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

The Mathematical or classical definition of probability is an a priori definition.

Probability of Occurrence of an Event

E represents the event.

Probability of Success

Probability of Success for an 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 non-occurrence of the Event.

Probability of Non Occurrence of the Event

Probability of Failure

∴ Probability of Failures for the Event

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

⇒ P(E) + P(E^c) = 1

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

Probability of Success + Probability of Failure = 1

∴ P(E) + P(E^c) = 1

⇒ Probability of Success + Probability of Failure = 1

• m, m^c are non-negative 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 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, m^c 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(Event^c) =

  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(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