### What is Probability?

**Probability** is a measurement of **how likely something is to happen**.

**Probability** can be expressed as a **fraction, a percentage, a decimal** or in an expression such as **1 in 10**.

Why all the different ways of saying the same thing? Well they all mean the same thing really. If you have read the lesson about converting between decimals, fractions and percentages, you will know that they are all related.

The expression **1 in 10** can easily join in and be written **as a fraction** simply by placing the 1 over the 10:

We can now convert it to a **decimal** by dividing the bottom number by the top number:

1 ÷ 10 = 0.1

And we can convert this to a **percentage**:

0.1 x 100 = 10%

We might want to **convert probabilities** between fractions, decimals and percentages if asked to compare probabilities, or simply if the question asks us to.

But the important thing to remember is that all these ways of expressing **probability** are ways of showing the **likelihood that something will happen**.

### Probability Lines

Here is a **probability line** showing the **likelihood** of some events:

On the **left** of the line is **0% probability** – so there is no chance of this happening.

On the **right** of the line is **100% probability** – so it is certain that events at this end will happen.

When you flip a coin the probability of it landing on heads is **1 in 2**. This can also be expressed as **a half or 50% or 0.5** (this is also called an **even chance** as it is just as likely not to happen as it is to happen).

It means that for every 2 coin tosses we make we would expect to see heads come up once. It doesn’t mean that we would actually see that happen, but the probability of it happening is 1 in 2. If we kept tossing a coin over and over again the average number of heads thrown would get closer and closer to being 50%.

When looking at **probability as a fraction**, the bottom number is the **number of possible outcomes** – in the case of the coin toss there are 2 (heads or tails). The top number is the **number of outcomes you are looking for**, in this case 1 (heads).

Example 1

**Question**: Express the probability as a fraction of **not rolling a 2 or a 3** on a fair six-sided dice?

On a six-sided dice, the probability of getting any one number is:

1 in 6

Because there is one outcome we are looking for (rolling any number) out of a total of six possible outcomes (rolling a one, a two, a three, a four, a five, or a six).

To work out the probability of **not throwing a 2 or a 3** on a dice, you just need to work out how many possible outcomes there are in total and how many you are looking for.

There are six possible outcomes as shown above, and four of these outcomes are not rolling a two or a three.

So as a fraction the probability is:

We can simplify this like any fraction to:

Example 2

**Question**: I have invented a teleportation device that will make me disappear from the room I am in and reappear in one of the 10 other rooms in my house. However my machine isn’t quite finished yet and which room I end up in is random. What is my chance of arriving in the bathroom written as a decimal?

There are 10 possible outcomes, and just 1 that we are looking for, so the probability of ending up in the bathroom is 1 in 10.

To write this as a decimal we divide the number of outcomes we are looking for by the total number of outcomes:

10 ÷ 1 = 0.1

### Probability Tables

**Probability tables** show the **total number of possible outcomes** of an event.

When probability questions get more complicated we sometimes need to **draw a table** showing **all the possible outcomes** so that we can keep count.

For example, if we are rolling **2 dice**, calculating the probability of getting **a total of 9** is difficult to do in your head because there are **lots of possible outcomes** to go through. In this situation you should draw a table like the one below, it will help you to answer the question.

Here is a probability table showing the possible totals when rolling **two dices**. The top heading shows the possible number rolled on one of the dices, the side heading shows the possible number rolled on the other:

1 | 2 | 3 | 4 | 5 | 6 | |

1 | 2 | 3 | 4 | 5 | 6 | 7 |

2 | 3 | 4 | 5 | 6 | 7 | 8 |

3 | 4 | 5 | 6 | 7 | 8 | 9 |

4 | 5 | 6 | 7 | 8 | 9 | 10 |

5 | 6 | 7 | 8 | 9 | 10 | 11 |

6 | 7 | 8 | 9 | 10 | 11 | 12 |

When asked: What is the probability when rolling two dice of the **total being 9?**

We can firstly count all the total outcomes which is **every result on the table** – a quick way to do this is to multiply the number of columns by the number of rows (not including the headings – the dark grey bits!):

6 x 6 = 36

Next we can count the number of outcomes we are looking for. There are **4** ways of getting a nine shown in the table.

So the probability of getting a total of nine with two dice is **4 in 36**.

This can be simplified by dividing both parts by 4 to get:

1 in 9