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

homesitemapprobabilitymultiplication rule 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.coin_toss_treeTo 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.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


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.



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


Find the probabilities along the branches for this event. multiplication_rule_tree_diagram_step_2 The probabilities along the branches are 12 on the left branch and 12 on the right branch.


Multiply the probabilities along the branches.
12 × 12 = 14


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.

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.
14 + 14 + 14 + 14 = 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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