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.

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 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).

Answer:

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

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

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See Also

What is probability? The multiplication rule of probability Simplifying fractions