# Probability

### Chapters

### Probability: Complement

# Probability: Complement

The `complement`

of an event is a list of all the ways that event **doesn't** happen. So, it's the list of all outcomes
of an experiment that **do not** form part of that event. Let's look at some examples.

## Examples

If the event is \(\{\text{Green}\}\), the complement is \(\{\text{Red}\}\).

If the event is roll an even number \(\{2,4,6\}\), the complement is rolling an odd number \(\{1,3,5\}\).

If the event is drawing a \(\{\text{Spade}\}\) from a standard deck of \(52\) playing cards, the complement is drawing one of \(\{\text{Heart, Diamond, Club}\}\).

If you take an event and its complement together, you get all of the possible outcomes for an experiment.

## Probability

The probability that an event (E) occurs is given by the formula:

### Example

What is the probability of rolling an even number on a 6-sided die?

**Number of Ways the Event Can Happen:** You can roll 2, 4 or 6, so there are 3 ways this can happen.

**Total Number of Outcomes:** You could roll 1, 2, 3, 4, 5 or 6, so there are 6 possible outcomes.

\(\text{Probability} = \dfrac{3}{6} = \dfrac{1}{2}\).

We use \(E'\) or \(E^c\) to denote the complement of the event \(E\) (or \(A'\) to denote the complement of event \(A\)). We always, always have:

### Example

Sam's dad loves black and green jelly babies. Event \(A\) is going to denote randomly drawing a black or green jelly baby from a bag of jelly babies with equal numbers of the five different colours: green, orange, black, red and yellow. So \(A = \{\text{black}, \text{green}\}\) and \(A' = \{\text{orange}, \text{red}, \text{yellow}\}\).

The number of ways Event \(A\) can occur is \(2\), and there are a total of \(5\) possible outcomes. So, the probability of event \(A\) is:

The number of ways the complement of Event \(A\) can occur is \(3\). As the total number of outcomes is \(5\), the probability of event \(A'\) is:

Let's just check:

## What is the Complement Good For?

Sometimes, it is just easier to work out the probability of the complement of an event and subtract it from \(1\).

### Example

If you toss a coin three times, what is the probability of getting at least one head?

**Solution:**

There are quite a few ways that this event \(E\) can occur: \(\{HHH,HHT,HTH,THH, \dots, TTH\}\), but only one way \(\{TTT\}\) that its complement can occur. Thus, it's easier to work out the probability of the complement occurring and subtract it from one. There are a total of \(8\) possible outcomes. So, the probability of the complement is

### Description

In this mini series, you will learn a bit more on the topic of **probability, **we will cover topics such as

- Ratios
- Fair dice
- Conditional probability
- Mutually exclusive events

and more

### Audience

Year 10 or higher students

### Learning Objectives

Explore more on the topic of probability

Author: Subject Coach

Added on: 28th Sep 2018

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