Definitions of probability

T. Dhasaratharaman*

Statistician, Kauvery Hospitals, India

There is a number of definitions of probability proposed by many authors.

However, the majority of definitions can be divided into three groups:

Definitions of probability as a quantitative measure of the ‘degree of certainty’ of the observer of the experiment.
Definitions that reduce the concept of probability to the more primitive notion of ‘equal likelihood’.
Definitions that take as their point of departure the ‘relative frequency’ of occurrence of the event in a large number of trials.

The classical definition of probability

Uses the sample space to determine the numerical probability that an event will happen, also called theoretical probability. The probability P(A) of an event A is equal to the number of possible simple events (outcomes) favourable to A divided by the total number of possible simple events of the experiment, i.e., P(A) = m/n

where, m = number of simple events into which the event A can be decomposed.

The classical definition of probability reduces the concept of probability to the concept of equiprobability of events, which is regarded as a primitive concept and hence not subject to the formal definition.

Statistical definition of probability

First of all, the question arises in a majority of cases, as to a reasonable way of selecting the ‘equally likely cases’.

Lengthy observations as to the occurrence or non-occurrence of an event A in a large number of repeated trials under the same set of conditions show that for a wide class of phenomena, the number of occurrences or non-occurrences of the event A is subject to a stable law; then it turns out that for sufficiently large N the ratio m/N in most of such series of observations, assumes an almost constant value. Since this constant is an objective numerical characteristic of the phenomena, it is natural to call it the statistical probability of the random event A under investigation.

“The probability of an event A can be approximated by the proportion of times that A occurs when the experiment is repeated a very large number of times.”

Kauverian Bookshelf

Applied Medical Statistics

Definitions of probability