12.02 Probability as a number

The likelihood of an event after a trial can be placed on a spectrum from 0 to 1 using fractions or decimals, or from 0\% to 100\% using percentages:

A probability can never be less than 0 or more than 1. The larger the number, the more likely it is, and the smaller the number, the less likely it is. We will now look at how to determine these numbers exactly.

In the  last lesson  we looked at the difference between an outcome and an event.

An outcome represents a possible result of a trial. When you roll a six-sided die, the outcomes are the numbers from 1 to 6.

An event is a grouping of outcomes. When you roll a six-sided die, events might include "rolling an even number", or "rolling more than 5".

Each outcome is always an event - for example, "rolling a 5" is an event.

But other events might not match the outcomes at all, such as "rolling more than 6".

If every outcome in a trial is equally likely, then the probability of one particular outcome is given by the equation:\text{Probability} = \dfrac{1}{\text{Size of sample space}}

Remember that the sample space is the list of all possible outcomes. We can multiply this number by 100\% to find the probability as a percentage.

For example, the sample space for rolling a standard 6-sided die is 1, \,2, \,3, \,4, \,5, \,6, which are all equally likely. So the probability of rolling a 5 is \dfrac{1}{6} since there is only one 5 on a die, and the size of the sample space is 6.

If the outcomes in a sample space are not equally likely, then we have to think about splitting the sample space up into "favourable outcomes" and the rest. Then we can use the formula:\text{Probability} = \dfrac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}

If every outcome is favourable, then we have a probability of 1. If there are no favourable outcomes, then probability is 0.

Examples

Example 1

A probability of \dfrac{4}{5}\, means the event is:

A

Impossible

B

Unlikely

C

Likely

D

Certain

Worked Solution

Create a strategy

Compare the value to 0 and 1.

Apply the idea

Probabilities range from 0 to 1. Events with probabilities close to 0 are unlikely, and events close to 1 are likely. A probability of \dfrac{4}{5}\, is close to 1 so the event is likely. The correct option is C.

Watch question walkthrough

Example 2

Select all events that have a probability of 25\% on this spinner:

Select all correct options.

A

Apple

B

Ball

C

Pig

D

Star

Worked Solution

Create a strategy

Write the percent into a fraction.

Apply the idea

The 25\% is equal to \dfrac{1}{4}. It means that it the sector will take up one quarter of the spinner. We can see that the apple and pig sectors both take up one quarter of the spinner. The correct options are A and C.

Watch question walkthrough

Example 3

A jar contains 10 marbles in total. Some of the marbles are blue and the rest are red.

a

If the probability of picking a red marble is \dfrac{4}{10}, how many red marbles are there in the jar?

Worked Solution

Apply the idea

We know that the probability of picking a red marble is \dfrac{4}{10}. That means 4 out of 10 marbles are red. So there are 4 red marbles.

Watch question walkthrough

b

What is the probability of picking a blue marble?

Worked Solution

Create a strategy

Find the number of blue marble and write the probability as a fraction.

Apply the idea

There are 10 marbles in total. We know that 4 of them are red and the rest are blue.

So, there are 10 - 4 = 6 blue marbles.

So the probability is of picking a blue marble is \dfrac{6}{10}=\dfrac{3}{5}.

Watch question walkthrough