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 ^{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
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.
StepbyStep:
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 ^{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}.
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