Statistical Methods

Unit 1 - Basic concepts of probability

1.4.Mathematical preliminaries
Complement of an event
Let A be an event. Then, complement of A is the event of non-occurrence of A. It is the event constituted by the outcomes which are not favorable to A The complement of A is denoted by A^I or A^c. While throwing a die, if A = {2, 4, 6} its complement A^I = {1, 3, 5}.

Sub-events
Let A and B be two events such that event A occurs whenever events B occurs. Then, events B is sub-event of event A.
While throwing a die, let A= {2, 4, 6} and B= {2}. Here B is a subset (sub-event) of event A, and is denoted by BA.

Union of events
Union of two or more events is the event of occurrence of at least one of these events. Thus union of two events A and B is the event of occurrence of at least one of them. The union of A and B is denoted by AB or A+B or A or B
Example 1
while tossing two coins simultaneously, let A= {HH} and B = {TT} be two events.
Then their union is AB = {HH, TT}
Here A is the event of occurrence of two heads and B is the events of occurrence of two tails their union AB is the event of occurrence of two heads or two tails.
Example 2
while throwing a die, let A ={2,4,6} B={3,6} and C={4,5,6} be three events.
Then, their union is ABC= {2, 3, 4, 5}

Intersection of events
Intersection of two or more events is the event of simultaneous occurrence of all these events. Thus intersection of two events A and B is the events of occurrence of both of them. The intersection of A and B is denoted by AB or AB or A and B.