Probability : Mathematical Classical A Priori Definition of Probability

A Priori

This is a Latin phrase which means "From what comes before"

» Meaning

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 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

There is an experiment with 'n' mutually exclusive, equally likely, and exhaustive elementary events. 'm' of the elementary events are favourable/favorable to the occurrence of event "A". Then the probability of occurrence of the event "A" (represented by P(A)) is given by the ratio of the "Number of favourable/favorable choices for the event" to the "Total number of possible choices in the experiment".

Probability of Occurrence of Event "A" =
Number of Favourable/Favorable Choices for the Event
Total Number of Possible Choices in the Experiment

⇒ P(A) =
m_A
n

• Probability of Success

The number of Favourable/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 success"

Therefore,

Probability of Success for Event "A" =
Number of Successes for the Event
Total Number of Possible Choices in the Experiment

⇒ P(A) =
m_A
n

Probability of Non-Occurrence of an Event

Number of Unfavourable/Unfavorable choices for the Event	=	Total Number of Possible Choices in the Experiment − Number of Favourable/Favorable Choices for the Event.
⇒ m_A^c	=	n − m_A

Where "A^c" represents the event of the non-occurrence of the Event "A",

Probability of Non-Occurrence of Event "A" =
Number of UnFavourable/UnFavorable Choices for the Event
Total Number of Possible Choices in the Experiment

⇒ P(A^c) =
m_A^c
n

Probability of Failure

The number of Unfavourable/Unfavorable choices is also identified as the number of failures for the event and the probability of non-occurrence of an event is also identified as "Probability of failures"

Therefore,

Probability of Failures for Event "A" =
Number of Failures for the Event
Total Number of Possible Choices in the Experiment

⇒ P(A^c) =
m_A^c
n

Probability of Occurrence + Probability of Non-Occurrence = 1

The events of occurrence (success) and non-occurrence (failure) are mutual complimentaries.
⇒ Probabilities of the occurrence (success) and non-occurrence (failure) of the events are complimentaries

We know, P(A^c) =
m_A^c
n

=
n − m_A
n

=

n
n
−
m_A
n

= 1 − P(A)

⇒ P(A) + P(A^c) = 1

The sum of the probabilities of occurrence and non-occurrence of an event is 1.

• Probability of Success + Probability of Failure = 1

Since the probability of occurrence of an event is identified as "Probability of success" and the probability of non-occurrence of the event is identified as "Probability of Failure"

P(A) + P(A^c) = 1

⇒ For an event, Probability of Success + Probability of Failure = 1

Probability (classical/mathematical definition) - Formula Interpretation

• "m", m^c are non-negative integers

The values m and m^c represent number of choices, they cannot be negative.

They can be either positive or zero [Zero is neither negative nor positive].

• "n" is a positive integer

n is 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 non-negative integers and n ≠ 0,

Therefore, the ratios

[= P(Event)] and

m^c

[= P(Event^c)] are non-negative rational numbers

• 0 ≤ m ≥ n

The value of "m" ranges between 0 and n.

Minimum value = zero, when there are no favourable/favorable choices for the event.
Maximum value = n, where all the possible choices form favourable/favorable choices for the event.

• 0 ≤ P(Event) ≥ 1

The value of "m", ranges between 0 and "n",

⇒ The value of

ranges between 0 {

} and 1 {

}

⇒ The value of P(A) ranges between 0 and 1 [0 ≤ P(A) ≤ 1].

• 0 ≤ P(Event^C) ≥ 1

P(A^c)	=	1 − P(A)
	=	1	[When P(A) = 0]
	=	0	[When P(A) = 1]

⇒ The value of P(A^c) ranges between 0 and 1

⇒ 0 ≤ P(A^c) ≤ 1.

• Impossible and Certain Events

Any event whose probability of occurrence is 0 is an impossible event.
[Event "A" is called an impossible event, where P(A) = 0]

Any event whose probability of occurrence is 1 is a certain event.
[Event "A" is called a certain event, where P(A) = 1]

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