# The Multiplication Rule on a Tree Diagram

## Probabilities of an Event and Another Event on a Tree Diagram

A tree diagram shows all the possible outcomes of an event and their probabilities.

For example, it shows us the probability of a single event, such as a single coin toss.

## 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 12.

• The probability of getting Tails is 12.

Imagine we wanted to find the probability that an event 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?

To find the probability of an event and another event, multiply along the branches.

# 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 12 on the left branch and 12 on the right branch.

# 3

Multiply the probabilities along the branches.

12 × 12 = 14

# 4

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

The probability of getting Heads and Heads is 14. 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.

14 + 14 + 14 + 14 = 1

## Slider

The slider below another real example of using the multiplication rule on a tree diagram.

Open the slider in a new tab