# Probabilistic Uncertainty

An experiment is a process of measurement or observation. The aim is to study the randomness in the experiment. A trial is a single performance of an experiment. Its result is called an outcome or a sample point. The set of all possible outcomes is called sample space S.

Let *S* be a finite sample space and *E* be an event in S, then the probability *P(E)* of the event *E* is a real number assigned to *E* (ratio of number of points in *E* upon number of points in S), which satisfies the following axioms:

- 1. P(E) > 0
- 2. P(S) = 1
- 3. P(Ej и
*E2)*= P(Ej) + P(E2) if Ej n E2 = Ф where*E*_{1}and*E*_{2}are two arbitrary events.

If the sample space is not finite and *E _{v} E_{2}* is an infinite sequence of mutually exclusive events in S,

*E*n

_{t}*Ej*=ф for

*i Фj,*then

## Elementary Properties of Probability

Using the same axioms, we have the following properties of probability Let us consider two arbitrary events *A* and B, then

## Conditional Probability

The conditional probability of an event *A* given the event *B* is denoted by P(A | B), which is defined as

where P(A n B) is the joint probability of *A* and B.

Similarly, the conditional probability of an event *A* given the event *B* is

From Equations 3.9 and 3.10, we have

Equation 3.11 is often quite useful to compute the joint probability and we obtain the Bayes rule as follows: