Basic Terminology
* Probability is defined as the likelihood of something occurring.
* A random experiment is an experiment that has more than one possible results and it is not possible to predict the result in advance.
* The possible result of an experiment is called its outcome.
* The set of all the outcomes of a random experiment is called sample space.
* A collection of the outcomes of a sample space is called an event.
* An element of a sample space is called a sample point. In other words every outcome of a random experiment is also called a sample point.
* A collection of the outcomes of a sample is called an event.
Events
* If S is a sample space, then event S is called a sure event.
* If S is a sample space, ten events ∅ is called an impossible event.
* An event of sample space with only one sample point is called a simple event.
* An event that has more than one sample point is called a compound event.
Algebra of Events
* The compliment of an event A is the event consisting of all the outcomes of the sample space that do not correspond to the occurrence of A.
* If events A and B are associated with a sample space S, then event “A or B” contains all the elements that are either in A or B or both.
* If events A and B are associated with a sample space S, then event “A and B” contains the elements that are common to both A and B.
* If event A and B are associated with a sample space S, hen event ‘A but not B’ is the event of all the elements of A, but not in B.
* Two events A and B in a sample space S are said to be mutually exclusive if A ∩ B
= ɸ.
* Two events A and B in a sample space S are said to be exhaustive if A U B = S.
Axiomatic Approach To Probability (Part I)
* If the axiomatic approach to probability, the probability P is real valued function, whose domain is the set of all the subsets of sample space S and range is [0, 1]
* Probability of any event, say A, is non-negative, i.e., P (A) ≥ 0.
* Probability of the sample space S is 1, i.e., P(S) = 1.
* If A and B are mutually exclusive events, then P (A U B) = P (A) + P (B).
* Probability of an event which cannot occur (impossible event) is zero i.e. P (ɸ) = 0.
* If A is an event in sample space S, then 0 ≤ P (A) ≤ 1.
* If E is an event in sample space S containing n sample point that are equally likely
then P (E) = \frac{1}{n}.
* S is a sample space containing n sample points and A is an event in S containing m sample points. If each outcome is equally likely,
Then P (A) = \frac{m}{n}.
Axiomatic Approach to Probability (Part II)
* If A and B are any two events of sample space S,
then P (A U B) = P (A) + P (B) – P (A ∩ B).
* If A is an event of a sample space, then A’ is the complement of event A, and P
(A’) = 1 – P (A).