Our life is full of uncertainty, of events that we do not know with what certainty will occur.

For example, we do not know in advance who will win the soccer game or what number will come up when rolling a die. Many people would like to predict the future to know what decision to make in a certain situation.

Among those people who have wanted to predict the future are several mathematicians, who with great dedication have devised ways to measure the possibility of an event occurring. That is, calculating the probability of an event occurring.

In this article we explain what an event is in probability, what is a possible, impossible and certain event. Additionally, we tell you how you can calculate the probability of each of these events. Not without first defining and giving examples of a fundamental concept such as sample space.

Be sure to read this interesting article in its entirety, it will be very useful to you.

## What is a sample space

Before starting, we must define a fundamental term: sample space.

The sample space is the set of all possible results of an activity or experiment. It is usually represented with a list or a diagram.

Let’s look at three examples of sample space:

**Example 1**

When rolling a die we have 6 possible results. For this reason the sample space is 6 elements.

### Example 2

You want to take, without seeing, a red ball from a box that has 3 green balls, 2 red balls, 2 blue balls and 1 black ball.

*BV:* green ball – *BR:* Red ball – *BA:* blue ball – *BN:* black ball

### Example 3

Suppose we flip a coin three times. What will be the number of possible outcomes of this experiment?

We can count the number of possible outcomes of this experiment as a set.

It can also be with a diagram like the following:

** **

**Tree diagram**

In this tree diagram we have represented all the possible results of this experiment that consists of tossing a coin three times.

What appears circled in blue is one of the possible results. This would be the case, in which in each of the three throws a head appears (C,C,C).

Another result would be Heads, Heads, Tails (C,C,X). We have also marked it in blue so that you can see it more easily.

**Tree diagram**

Following the direction of the arrows you will find all the possible results of flipping a coin three times.

## What is an event in probability

An event is a set of one or more outcomes in a probability experiment.

### Example 1

In the above experiment, tossing a coin three times, any of the possible outcomes is an event.

For example, **obtain** **three faces,** by flipping the coin three times, **it is an event**. Also **get a tail and two heads,** by flipping the coin three times, **it is an event**.

### Example 2

If we throw a die and a coin at the same time we obtain a sample space that can be represented in a tree diagram.

Let’s see:

**Tree diagram**

By counting we find that the sample space of this experiment consists of 12 elements.

* AND*={1C, 1X, 2C, 2X, 3C, 3X, 4C, 4X, 5C, 5X, 6C, 6X}

Any subset of this sample space is an event. For example, getting one and heads (**1 C**) is an event. It is also an event to get two and tails (**2X**) or get three and heads (**3C**).

### How to calculate the probability of an event

The probability of an event is a number that tells us the level of certainty we have that that event will occur.

This number that indicates the probability of an event occurring is a quotient that is calculated by dividing the number of favorable cases by the number of possible cases, and is always between 0 and 1. Let’s look at a simple example:

Look at this roulette. We are going to calculate the probability that when we turn it it will fall into the color red. We will call this probability P(red).

The roulette wheel is divided into 12 equal sectors. Each sector with a different color.

● The number of favorable cases is 1 (there is a single red sector).

● The number of possible cases is 12 (there are 12 different colors).

P(red) =number of favorable casesnumber of possible cases = 112 = 0.083

The probability that when you spin the roulette wheel it lands on the color is 0.083.

## What is a possible event and examples

An event is possible if it is among the possible outcomes of the experiment. That is, if it is included in the sample space of the experiment.

The probability of a possible event is said to be a number between 0 and 1.

### Example 1

Getting a three when throwing a die is a possible event because the 3 is part of the sample space of the experiment. Remember that the sample space for rolling a die is:

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

Rolling a 3 when rolling a die is a possible event. This is because 3 is part of the sample space of the die-rolling experiment. The probability of rolling a 3 when rolling a die is:

P= 16 = 0.17

### Example 2

We have the following Spanish decks.

You want to draw the King of Pentacles card from this bunch of cards without looking. Is this event possible? Because?

Before answering the question, we are going to determine the sample space of this experiment.

E={Two of Wands, Three of Swords, Two of Pentacles, King of Pentacles}

As we see, the sample space of this experiment consists of 4 elements.

Since the King of Pentacles card is part of the sample space, then the event of drawing the King of Pentacles card is a possible event.

In this experiment the probability of drawing the King of Pentacles card is:

P= 14 = 0.25

## What is an impossible event and examples

An event is impossible if it is not among the possible results of the experiment. That is, if it is not included in the sample space of the experiment.

The probability of an impossible event is said to be equal to zero.

### Example 1

Rolling an 8 when rolling a die is an impossible event, because 8 is not part of the possible outcomes of the die-rolling experiment.

As we already know, the sample space of the experiment of throwing a die is:

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

The 8 is not included in the sample space, which is why rolling an 8 when rolling a die is impossible.

The probability of rolling an 8 when rolling a die is:

P= 06 = 0

### Example 2

Taking a black ball without looking from a box that has 2 red balls, 2 green balls and 1 blue ball is an impossible event.

*BV:* green ball – *BR:* Red ball – *BA:* blue ball

There is no black ball in the box, so the event of drawing a black ball without looking is impossible in this experiment.

The probability of drawing a black ball is:

P= 05 = 0

## What is a safe event and examples

A safe event is one that always occurs.

We can also say that a safe event is one that is made up of all the possible results contained in the sample space.

Let’s look at some examples.

### Example 1:

As you can see in the box there are 6 chips with the numbers 2, 4, 6, 8, 10 and 12. They are all even numbers.

We can say that **It is a certain event that when you take a chip you will get an even number**.

We can calculate the probability of this event like this:

*P(even)*= number of favorable casesnumber of possible cases = 66 = 1

There are 6 possible cases, and all 6 cases are favorable.

From here we can deduce that…

The probability of a certain event is equal to 1.

Now, if we go back to the previous box we can also conclude that **choosing a red tile is a sure event because all the tiles are red**.

Let’s see how **calculate the probability of occurrence of this event**:

*Q*(red)= 66 = 1

As you could see, **the probability of taking a red token is equal to 1**so we also verify that **It’s a sure thing**.

### Example 2:

**It is a certain event that the roulette wheel will land on a number less than 9**. This is because all numbers are less than 9.

By calculating the probability of this event we have:

*Q*(n<9)= 88 = 1

This also indicates that the event is safe, because **the probability of occurrence is 1**.

## Exercises on certain, possible and impossible events for primary school

In this section we are going to develop exercises to practice what we have learned about safe, possible and impossible events.

### Exercise 1:

Answer the following questions:

**Is it a certain event that an even number will come up when we roll the die? Because?**

Answer:

It is not certain that an even number will come up because there are faces with odd numbers (1,3 and 5).

Furthermore, when calculating the probability of the event we have:

Number of favorable cases: 3 (Remember that there are 3 even numbers on the die: 2, 4 and 6) Number of possible cases: 6 (The die has 6 sides)

*P(even)*= number of favorable casesnumber of possible cases = 36 = 12 = 0.5

**The probability that we get an even number is not 1, so we also check that it is not a certain event.**

**When we roll the die, is it a possible event that we get a number less than 5? Because?**

Answer:

**Yes it is a possible event** because there are 4 faces with numbers less than 5 which are 1, 2, 3 and 4.

We can also check this by calculating the probability of occurrence of this event:

Number of favorable cases: 4 (1, 2, 3 and 4 are less than 5)

Number of possible cases: 6

*P(n<5)*= number of favorable casesnumber of possible cases = 46 = 23 = 0.66

**In this case the probability of this event is greater than 0 and less than 1**. Therefore, it is a possible event to obtain a number less than 5 when rolling the die.

**Is it an impossible event to get the number 7 when rolling the dice? Why?**

Answer:

Yes, it is an impossible event because the numbers go up to 6. So 7 is not included in the possible results obtained when throwing a traditional dice.

Let’s calculate the probability that we get a 7:

Number of favorable cases: 0 (there are no 7 dots on any side of the die)

Number of possible cases: 6

*P(n=7)*= number of favorable casesnumber of possible cases = 06 = 0

As we see,** the probability of this event is 0**. Therefore it is an impossible event.

### Exercise 2:

From a deck of Spanish cards we have chosen only the jack, knight and king of cups cards:

**If we choose a card without looking, is it safe, possible or impossible for the chosen card to have a number greater than 12?**

It is impossible for the card to have a number greater than 12 because we have cards with the numbers 10, 11 and 12, (two less than 12 and one equal to 12).

If we want to calculate the probability that the card has a number greater than 12, it would be like this:

Number of favorable cases: 0 (no number greater than 12)

Number of possible cases: 3

*Q*(n<12)= 03 = 0

So we check that **The event is impossible because the probability of its occurrence is 0.**

**If someone takes a card, without seeing, it is certain that the card is a cup card. **

**Is this true or false? Because?**

It’s true. **It’s a sure thing **that if we choose any card from this group of decks, without seeing, we will draw one of cups.

This is because the three cards are of that suit of the deck.

To know the probability of this occurring, we do the following:

Number of favorable cases: 3 (all 3 cards are cups)

Number of possible cases: 3

*Q*(cup)= 33 = 1

**The probability of it being a cup is 1**that’s why we also know that it is a safe event.

*Before concluding, we leave you a few more resources so you can practice what you have learned.*