# A Priori

A Priori 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

• 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

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

Probability of Occurrence of an Event

=
 Number of Favorable Choices for the Event Total Number of Possible Choices for the Experiment
⇒ P(E) =
 mE n

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 success

Probability of Success for an Event

=
 Number of Successes for the Event Total Number of Possible Choices for the Experiment
⇒ P(E) =
 mE n

# Probability of Non-Occurrence of the Event

Number of Unfavorable choices for the Event

= Total Number of Possible Choices in the Experiment − Number of Favorable Choices for the Event.

⇒ mEc = n − mE

Where Ec represents the event of the non-occurrence of the Event.

Probability of Non Occurrence of the Event

=
 Number of Unfavorable Choices for the Event Total Number of Possible Choices for the Experiment
⇒ P(Ec) =
 mEc n

## Probability of Failure

The number of 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 failure

∴ Probability of Failures for the Event

=
 Number of Failures for the Event Total Number of Possible Choices for the Experiment
⇒ P(E) =
 mEc n

# Probability of Occurrence + Probability of Non-Occurrence = 1

The events of occurrence (success) and non-occurrence (failure) are mutual compliments.

⇒ Probabilities of the occurrence (success) and non-occurrence (failure) of the events are compliments

P(Ec) =
 mEc n
=
 n − mE n
=
 n n
 mE n
= 1 − P(E)

⇒ P(E) + P(Ec) = 1

The sum of the probabilities of occurrence and non-occurrence 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 non-occurrence of the event is identified as Probability of Failure.

∴ P(E) + P(Ec) = 1

⇒ Probability of Success + Probability of Failure = 1

# Formula Interpretation

• ## m, mc are non-negative integers

The values m and mc 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 non-negative 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, mc and n are non-negative integers and n ≠ 0.
Therefore, the ratios  m n
and  mc n
are non-negative rational numbers
Since P(Event) =  m n
and P(Eventc) =  mc n
, we can say that the probabilities are positive rational numbers.
• ## 0 ≤ P(Event) ≥ 1

The value of m, ranges between 0 and n.
⇒ 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(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

• ## 0 ≤ P(Eventc) ≥ 1

P(Ac) = 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(Ac) ranges between 0 and 1

⇒ 0 ≤ P(Ac) ≤ 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