Events in Probability

The entire possible set of outcomes of a random experiment is sample space sample space of that experiment. The likelihood of occurrence of an event is known as probability. The probability of occurrence of any event lies between 0 and 1.

The sample space for tossing of three coins simultaneously is given by:

S = {(T , T , T) , (T , T , H) , (T , H , T) , (T , H , H ) , (H , T , T ) , (H , T , H) , (H , H, T) ,(H , H , H)}

Suppose, if we want to find only the outcomes which has at least two heads; then the set of all such possibilities can be given as:

E = { (H , T , H) , (H , H ,T) , (H , H ,H) , (T , H , H)}

Thus, an event is a subset of the sample space, i.e., E is a subset of S.

There could be a lot of events associated with a given sample space. For any event to occur,the outcome of the experiment must be an element of the set of event E.

Types of Events in Probability:

1. Impossible Events:If the probability of occurrence of an event is 0, such an event is called as impossible event and if the probability of occurrence of an event is 1, such an event is called as sure event. In other words,the empty set ϕ is an impossible event and the sample space S is a sure event.
2. Simple Event:Any event consisting of a single point of the sample space is known as a simple event.For example: If S = {56 , 78 , 96 , 54 , 89 } and E = {78} then E is a simple event.
3. Compound Event: Contrary to simple event, if any event consists of more than one single point of the sample space then such an event is called as compound event. Considering the same example again If S = {56 ,78 ,96 ,54 ,89 } and $E_1$ = {56 ,54 }, $E_2$ = {78 ,56 ,89 } then, $E_1$ and $E_2$ represent two compound events.
4. Independent Events and Dependent events: If the occurrence of any event is completely unaffected by the occurrence of any other event, such events are known as independent event in probability and the events which are affected by other events are known as dependent events.
5. Mutually exclusive events:If occurrence of one event excludes the occurrence of other event, such events are mutually exclusive events i.e. two events don’t have any common point.For example:If S = {1 , 2 , 3 , 4 , 5 , 6} and $E_1$ and $E_2$ are two events such that $E_1$ consists of numbers less than 3 and $E_2$ consists of numbers greater than 4.That is, $E_1$ = {1,2} and $E_2$ = {5,6} . Then, $E_1$ and $E_2$ are mutually exclusive.
6. Exhaustive events: A set of events is called as exhaustive if all the events together consume the entire sample space.