Probability

Probability is a numerical measure of the likelihood that an event will occur. Thus probabilities can be used as measures of the degree of uncertainty associated with events.

Probability values are always assigned on a scale from 0 to1 .A probability near zero indicates an event will unlikely to occur; a probability near one indicates an event is almost certain to occur. Other probabilities between 0 and 1 represent degrees of likelihood that an event will occur. For example if we consider the event rain tomorrow. When the weather report indicates a near zero probability of rain it means almost no chance of rain. However, if a .90 probability of rain is reported, we know that rain will occur. A .50 probability indicates that rain is just as likely to occur as not.

In discussing probability, we define Experiment as a process that generates well-defined outcomes.

By specifying all possible experimental outcomes, we identify sample space for an experiment.

An experimental outcome is also called a sample point to identify it as an element of the sample space.

The probability that an event occurs is defined as

P(A)=The number of distinct ways that the event can occur/ The number of all possible outcomes.

The probability of an event is the measure of the chance that the event will occur as a result of an experiment. The probability of an event A is the number of ways event A can occur divided by the total number of possible outcomes. The probability of an event A, symbolized by P(A), is a number between 0 and 1, inclusive, that measures the likelihood of an event in the following way:

• If P(A) > P(B) then event A is more likely to occur than event B.
• If P(A) = P(B) then events A and B are equally likely to occur.

It has to satisfy the following three Axioms, in order to be both meaningful and useful.

AXIOM 1

0 ≤ P (E) ≤ 1 The probability of every event, that is , an outcome or a set of outcomes, is a number between 0 and 1, both inclusive.

AXIOM 2 P (S)

The probability of the whole sample space considered as an event (=subset of itself), has to be 1.One of the outcomes listed under S is bound to happen.

AXIOM 3

P ( E₁ ∪ E₂ ) = P ( E₁ ) + P ( E₂ ) , where E1 and E2 are mutually exclusive.