The three axioms of probability are,
- The first axiom states that the probability that the outcome of the experiment is contained in E is some number between 0 and 1.
- The second axiom states that with probability 1, the outcome will be a member of the sample space S. This axiom is also known as the axiom of certainty.
- The third axiom is known as the axiom of additivity. It states that for any set of mutually exclusive events, the probability that at least one of these events occurs is equal to the sum of their respective probabilities.
We shall suppose that for each event E of an experiment having a sample space S, there is a number, denoted by P(E), that is in accord with the following three axioms:
Axiom 1:
The first axiom states that the probability that the outcome of the experiment is contained in E is some number between 0 and 1.
0 ≤ P(E) ≤ 1.
If the probability of the occurrence of an event is equal to 1 we say that an event is a sure event and if the occurrence of the event is impossible then the probability of the event is equal to 0.
According to, to the first axiom the probability of occurrence of any event is a non-negative real number less than or equal to 1.
Axiom 2:
Let S denote the sample space then we have that P(S) = 1.
The second axiom states that with probability 1, the outcome will be a member of the sample space S. This axiom is also known as the axiom of certainty.
Axiom 3:
The third axiom is known as the axiom of additivity. It states that for any set of mutually exclusive events, the probability that at least one of these events occurs is equal to the sum of their respective probabilities.
For any sequence of mutually exclusive events E1,E2,…En we have that,
P(∪Ei) = ∑ P(Ei).
Two consequences of the probability axioms:
Two corollaries that follow straightforwardly from the above three axioms are:
- Let E’ denote the complement of an event E. The complement rule states that P(E’) = 1 – P(E).
- If E and F are two events then we have that, P(E∪F) = P(E) + P(F) – P(E∩F).
