Mathematical Classical A Priori Definition of Probability
A Priori
In the sense in which it is used in the subject of probability, it means
 Derived by logic, without observed facts
 Involving deductive reasoning from a general principle to a necessary effect; not supported by fact
 Based on hypothesis or theory rather than experiment
How do we know that the six elementary events of getting 1, 2, ..., 6 on tossing a die are equally likely?
We did not conduct any experiment.
Then how did we conclude so?
We came to such a conclusion on the acceptance of the notion that the die behaves in an unbiased manner. Moreover, we also accept and make use of the notion that the six elementary events are equally likely.
All this is based on conclusions drawn by earlier studies or by logical reasoning.
We have not conducted any study to attribute the probability for any of these elementary events.
A Priori Probabilities
Probabilities which are based on reasoning and generally accepted principles or notions are called a priori probabilities.The Mathematical or classical definition of probability is an a priori definition.
Mathematical/Classical/A Priori Definition of Probability
Probability of Occurrence of an Event
= 

⇒ P(E)  = 

E represents the event.
Probability of Success
The number of Favorable choices is also identified as the number of successes for the event and the probability of occurrence of an event is also identified as Probability of successProbability of Success for an Event
= 

⇒ P(E)  = 

Probability of NonOccurrence of the Event
= Total Number of Possible Choices in the Experiment − Number of Favorable Choices for the Event.
⇒ m_{E}^{c} = n − m_{E}
Where E^{c} represents the event of the nonoccurrence of the Event.
Probability of Non Occurrence of the Event
= 

⇒ P(E^{c})  = 

Probability of Failure
The number of Unfavorable choices is also identified as the number of failures for the event and the probability of nonoccurrence of an event is also identified as Probability of failure∴ Probability of Failures for the Event
= 

⇒ P(E)  = 

Probability of Occurrence + Probability of NonOccurrence = 1
⇒ Probabilities of the occurrence (success) and nonoccurrence (failure) of the events are compliments
P(E^{c})  = 
 
= 
 
= 
 
=  1 − P(E) 
⇒ P(E) + P(E^{c}) = 1
The sum of the probabilities of occurrence and nonoccurrence of an event is 1.
Probability of Success + Probability of Failure = 1
The probability of occurrence of an event is identified as Probability of success and the probability of nonoccurrence of the event is identified as Probability of Failure.∴ P(E) + P(E^{c}) = 1
⇒ Probability of Success + Probability of Failure = 1
Formula Interpretation
m, m^{c} are nonnegative integers
The values m and m^{c} represent number of choices and as such they cannot be
 negative
 fractional values
They can be either positive integers or zero.
Zero being neither negative nor positive, we can say they are nonnegative integers.
n is a positive integer
n represents the total number of choices and as such it cannot be
 negative
 fractional values
It can only be a positive integer.
It cannot be Zero as it amounts to saying there are no outcomes in the experiment.
Probabilities are positive Rational Numbers
The values m, m^{c} and n are nonnegative integers and n ≠ 0.Therefore, the ratios
andm n
are nonnegative rational numbersm^{c} n Since P(Event) =
and P(Event^{c}) =m n
, we can say that the probabilities are positive rational numbers.m^{c} n 0 ≤ P(Event) ≥ 1
The value of m, ranges between 0 and n.⇒ The value of
ranges between 0 {m n
} and 1 {0 n
}.n n ⇒ The value of P(Event) ranges between 0 and 1
⇒ 0 ≤ P(A) ≤ 1
0 ≤ m ≥ n
The value of m ranges between 0 and n.
 Minimum value = zero
when there are no favorable choices for the event
 Maximum value = n
where all the possible choices form favorable choices for the event
 Minimum value = zero
0 ≤ P(Event^{c}) ≥ 1
P(A^{c}) = 1 − P(A)It is
= 1, When P(A) = 0
= 0, When P(A) = 1
between 0 and 1 when 0 < P(A) > 1
⇒ The value of P(A^{c}) ranges between 0 and 1
⇒ 0 ≤ P(A^{c}) ≤ 1.
Impossible Event
Any event whose probability of occurrence is 0 is an impossible event.Event A is called an impossible event, when P(A) = 0
Certain Event
Any event whose probability of occurrence is 1 is a certain event.Event A is called a certain event, where P(A) = 1