Mathematical Classical A Priori Definition of Probability

A Priori

Meaning

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

In the sense in which it is used in the subject of probability, it means

  1. Derived by logic, without observed facts.
  2. Involving deductive reasoning from a general principle to a necessary effect; not supported by fact.
  3. 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 do not conduct any study or experiment to attribute the probability for any of these events.

We come to such a conclusion on the acceptance of the notion that the die behaves in an unbiased manner and that the six elementary events are equally likely.

Our conclusions are based on conclusions drawn by earlier studies or by logical reasoning.

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

If there is an experiment with 'n' mutually exclusive, equally likely, and exhaustive elementary events and '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)=
mA
n

Probability of success

The number of Favourable/Favorable choices for the event is also identified as the number of successes for the event and the probability of occurrence of an event is also identified as the "Probability of success" for the event

Therefore,

Probability of Success for Event "A" =
Number of Successes for the Event
Total Number of Possible Choices in the Experiment
⇒ P(A)=
mA
n

Probability of Non-occurrence of an Event

Where Ac represents the event of the non-occurrence of the Event A,

Number of unfavourable/unfavorable
choices for the event

=

Total number of possible choices in the experiment
- Number of favourable/favorable choices for the event.
⇒ mAc = n - mA

Probability of Non-Occurrence of Event A =
No. of unfavourable choices for the event
Total number of possible choices in the experiment
⇒ P(Ac) =
mAc
n

Probability of Failure

The number of unfavourable/unfavorable choices for the event is also identified as the number of failures for the event and the probability of non-occurrence of the event is also identified as "Probability of failure" for the event

Therefore,

Probability of Failure of Event A =
Number of failures for the event
Total number of possible choices in the experiment
⇒ P(Ac) =
mAc
n

Probability of occurrence + Probability of Non-occurrence = 1

The events of occurrence (success) and non-occurrence (failure) are mutual complimentaries.
⇒ Probabilities of occurrence and non-occurrence of an event are complimentaries
    (Or) Probabilities of success and failure for an event are complimentaries

We know, P(Ac) =
mAc
n
=
n - mA
n
=
n
n
-
mA
n
= 1 - P(A)

⇒ P(A) + P(Ac) = 1

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

⇒ For any event, Probability of occurrence + Probability of non-occurrence = 1

Probability of success + Probability of failure = 1

Since 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"

For any event,

Probability of occurrence + Probability of non-occurrence = 1

⇒ Probability of success + Probability of failure = 1

Probability (classical/mathematical definition) - Formula Interpretation

m, mc are non-negative integers

The values m and mc 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, mc and n are non-negative integers and n ≠ 0 (n > 0),

Therefore,

the ratios
m
n
[P(Event)] and
mc
n
[P(Eventc)] are non-negative rational numbers

0 ≤ m ≤ n

The value of m ranges from 0 to n.
  • Minimum value = zero.

    m = 0, when there are no favourable/favorable choices for the event.

  • Maximum value = n.

    m = n, when all the possible choices form favourable/favorable choices for the event.

0 ≤ P(Event) ≤ 1

The value of m, ranges from 0 to n,

Therefore,

the value of
m
n
ranges between 0 {
0
n
} and 1 {
n
n
}

⇒ The value of P(Event) ranges between 0 and 1 (0 = P(Event) = 1).

0 ≤ P(EventC) ≤ 1

P(Eventc) = 1 - P(Event)
= 1(When P(Event) = 0)
= 0(When P(Event) = 1)

⇒ The value of P(Eventc) ranges between 0 and 1 (0 = P(Eventc) = 1)

Certain Event

Any event whose probability of occurrence is 1 is a certain event.

Event "A" is called a certain event, if P(A) = 1.

Impossible Event

Any event whose probability of occurrence is 0 is an impossible event.

Event "A" is called an impossible event, if P(A) = 0.

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