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 E1 and E2 are two arbitrary events.

If the sample space is not finite and Ev E2 is an infinite sequence of mutually exclusive events in S, Et n 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:

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