Contents

# Certain|Definition & Meaning

## Definition

**Certain** is a term used in **probability** to define an event that is **sure** to occur. It comes from the word “**certainty**” which means something that is definite, firm, and **reliable**. A **certain event** is an event in which there is **no doubt** that the event will not take place.

**Figure 1** shows a certain event.

Choosing a purple ball from a box of purple balls is a certain or sure event.

## Probability

Probability is a **numerical** quantity that defines the **degree** of how much there is a **likelihood** of an **event** taking place.

The probability of a particular outcome is the** ratio** of the number of **favorable outcomes** to the **total number of outcomes** that can occur.

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

**Probability of an Event To Occur **= P(E) = n(E) / n(S)

Where,

**n(E)** = Number of Ways the Event Can Occur

**n(S)** = Total Number of Events

**Figure 2** shows the procedure to find probability.

## Random Device

In **probability**, random devices are used to produce **random outcomes**. The experiment can result in all kinds of different **events** present in the sample space of a **random device**.

**Figure 3** shows the random devices used in probability.

## Sample Space

Sample space is the set of **all possibilities** which can be obtained from a particular **random device**. It is written in **curly **brackets and denoted by **S**.

For **example**, the sample space of a **die** are the **six numbers **which can be written as:

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

## Event

An event is a **favorable outcome** required from a **random** experiment. It is the **subset** of the **sample space **and is mostly denoted by **E**.

For **example**, the event for the numbers **one** and **two** turned up when a **die** is rolled can be written as:

**E = { 1, 2 }**

The **probability** for this event can be obtained by putting the values of **n(S)** and** n(E)**.

Here,

**n(S) = 6**, **n(E) = 2**

Where **n(S)** is the number of outcomes in the sample space and **n(E)** is the number of desired outcomes for an event.

So, the **probability** of the particular event** P(E)** will be:

**P(E) = n(E) / n(S) = 2 / 6 = 1 / 3 = 0.3**

## Types of Events

The value of **probability** explains what type the **event** will be. In mathematics, the probability is measured from **0 to 1** which shows the extent of the **chance** of an event occurring.

Following are the **type **of** events** classified by the measure of probability.

### Impossible Event

An event is** impossible** to occur if its probability is **zero**. For example, the number **seven** occurring on a die has **n(E) = 0** as the outcome is not included in the sample space.

The **probability** can be calculated as follows:

**P(E) = n(E) / n(S) = 0 / 6 = 0**

### Unlikely Event

An event is **unlikely** to happen if its **probability** is between **0** and **0.5**. The chance of an unlikely event occurring is **small** and increases from 0 to 0.5.

For **example**, a **three** occurring on a die has **n(E) = 1** and **n(S) = 6**. The probability **P(three)** can be calculated as:

**P(three) = n(E) / n(S) = 1 / 6 = 0.16**

As the probability is **less than 0.5** and more close to **0**, the event is **very unlikely** to occur.

### Evenly Chanced Events

Events are said to be **evenly chanced** when they have the **same probability** values. For example, all the numbers on a die have an **equal chance** of occurring i.e. **1 / 6** or **0.16**.

A** coin** has the **sample space**:

**S = { Head, Tail }**

So,

**n(S) = 2**

For both, the head **n(H)=1** and tail **n(T)=1**, the probabilities will be equal hence they are equally likely to occur.

**P(H) = n(H) / n(S) = 1 / 2 or 0.5**

**P(T) = n(T) / n(S) = 1 / 2**

As shown in **figure 4**, the evenly chanced events divide the probability scale into **two equal** parts.

### Likely Event

A **likely** event has a probability between **0.5 and 1**. It is an event that has a **greater chance** of occurrence than an **unlikely** event.

For example, if the favorable outcome is **E = { 1,2,3,4,5 }** to occur on a die, the number of **desired** outcomes **n(E)** and the number of outcomes in the sample space **n(S)** are:

**n(E) = 5**

**n(S) = 6**

The **probability** for the event** P(E)** is given as:

**P(E) = n(E) / n(S) = 5 / 6 = 0.83**

Hence, the numbers **1**, **2**, **3**, **4**, and** 5** are very **likely** to occur on the dice.

The **likely** and **unlikely** event range is shown on the probability scale in **figure 5**.

### Certain Event

An **event** is said to be **certain** if the value of **probability** is **1**. This shows that the event is **sure to happen**.

An example of a certain event is obtaining a **head** or **tail** on a **coin**. Both **n(E)** and **n(S)** will be equal to **2**, hence the probability **P(E)** will be:

**P(E) = n(E) / n(S) = 2 / 2 = 1**

**Figure 6** shows the **certain** event and the **impossible** event as the boundaries on the probability scale.

## An Example of a Certain Event

**Two dice** are rolled. What is the probability of getting a **product** of **8** from the two numbers turned up on the two dice?

### Solution

One die has **6** possible outcomes. As two dice are rolled, there are **(6)(6) = 36** total possibilities that can occur. So,

**n(S) = 36**

Let **E** be the **event** for the **two dice** to turn up such numbers whose **product** is **8**. These are **(2,4)** and **(4,2)** on the dice. So,

**E = { (2,4), (4,2) }**

Therefore,

**n(E) = 2**

So, the probability **P(E)** will be:

**P ( E = product of 8 ) = n(E) / n(S) = 2 / 36 = 1 / 18 = 0.05**

As it is** close** to** zero** and between **0** and **0.5**, the event is very much **unlikely** to occur.

*All the images are created using GeoGebra.*