# The Multiplication Rule on a Tree Diagram

(KS3, Year 7)

## A Single Coin Toss

A single coin toss can be shown on a tree diagram.To find the probability of an event, read the probability on each branch:-
The probability of getting
**Heads**is.^{1}⁄_{2} -
The probability of getting
**Tails**is.^{1}⁄_{2}

**and**another event happens. For example, a double coin toss.

## A Double Coin Toss

A double coin toss can be shown on a tree diagram.Imagine we wanted to find the probability of getting**Heads and Heads**in a double coin toss.

## How to Find the Probability of an Event __and__ Another Event on a Tree Diagram

## Question

What is the probability of getting**Heads and Heads**in a double coin toss?

**and**another event, multiply

*along*the branches.

## Step-by-Step:

## 1

Find the event given in the question.
The event

**Heads and Heads**is found in the top branches.## 2

Find the probabilities along the branches for this event.
The probabilities along the branches are

**on the left branch and**^{1}⁄_{2}**on the right branch.**^{1}⁄_{2}## 3

Multiply the probabilities along the branches.

^{1}⁄

_{2}×

^{1}⁄

_{2}=

^{1}⁄

_{4}

## 4

Simplify the fraction if possible. (The fraction in our example is already as simple as possible).

## Answer:

The probability of getting**Heads and Heads**is

^{1}⁄

_{4}.

Using notation, if

**H**is the event of a

**Heads**coming up, the probability of the event happening twice is

**P(HH)**. We can also express this as a number (0.25) or a percentage (25%).

## Finding All Probabilities from a Tree Diagram

All the probabilities in a tree diagram can be found in the same way. Consider the tree diagram for a double coin toss. If we multiply across all branches, we find the probability of each outcome.**Note:**If we add the probabilities of each outcome, they add to 1.

^{1}⁄

_{4}+

^{1}⁄

_{4}+

^{1}⁄

_{4}+

^{1}⁄

_{4}= 1

## What Is Probability?

A probability is a measure of how likely (how*probable*) an event is to happen. A probability is expressed as a number between 0 (impossible) and 1 (certain). The formula for finding a probability is shown below:

## Top Tip

## A Useful Check

There is a useful way to check that the probabilities on a tree diagram are all correct. The probabilities of each final outcome add up to 1:## Worksheet

This test is printable and sendable