The Multiplication Rule on a Tree Diagram
Probabilities of an Event and Another Event on a Tree Diagram
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 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.
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
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.
Find the probabilities along the branches for this event.
The probabilities along the branches are 1⁄2 on the left branch and 1⁄2 on the right branch.
Multiply the probabilities along the branches.
1⁄2 × 1⁄2 = 1⁄4
Simplify the fraction if possible. (The fraction in our example is already as simple as possible).
The probability of getting Heads and Heads is 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
The slider below another real example of using the multiplication rule on a tree diagram.Open the slider in a new tab