Basic Probability Theory

Probability is a value that used to measure the rate of occurrence of an random event. The experiment is a terms that widely used in probability theory. Simple examples of experiment in probability theory is tossing of a pair coin and the experiment of throwing a dice. Two main basic part of probability is Sample Space and Event.

In an experiment known terms of Sample Space, Sample Space (S) is the set of all possible outcomes of an experiment. Examples of experiments throwing a six-sided dice, so the sample space it is

S = {1, 2, 3, 4, 5, 6}

Event  is a subset of the sample space. Events usually denoted by the letters A, B and others. Examples, appear side the dice is odd number in an experiment of throwing a dice. 

A = {Event appear odd number} = {1,3,5}

The Probability of an event, such as event A is written as P(A), then the probability of event A is defined as follows:

Where  x is : the number of elements in the Event or occurrence frequency
           n    is : all the elements in the sample space of events or number of observations

Value or range of an Odds from an event is between 0 ≤ P(A) ≤1

Pattern of Events in Probability Theory

Complement or Opposite

Let [S] be the sample space of an event, and [A] is a subset of [S] then even [not A] or the complement of the event [A] is a subset of [S] which are not members of the even [A].

Intersection or Joint Probability
Intersection of two events are events that contain all the same elements from events A and B.
Intersection of two events denoted by A ∩ B
Example :
Event A = {1,2,3,4,5}, event B = {2,4,6,8}  so A ∩ B : {2,4}
P(A and B) = P (A ∩ B) = P(A)P(B)

Union
Union is the set of events that includes all the elements  in set A, B or both. The union of two events A and B is denoted as A ∪ B
Example: A = {2,3,5,8} and B = {3,6,8} then A ∪ B = {2,3,5,6,8}
P(A ∪ B) = P(A)+P(B) - P (A ∩ B) if the two event are not mutually exclusive
P(A ∪ B) = P(A)+P(B) if the two event are mutually exclusive

Mutually exclusive
Mutually exclusive are two events occur that do not have intersections.

Conditional Probability
Two events said have a conditional probability if the one event become condition to  occurance of the other event. The conditional probability is written P(A|B) and it's read "the probability of event A is given by event B". It's defined by