Classification Of Events

TYPES OF EVENTS

As explained earlier :

“An event is a collection of one or more of the outcomes of an experiment”

OR 

In simple words,

“An event is a subset of a sample space”

An event can be classified into various types as follows :-

(1)  SIMPLE EVENT :- 

A simple event has only one sample point of the sample space. Each of the final outcomes from experiment is called a simple event. In other words, a simple event includes one and only one outcomes.

For example, consider the experiment of rolling a die once, therefore sample space S is given by

S= {1, 2, 3, 4, 5, 6}.

Now event like Getting a ‘3’ , Getting ‘5’, Getting ‘6’ are simple events.

A Simple Event is also called an Elementary Event.

(2) COMPOUND EVENT :-

A compound event is a collection of more than one outcome of an experiment.

For example, consider the experiment of rolling a die once, therefore sample space S is given by

S= {1, 2, 3, 4, 5, 6}. Then, events like

Getting an even number.

(Possible outcomes :- 2, 4, 6)

                   or

Getting a number ≥ 3.

(Possible outcomes :- 3, 4, 5, 6) are compound events

A Compound Event is also called a Composite Event.

(3) IMPOSSIBLE EVENT :- 

Every non empty subset A of a sample space S , is called an Event. The empty set Φ is also a subset of the sample space S, therefore, it also represents an event. Now consider an experiment in which two dice are thrown. The sample space S of this experiment consists of 36 points

S = {(1, 1), (1, 2),….......…, (1, 6),

         (2, 1), (2, 2),……….…, (2, 6),

         (3, 1),…………….....…., (3, 6),

         ........………………………………,

        (6, 1),….……………....., (6, 6) }.

If A is an event such that,

A= the event that the sum of the numbers on the faces is greater than 12, then, A= Φ

Since no outcome of this experiment is a member of Φ , the event represented by Φ cannot occur at all. We call the event Φ as an Impossible Event. Thus,

An event corresponding to the empty set is called an impossible event or null event

(4) SURE EVENT :- 

Consider the experiment of rolling a die once.

Here Sample Space S is given by

S = {1, 2, 3, 4, 5, 6}

Now let A be the event 'the number turns up is even or odd'.

So, here the event A is that the number on the face is one of the numbers 1, 2, 3, 4, 5, 6, then

A = {1, 2, 3, 4, 5, 6}

Here A is a Sure (or Certain) Event.

Thus

The event corresponding to the entire Sample Space is called a Sure (or Certain) Event.

(5) MUTUALLY EXCLUSIVE EVENTS :- 

Consider the eaperiment of throwing a dice.

Let A be the event, 'the number obtained is less than 4 '. Then,

A = {1, 2, 3} .

Let B be the event, 'the number obtained is atleast 5 '. Then,

B = {5, 6}.

Clearly A ∩ B = Φ.

Thus the joint occurance of A and B is thus an impossible event.

The events A & B are called Mutually Exclusive Events

Thus, "Two events A and B are called mutually exclusive events if the occurance of any one of them excludes the occurance of the other event i.e. if they cannot occur simultaneously"

In terms of sets,Two events A and B are called mutually exclusive events if            A ∩ B = Φ and n(A ∩ B) = 0.

(6) EXHAUSTIVE EVENTS:-

If two events A and B are such that A U B = S then P(A U B) = 1 and the events A and B are said to be Exhaustive.

For Example. 

Let S be the sample space when an ordinary die is thrown.

If A is the event the number is less than 5 and B the event the number is greater than 3, then the events A and B are exhaustive as A U B = S.

Thus,

If E1 , E2 .........., En are the subsets of a sample space S, and 

if E1 U E2 U E3 U .....U Em = S, then E1, E2,..........En form a set of Exhaustive Events.