# Event in Probability

In this chapter we will discuss the concept of events, its types and examples.

The concept is extensively used in probability, so make sure you understand this lesson.

## What is an event ?

The set of desired outcome from the experiment is called an event.

Consider the experiment of tossing an unbiased coin.

The possible outcome for this experiment is Head or Tails.
Sample Space ( S ) = { Head, Tails }

Now suppose you desire the outcome of ” Heads” for the above experiment. This outcome is shown below by event A;

Note that the set A is subset of sample space (S).

Now consider another event of rolling a dice.

The sample space is given as;
S = {1, 2, 3, 4, 5, 6}

Suppose you are interested in an event of getting even number from the dice. The event can be expressed in the form of set as;
A = { 2, 4, 6 }

Conclusion
The desire of particular outcome from the experiment is called even in probability.

### Classification of events

There are basically two types of events;

(a) Simple event

(b) Compound event

Simple event

The event which contain only one desired outcome is called simple event.

For example, in the experiment of throwing a dice, the desired outcome of getting “2” is a simple event.

A = { 2 }

Compound event

The event in which we have more than 1 desired outcome is called compound event.

For example, in the experiment of throwing a dice, the desired outcome of getting ” odd number” is an compound event.

A = { 1, 3, 5 }

Note that in above set A, we desire number 1, 3, & 5 from the dice.

I hope you understood the basics concept of events. Let us now look at more complex form of events in probability.

### Types of events

Sure event

The outcome which will always occur after the experiment is called an sure event.

The set of sure event is equal to the sample space of an experiment.

For example;
Consider the experiment of tossing a coin and let A be the event of getting ” heads or tails “.

The set A is expressed as;
A = { heads, tails }

No matter how many times the experiment is performed we will always get either of the two outcome.

Impossible event

The event which will never occur after the experiment is called impossible event.

Consider the experiment of rolling a dice and Let A be the event of getting number 7.

We know that dice contain number from 1 to 6 and we will never get number 7 as a result.

Hence, A is an impossible event.

Equally likely events

When the occurrence of one event is as likely to happen as other event then the events are ” Equally likely events “

Here the probability of happening of one event is same as the second event.

For example;
Consider the experiment of tossing an unbiased coin.

Let A = event of getting head
B = event of getting tail

We know that after tossing coin, event A & B are equally likely to happen.
Hence, A & B are equally likely events.

Mutually exclusive events

Two events are said to be mutually exclusive if the occurrence of one events prevent the occurrence of other events.

Consider the experiment of rolling a dice.
A = { getting even number }
B = { getting odd number }

We know that if event A occur then event B cannot happen and vice- versa.

The mutually exclusive event is expressed as A ∩ B = 𝜙; It means that there is no common element between set A & B.

Note:
In mutually exclusive event, there is no common element between set A and set B.

Exhaustive events

Two events are said to be exhaustive when their union will results in formation of sample space.

Hence, on combining exhaustive events, all the possible outcome of the experiment is covered.

For example;
Consider the experiment of drawing random card from the deck of 52 cards.

We know that the 52 cards are made of red and black cards.

Given below are the events;
A = getting red cards
B = getting black card

We know that on combining both the events A & B, we will cover all the possible outcome of sample space.

Hence, event A & B are exhaustive.

The exhaustive event is expressed as A U B = S.
It mean that union of set A & B will result in formation of sample space.

Let us consider another example of throwing an unbiased coin.

A = event of getting head
B = event of getting tail

On combining both event A & B, we have covered all the possible outcome of the experiment.

Complimentary event

Two events A & B are complimentary if negation of event A results in event B.

Suppose you are in station waiting for a train. The exact time for arrival of train is 1: 00 PM.

Consider the below two events;
A = { Train arrive at time }
B = { Train is late }

Here A & B are complimentary because negation of A results in B and vice-versa.

i.e. If train arrive on time it means that it is not late.

Consider another example of throwing a dice.

Given below are two events;
A = { getting even number }
B = { getting odd number }

Both the events are complimentary because if A happen then B cannot happen and vice-versa.

i.e. if we get even number (event A ) then we can’t get an odd number (event B )

Note:
Union of complimentary events results in sample space.