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The Multiplication Rule on a Tree Diagram
(KS3, Year 7)

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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.coin_toss_treeTo 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.
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.coin_toss_twice_treeImagine 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.

Step-by-Step:

1

Find the event given in the question. The event Heads and Heads is found in the top branches. multiplication_rule_tree_diagram_step_1

2

Find the probabilities along the branches for this event. multiplication_rule_tree_diagram_step_2 The probabilities along the branches are 1⁄2 on the left branch and 1⁄2 on the right branch.

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.

multiplication rule tree diagram answer Using notation, if H is the event of a Heads coming up, the probability of the event happening twice is P(HH). multiplication_rule_tree_diagram_answer_notation 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.

multiplication rule tree diagram all probabilities Note: If we add the probabilities of each outcome, they add to 1.
1⁄4 + 1⁄4 + 1⁄4 + 1⁄4 = 1

Lesson Slides

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

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: probability_formula_mini

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: outcome_add_to_1
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This page was written by Stephen Clarke.

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