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# Event|Definition & Meaning

## Definition

The** outcome(s)** obtained after conducting a random **experiment** is known as an event. It is the **set** of all the results or **consequences** of the experiment. An event is a **subset** of the given **sample space** for the experiment.

**Figure 1** shows the event **E** and the sample space** S** in the **Venn** diagram.

All the elements of **E** are present in the superset **S**.

A **sample space** contains all of the possible **outcomes** or results of an experiment whereas an **event** is an outcome that actually occurred.

## Random Experiment

A **random** experiment is defined as a **procedure** that produces a result that could not be **predicted** before the experiment.

It can be **repeated** many times but its **outcomes** are known in the sample space. An **event** is the result of a random experiment.

## Probability

**Probability** is defined as how much there is a **chance** of an **event** occurring from a known set of **outcomes**. The probability is measured from **0** to **1**.

The mathematical **formula** for probability is given as

P(E) = n(E) / n(S)

Where **P(E)** is the **probability** of an event to occur, **n(E)** is the **number** of **favorable** outcomes specified in an event** E** and **n(S)** is the total number of possible **outcomes** in the sample space **S**.

For example, a **die** is rolled. The probability of an **even** number occurring is demonstrated in **figure 2**.

## Types of Events Based on Measure of Probability

Events can be classified on a basis of the **values** of probability. They are discussed below.

### Impossible Event

As the name suggests, this type of event is **impossible** to occur in an experiment. This is because it is not included in the **sample space** of the experiment and has a **probability** of **0**.

For example, an **eight** occurring on a** die** is an impossible event.

### Unlikely Event

This type of event is **unlikely** to take place after an **experiment**. Its probability range is between **0** and **0.5** and its likelihood to occur increases as the **probability** increases.

For example, getting a **two** when a **die** is rolled is a very unlikely event as its **probability** is **0.16**.

### Equally Likely Events

These events have the **same** chance of occurrence, hence have **equal probability** values. They are also called **evenly chanced** events.

For example, a **head** or a **tail** occurring on a tossed **coin** is an equally likely event as both have the same probability i.e. **0.5**.

### Likely Event

An event is said to be **likely** when it has a probability **greater** than **0.5** but less than **1**. For example, the chance of numbers** 2**, **3**, **4**, and **5** occurring on a **die** is a likely event as its **probability** is **0.67**.

A likely event set has a **maximum** number of **elements** from its sample space but not all the elements.

### Sure Event

A sure event is a **certain** event with a probability of **one**. It is sure to occur as it includes all the **elements** from the sample space.

For example, getting a **red ball** from a bag of red balls is a sure event.

**Figure 3** shows the probability** scale **and the types of **events**.

## Other Types of Events

Some other types of **events** are discussed below:

### Simple and Compound Events

A **simple** event consists of **one** element of the sample space whereas a **compound** event contains more than one **element**. If

S = { 1, 3, 4, 7, 8, 11, 14 }

and

P = { 4 } , Q = { 1, 7, 11 }

Then **P** is a **simple** event and **Q** is a **compound** event.

### Independent and Dependent Events

Two events not **affecting** each other are known as **independent** events. Contrarily, if the **occurrence** of an event is affected by the occurrence of the other event, then the two events are called **dependent** events.

### Exhaustive Events

Events are called **exhaustive** when they occupy all the elements of the **sample space**. If

S = { 11, 12, 13, 14 }

Then

U = { 11, 13 } and V = { 12, 14 }

are **exhaustive** events.

### Mutually Exclusive Events

Two events are said to be **mutually** exclusive if they cannot occur **simultaneously**.

If

S = { 6, 7, 8, 9, 10 }

Then

C = { 6, 7 } and D = { 9, 10 }

are mutually **exclusive** events.

**Figure 4** shows the **Venn** diagram for the above two **mutually** exclusive events.

Both the events don’t have an **intersection** i.e. they don’t share an outcome.

### Complementary Events

An event** E** has a **complementary** event **E´** if E´ has all the **elements** of the sample space **not** in E. So,

E**´** = S – E

Also,

P(E) + P(E**´**) = 1

**E** and **E´** together make the sample space hence are **exhaustive** events. As both don’t have a **common** element, they are also mutually **exclusive**. If

S = { 25, 26, 27, 28, 29 }

and

F = { 26, 27 }

Then

F**´** = { 25, 28, 29 }

is the **complement** of **F**.

### “AND” Events

The AND operator denotes the intersection or **common** points between the events. **AND** is denoted by the **intersection** symbol “∩”. If

Q = { 1, 2, 3 }

And

R = { 3, 4, 5 }

Then

Q AND R = Q ∩ R = { 3 }

**Figure 5** shows the two events **Q**, **R**, and their intersection **Q ∩ R**.

### “OR” Events

The OR is the **union(∪)** of elements of events. If

L = { 10, 20, 30 }

and

M = { 40, 50 }

Then

L OR M = L ∪ M = { 10, 20, 30, 40, 50 }

**Figure 6** shows the **Venn** diagram for events **L** and **M**.

## A Solved Problem of Event Classification

A deck of **cards** is shuffled. What is the probability that a **heart** card will occur when a card is picked? What **type** of **event** is it?

### Solution

There are **52** cards in a **deck** and it has a total of **13 heart** cards. So,

n(H) = 13

n(S) = 52

The **probability** of picking a heart card is

P(Q) = n(H) / n(S)

P(Q) = 13 / 52 = 0.25

So, **event H** is an **unlikely** event as its probability is between **0** and **0.5**.

*All the images are created using GeoGebra.*