# Probability Definitions

The term probability has been interpreted in terms of four definitions :

1. Classical Definition of Probability: the classical definition states that if an experiment consists of ’$$S$$’ outcomes that are mutually exclusive, exhaustive, and equally likely and $$S_A$$ of them ate the favorable outcomes of an event A then the probability of the event is

$$P\left( A \right)\; =\; \frac{S_A}{S}$$

In other words, the probability of event A is equal to the ratio of the number of favorable outcomes $$S_A$$ to the total number of outcomes.

2. Axiomatic Definition of Probability: In the axiomatic definition of probability, the probability of outcome A is defined by a number assigned to A, such a number satisfies the following axioms:

a) $$P\left( A \right)\; \geq \; 0$$  i.e.,  $$P(A)$$ should be non-negative.

b) The probability of certain event A = 1. i.e., $$P(A) = 1$$

c) If the two events A and B are mutually exclusive then the probability of the event $$(A \cup B)$$ is $$P\left( A\cup B \right)\; =\; P\left( A \right)\; +\; P\left( B \right)\;$$

3. Empirical Definition of Probability: In $$N$$ trials of a random experiment of an event is found to occur m times then the relative frequency of occurrence of the event is $$\frac{m}{N}$$ is the limiting value approaches to $$P$$. When N increases to infinity then $$P$$ is called the probability of event $$A$$.

i.e., $$P(A) =\lim_{x \to \infty} (\frac{m}{N})$$

4. Subjective Definition of probability: In subjective interpretation of probability the number $$P(A)$$ is assigned to a statement which is a measure of our state of knowledge or belief concerning the truth of $$A$$. These kinds of probability are more often used in our day-to-day life and conversation.