Modern Definition of Probability. Set Theoretic Approach 
Modern Definition of Probability. Set Theoretic Approach 
Sample Space » Sample Points 
Sample Space : S = {H, T}
Sample Space : O = {1, 2, 3, 4, 5, 6}
Sample Space : S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
S = {  (1,1),  (1,2),  ..,  ..,  ..,  (1,6)  
(2,1),  (2,2),  ..,  ..,  ..,  (2,6)  
...  
...  
(6,1),  (6,2),  ..,  ..,  ..,  (6,6)  } 
S = {  (1,1,1),  (1,1,2),  ..,  ..,  ..,  (1,1,6)  
(1,2,1),  (1,2,2),  ..,  ..,  ..,  (1,2,6)  
...  
...  
(6,6,1),  (6,6,2),  ..,  ..,  ..,  (6,6,6)  } 
Sample space : O = {1, 2, 3, 4, 5, 6}
A, H and F are all Subsets of O
Sample space : S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
C and G are Subsets of S.
Experiment : Tossing 3 coins
Sample space : S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Each of the elementary events in the sample space form a sample point.A subset of the sample space with just a single element (a sample point) forms an elementary event for the experiment.
The term sample point is used in two senses
'HHH' representing the event of getting all heads is a sample point.
M = {'HHH'}, Set 'M' is a sample point.
Limitations of Classiscal/Mathematical/Apriori Definition of Probability 
The foundations of the Modern Probability theory were laid by Andrey Nikolaevich Kolmogorov who combined the notion of sample space (introduced by Richard von Mises) and the measure theory and presented his axiom system for probability theory 1933. This forms the axiomatic basis for modern probability theory.
Modern Definition of Probability : Axiomatic Approach 
i  f(x) ∈ [0, 1] for all x ∈ Ω For every value of x in the sample space Ω, the probability function f(x) lies between zero & one. The probability of the null event is 0.  
ii. 
The probability of an event is the sum of the probabilities of the individual elementary events forming the event. The probability of the entire sample space is 1. 
Modern Definition of Probability : Set Theoretic Approach 
Where
= 
 (Or) 
 
⇒ P(A)  = 


⇒ Occurrence of an elementary event from the sample space is a certainity.
Probability of a certain event is 1 ⇒ P(S) = 1.
The probability of occurrence of event "S"  = 
 
⇒ P(S)  = 
 

=  1 
Probability Axioms » Kolmogorov Axioms 
Unlike theorems, axioms cannot be derived by principles of deduction, nor are they demonstrable by formal proofs  simply because they are starting assumptions  there is nothing else they logically follow from (otherwise they would be called theorems). In many contexts, "axiom," "postulate," and "assumption" are used interchangeably.
P(E_{1} U E_{2} U ...) =  Σ i  P(E_{i}) 
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