Modern Definition of Probability. Set Theoretic Approach
Sample Space  Sample Points
Sample Space
The sample space of a random experiment is the set of all possible elementary events or outcomes for the experiment. It is also called the universal sample space.
 It is a nonempty set.
 It is denoted as  S, Ω or U [for universe].
Examples
 For the experiment of throwing a coin,
S = {H, T}
S represents the sample space.
H the elementary event of getting a head
T the elementary event of getting a tail.
For the experiment of rolling a dice,
Ω = {1, 2, 3, 4, 5, 6}
Ω represents the sample space,
1 the elementary event of the number 1 appearing on the top of the dice,
2 the elementary event of the number 1 appearing on the top of the dice,
...
6 the elementary event of the number 6 appearing on the top of the dice.
For the experiment of tossing 3 coins,
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
S represents the sample space,
HHH the elementary event of all three coins showing up heads,
HHT the elementary event of the 1^{st} and 2^{nd} coins showing up heads and the 3^{rd} coin showing up tails,
...
TTT the elementary event of all three coins showing up tails.
For the experiment of rolling 2 dice
S = { (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
...
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)} For the experiment of throwing 3 dice
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)}
Event
Any subset of the sample space forms an an event in relation to the experiment.Examples
In the experiment of rolling a dice, where the sample space is
Ω = {1, 2, 3, 4, 5, 6}
Event of getting an even number (say Event A), would be represented as
A = {2, 4, 6}
Event of getting a prime number (say Event H), would be represented as
H = {2, 3, 5}
Event of getting a multiple of 3 (say Event F), would be represented as
F = {3, 6}
The sets A, H and F are all subsets of Ω
In the experiment of tossing 3 coins, where the sample space is
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Event of getting two heads (say Event C), would be represented as
C = {HHT, HTH, THH}
Event of getting all three of the same kind (say Event G), would be represented as
G = {HHH, TTT}
The sets C and G are all subsets of S
Sample Points
Each elementary event within a sample space is called 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
 An element of a set.
 A set with a single element.
Modern Definition of Probability
Combinatorics
A branch of pure mathematics concerning the study of discrete (and usually finite) objects. Graph theory, Permutations, Combinations are some topics that fall within the purview of combinatorics.Pure mathematics
That part of mathematical activity that is done without explicit or immediate consideration of direct application. The pure status may not be permanent as what is pure in one era often becomes applied later.Applied mathematics
A branch of mathematics that concerns itself with the application of mathematical knowledge to other domains.Limitations of Classical Definition
 The classical theory assumes finite outcomes in the experiment. This renders it inapplicable to some important random experiments, such as tossing a coin until a head appears, which give rise to the possibility infinite set of outcomes.
 Classical definition dealt with discrete variables. Analytical considerations compelled the incorporation of continuous variables into the theory wherein the classical definition was found lacking.
 Another limitation of the classical definition was the condition that each possible outcome is equally likely. This rendered the definition circular  since probability is used to define the idea of probability.
Evolution of modern definition
The limitations of the classical definition led to the evolution of the modern definition of probability which is based on the concept of sets and has an axiomatic approach.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
Each element x ∈ Ω has a related probability value attached to it such that it satisfies the following properties. 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.
 $\sum _{x\in \Omega}f(x)=1$
The probability of the entire sample space is 1.
 The Probability of an Event (E) which is a subset of the sample space (Ω) is given by $P(E)={\displaystyle \sum _{x\in E}f(x)}$
The probability of an event is the sum of the probabilities of the individual elementary events forming the event.
Modern Definition of Probability  Set theoretic Approach
Probability of occurrence of an event is given by the ratio of the cardinality of the set of all the elementary events favorable/favourable to the occurrence of the event and the set of all possible elementary events in relation to the experiment i.e. the sample space.Where
 S represents the sample space and
 A is a subset of S
The probability of occurrence of event A
= 

n(A) 
n(S) 
Probability Axioms  Kolmogorov Axioms
Axiom
In mathematics, an axiom is any starting assumption from which other statements are logically derived. It can be a sentence, a proposition, a statement or a rule that enables the construction of a formal system.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.
Probability Axioms  Kolmogorov Axioms
The modern definition of probability is based on the Kolmogorov axiomsFirst Axiom
The probability of an event is a nonnegative real numberP (E) ≥ 0 ∀ E ∈ Ω
Second Axiom
The probability that any one of the elementary events in the entire sample set will occur is 1P (Ω) = 1
Third Axiom
Any countable sequence of pair wise disjoint events E_{1}, E_{2}, ... satisfiesP(E_{1} U E_{2} U ...) = $\sum _{x\in i}f(x)}=P({E}_{i})$