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The Multiplication Rule on a Tree Diagram
(KS3, Year 7)
The Lesson
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}.
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?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 ^{1}⁄_{2} on the left branch and ^{1}⁄_{2} on the right branch.
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
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: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:- Do you disagree with something on this page?
- Did you spot a typo?