ,?? ? 1, 2,… , 6   
where  ?? ,?? )  ?? ,?? )  indicates that the flip of the coin resulted in head (tail) on the 
upper face and the cast of the die resulted in ??  ?? = 1, 2,… , 6) dots on the upper 
face.  
(iii) Consider an experiment where a coin is tossed repeatedly until a head is observed. 
In this case the sample space may be taken as ?? = 1, 2,… (or 
 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   3
   
 
?? = {T, TH, TTH,… }), where ?? ? ?? (or TT? TH ? ?? with  ?? - 1) Ts and one H) 
indicates that the experiment terminates on the ?? -th trial with first ?? - 1 trials 
resulting in tails on the upper face and the ?? -th trial resulting in the head on the 
upper face. 
(iv)  In the random experiment of measuring lifetimes (in hours) of a particular brand 
of batteries manufactured by a company one may take ?? = 0,70,000 , where we 
have assumed that no battery lasts for more than 70,000 hours. _ 
Definition 1.2  
(i) Let ?? be the sample space of a random experiment and let ?? ? ?? . If the outcome of 
the random experiment is a member of the set ?? we say that the event ?? has occurred. 
(ii)  Two events ?? 1
and ?? 2
are said to be mutually exclusive if they cannot occur 
simultaneously, i.e., if ?? 1
n?? 2
= ?? , the empty set. _ 
In a random experiment some events may be more likely to occur than the others. For 
example, in the cast of a fair die (a die that is not biased towards any particular 
outcome), the occurrence of an odd number of dots on the upper face is more likely 
than the occurrence of 2 or 4 dots on the upper face. Thus it may be desirable to 
quantify the likelihoods of occurrences of various events. Probability of an event is a 
numerical measure of chance with which that event occurs. To assign probabilities to 
various events associated with a random experiment one may assign a real number 
?? ?? ) ? 0,1  to each event ?? with the interpretation that there is a  100 ×?? ?? ) % 
chance that the event ?? will occur and a  100 × 1-?? ?? )  % chance that the 
event ?? will not occur. For example if the probability of an event is 0.25 it would 
mean that there is a 25% chance that the event will occur and that there is a 75% 
chance that the event will not occur. Note that, for any such assignment of 
possibilities to be meaningful, one must have ?? ?? ) = 1. Now we will discuss two 
methods of assigning probabilities. 
1.1.1 Classical Method 
This method of assigning probabilities is used for random experiments which result in 
a finite number of equally likely outcomes. Let ?? = ?? 1
,… ,?? ??  be a finite sample 
space with ??  ? N) possible outcomes; here N denotes the set of natural numbers. For 
?? ? ?? , let  ??  denote the number of elements in ?? . An outcome ?? ? ?? is said to be 
favorable to an event ?? if ?? ? ?? . In the classical method of assigning probabilities, 
the probability of an event ?? is given by 
    ?? ?? ) =
number of outcomes favorable to ?? total number of outcomes
=
 ?? 
 ?? 
=
 ?? 
?? . 
 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   4
   
 
Note that probabilities assigned through classical method satisfy the following properties 
of intuitive appeal: 
(i) For any event ?? ,?? ?? ) = 0; 
(ii) For mutually exclusive events ?? 1
,?? 2
,… ,?? ??  (i.e.,?? ?? n?? ?? = ?? , whenever ?? , ?? ?
 1,… ,?? ,?? ? ?? )   
??  ?? ?? ?? ?? =1
 =
  ?? ?? ?? ?? =1
 
?? =
  ?? ?? 
?? ?? =1
?? = 
 ?? ?? 
?? ?? ?? =1
= ?? ?? ?? );
?? ?? =1
 
(iii) ?? ?? ) =
 ?? 
 ?? 
= 1. 
Example 1.2 
Suppose that in a classroom we have 25 students (with registration numbers 1, 2,… , 25) 
born in the same year having 365 days. Suppose that we want to find the probability of 
the event ?? that they all are born on different days of the year. Here an outcome consists 
of a sequence of 25 birthdays. Suppose that all such sequences are equally likely. Then 
 ?? = 365
25
, E = 365 × 364 ×? × 341 =
365
?? 25
 and  ?? ?? ) =
 ?? 
 ?? 
=
365
?? 25
365
25
· _ 
The classical method of assigning probabilities has a limited applicability as it can be 
used only for random experiments which result in a finite number of equally likely 
outcomes. _ 
1.1.2 Relative Frequency Method 
 
Suppose that we have independent repetitions of a random experiment (here independent 
repetitions means that the outcome of one trial is not affected by the outcome of another 
trial) under identical conditions. Let ?? ?? ?? ) denote the number of times an event ?? occurs 
(also called the frequency of event ?? in ?? trials) in the first ?? trials and let ?? ?? ?? ) =
?? ?? ?? )/?? denote the corresponding relative frequency. Using advanced probabilistic 
arguments (e.g., using Weak Law of Large Numbers to be discussed in Module 7) it can 
be shown that, under mild conditions, the relative frequencies stabilize (in certain sense)  
as ?? gets large (i.e., for any event ?? , lim
?? ? 8
r
N
 E) exists in certain sense). In the relative 
frequency method of assigning probabilities the probability of an event ?? is given by 
 
?? ?? ) = lim
?? ? 8
?? ?? ?? ) = lim
?? ? 8
?? ?? (?? )
?? · 
Page 5


 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   1
   
 
MODULE 1 
PROBABILITY 
LECTURES 1-6 
Topics  
1.1 INTRODUCTION  
1.1.1 Classical Method 
1.1.2 Relative Frequency Method 
1.2 AXIOMATIC APPROACH TO PROBABILITY AND  
         PROPERTIES OF PROBABILITY MEASURE 
1.2.1 Inclusion-Exclusion Formula 
1.2.1.1 Boole’s Inequality 
1.2.1.2 Bonferroni’s Inequality 
1.2.2 Equally Likely Probability Models 
1.3 CONDITIONAL PROBABILITY AND 
         INDEPENDENCE OF EVENTS        
1.3.1 Theorem of Total Probability 
1.3.2 Bayes’ Theorem 
 
1.4 CONTINUITY OF PROBABILITY MEASURES 
 
MODULE 1 
PROBABILITY 
LECTURE 1 
Topics 
1.1 INTRODUCTION  
1.1.1 Classical Method 
1.1.2 Relative Frequency Method  
 
 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   2
   
 
1.1 INTRODUCTION  
In our daily life we come across many processes whose nature cannot be predicted in 
advance. Such processes are referred to as random processes.  The only way to derive 
information about random processes is to conduct experiments. Each such experiment 
results in an outcome which cannot be predicted beforehand. In fact even if the 
experiment is repeated under identical conditions, due to presence of factors which are 
beyond control, outcomes of the experiment may vary from trial to trial. However we 
may know in advance that each outcome of the experiment will result in one of the 
several given possibilities. For example, in the cast of a die under a fixed environment the 
outcome (number of dots on the upper face of the die) cannot be predicted in advance and 
it varies from trial to trial. However we know in advance that the outcome has to be 
among one of the numbers 1, 2,… , 6. Probability theory deals with the modeling and 
study of random processes. The field of Statistics is closely related to probability theory 
and it deals with drawing inferences from the data pertaining to random processes. 
Definition 1.1 
(i) A random experiment is an experiment in which: 
(a) the set of all possible outcomes of the experiment is known in advance; 
(b) the outcome of a particular performance (trial) of the experiment cannot be 
predicted in advance; 
(c) the experiment can be repeated under identical conditions. 
(ii) The collection of all possible outcomes of a random experiment is called the 
sample space. A sample space will usually be denoted by ?? . _    
Example 1.1 
(i) In the random experiment of casting a die one may take the sample space as 
?? = 1, 2, 3, 4, 5, 6 , where ?? ? ?? indicates that the experiment results in ??   ?? =
1,…,6) dots on the upper face of die. 
(ii) In the random experiment of simultaneously flipping a coin and casting a die one 
may take the sample space as 
 
?? = ?? ,?? × 1, 2,… , 6 =  ?? ,?? ): ?? ? ?? ,?? ,?? ? 1, 2,… , 6   
where  ?? ,?? )  ?? ,?? )  indicates that the flip of the coin resulted in head (tail) on the 
upper face and the cast of the die resulted in ??  ?? = 1, 2,… , 6) dots on the upper 
face.  
(iii) Consider an experiment where a coin is tossed repeatedly until a head is observed. 
In this case the sample space may be taken as ?? = 1, 2,… (or 
 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   3
   
 
?? = {T, TH, TTH,… }), where ?? ? ?? (or TT? TH ? ?? with  ?? - 1) Ts and one H) 
indicates that the experiment terminates on the ?? -th trial with first ?? - 1 trials 
resulting in tails on the upper face and the ?? -th trial resulting in the head on the 
upper face. 
(iv)  In the random experiment of measuring lifetimes (in hours) of a particular brand 
of batteries manufactured by a company one may take ?? = 0,70,000 , where we 
have assumed that no battery lasts for more than 70,000 hours. _ 
Definition 1.2  
(i) Let ?? be the sample space of a random experiment and let ?? ? ?? . If the outcome of 
the random experiment is a member of the set ?? we say that the event ?? has occurred. 
(ii)  Two events ?? 1
and ?? 2
are said to be mutually exclusive if they cannot occur 
simultaneously, i.e., if ?? 1
n?? 2
= ?? , the empty set. _ 
In a random experiment some events may be more likely to occur than the others. For 
example, in the cast of a fair die (a die that is not biased towards any particular 
outcome), the occurrence of an odd number of dots on the upper face is more likely 
than the occurrence of 2 or 4 dots on the upper face. Thus it may be desirable to 
quantify the likelihoods of occurrences of various events. Probability of an event is a 
numerical measure of chance with which that event occurs. To assign probabilities to 
various events associated with a random experiment one may assign a real number 
?? ?? ) ? 0,1  to each event ?? with the interpretation that there is a  100 ×?? ?? ) % 
chance that the event ?? will occur and a  100 × 1-?? ?? )  % chance that the 
event ?? will not occur. For example if the probability of an event is 0.25 it would 
mean that there is a 25% chance that the event will occur and that there is a 75% 
chance that the event will not occur. Note that, for any such assignment of 
possibilities to be meaningful, one must have ?? ?? ) = 1. Now we will discuss two 
methods of assigning probabilities. 
1.1.1 Classical Method 
This method of assigning probabilities is used for random experiments which result in 
a finite number of equally likely outcomes. Let ?? = ?? 1
,… ,?? ??  be a finite sample 
space with ??  ? N) possible outcomes; here N denotes the set of natural numbers. For 
?? ? ?? , let  ??  denote the number of elements in ?? . An outcome ?? ? ?? is said to be 
favorable to an event ?? if ?? ? ?? . In the classical method of assigning probabilities, 
the probability of an event ?? is given by 
    ?? ?? ) =
number of outcomes favorable to ?? total number of outcomes
=
 ?? 
 ?? 
=
 ?? 
?? . 
 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   4
   
 
Note that probabilities assigned through classical method satisfy the following properties 
of intuitive appeal: 
(i) For any event ?? ,?? ?? ) = 0; 
(ii) For mutually exclusive events ?? 1
,?? 2
,… ,?? ??  (i.e.,?? ?? n?? ?? = ?? , whenever ?? , ?? ?
 1,… ,?? ,?? ? ?? )   
??  ?? ?? ?? ?? =1
 =
  ?? ?? ?? ?? =1
 
?? =
  ?? ?? 
?? ?? =1
?? = 
 ?? ?? 
?? ?? ?? =1
= ?? ?? ?? );
?? ?? =1
 
(iii) ?? ?? ) =
 ?? 
 ?? 
= 1. 
Example 1.2 
Suppose that in a classroom we have 25 students (with registration numbers 1, 2,… , 25) 
born in the same year having 365 days. Suppose that we want to find the probability of 
the event ?? that they all are born on different days of the year. Here an outcome consists 
of a sequence of 25 birthdays. Suppose that all such sequences are equally likely. Then 
 ?? = 365
25
, E = 365 × 364 ×? × 341 =
365
?? 25
 and  ?? ?? ) =
 ?? 
 ?? 
=
365
?? 25
365
25
· _ 
The classical method of assigning probabilities has a limited applicability as it can be 
used only for random experiments which result in a finite number of equally likely 
outcomes. _ 
1.1.2 Relative Frequency Method 
 
Suppose that we have independent repetitions of a random experiment (here independent 
repetitions means that the outcome of one trial is not affected by the outcome of another 
trial) under identical conditions. Let ?? ?? ?? ) denote the number of times an event ?? occurs 
(also called the frequency of event ?? in ?? trials) in the first ?? trials and let ?? ?? ?? ) =
?? ?? ?? )/?? denote the corresponding relative frequency. Using advanced probabilistic 
arguments (e.g., using Weak Law of Large Numbers to be discussed in Module 7) it can 
be shown that, under mild conditions, the relative frequencies stabilize (in certain sense)  
as ?? gets large (i.e., for any event ?? , lim
?? ? 8
r
N
 E) exists in certain sense). In the relative 
frequency method of assigning probabilities the probability of an event ?? is given by 
 
?? ?? ) = lim
?? ? 8
?? ?? ?? ) = lim
?? ? 8
?? ?? (?? )
?? · 
 
NPTEL- Probability and Distributions 
 
 
Dept. of Mathematics and statistics Indian Institute of Technology, Kanpur                                   5
   
 
 
Figure 1.1. Plot of relative frequencies (?? ?? ?? )) of number of heads against number of 
trials (N) in the random experiment of tossing a fair coin (with probability of head in each 
trial as 0.5). 
In practice, to assign probability to an event ?? , the experiment is repeated a large (but 
fixed) number of times (say ?? times) and the approximation ?? ?? ) ˜ ?? ?? ?? ) is used for 
assigning probability to event?? . Note that probabilities assigned through relative 
frequency method also satisfy the following properties of intuitive appeal: 
(i) for any event ?? ,?? ?? ) = 0; 
(ii) for mutually exclusive events ?? 1
,?? 2
,… ,?? ?? 
??  ?? ?? ?? ?? =1
 = ?? ?? ?? )
?? ?? =1
; 
(iii) ?? ?? ) = 1. 
Although the relative frequency method seems to have more applicability than the 
classical method it too has limitations. A major problem with the relative frequency 
method is that it is imprecise as it is based on an approximation  ?? ?? ) ˜ ?? ?? ?? ) . 
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