# 10.02 Probability with numbers

Lesson

## Probability with numbers

Now that we know how to describe events with language, we will now investigate using numbers to calculate probabilities. If we can split up the sample space into equally likely outcomes and can identify the favourable outcomes making up an event, we can use the formula: \text{Probability} = \dfrac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}

We can also use a useful fact about complementary events - since exactly one of them must happen, their probabilities always add to 1. This means if we know the probability of an event, the probability of the complementary event will be one minus the probability of the original: \text{Probability of complementary event} = 1 - \text{Probability of event}

### Examples

#### Example 1

Consider this list of numbers: 2,\,2,\,2,\,3,\,3,\,3,\,4,\,4,\,5,\,5,\,5,\,7,\,7,\,7,\,7,\,9,\,9

a

A number is chosen from the list at random. What is the probability it is an odd number?

Worked Solution
Create a strategy

We can use the rule: \text{Probability} = \dfrac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}

Apply the idea

There are 12 odd numbers in the list, and 17 numbers in total.\text{Probability} = \dfrac{12}{17}

b

A number is chosen from the list at random. What number has the highest probability of being chosen?

Worked Solution
Create a strategy

Choose the number that appears more often than any other number.

Apply the idea

The number that appears the most times in the list is 7 which appears 4 times.

So, 7 has the highest probability of being chosen.

#### Example 2

The probability of the local football team winning their grand final is 0.36.

What is the probability that they won't win the grand final?

Worked Solution
Create a strategy

We can use the rule: \text{Probability of complementary event} = 1 - \text{Probability of event}

Apply the idea
Idea summary

If a sample space can be split up into equally likely outcomes, then we can use the following formula: \text{Probability} = \dfrac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}

If two events are complementary then their probabilities will add to 1. This means: \text{Probability of complementary event} = 1 - \text{Probability of event}

### Outcomes

#### MA4-21SP

represents probabilities of simple and compound events