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D2.1 · Solve probability problems using tree and Venn diagrams
Learn to solve probability problems using tree and venn diagrams through clear examples and targeted practice.
Ontario Grade 8 Mathematics
Data
Count outcomes without skipping a branch
A probability tree shows possible stages of a chance experiment. Each branch from one point names a possible outcome and its probability. Branches from the same point must cover all possibilities without overlap, so their probabilities total one. In this lesson we use a fair coin twice. The tree is a model of the stated experiment, not a guarantee that short real sequences will contain equal numbers of heads and tails.
What you will learn
- Read the branches in a probability tree.
- List the outcomes of two coin tosses.
- Find the probability of a chosen outcome.
First stage
For a fair coin, heads and tails each have probability one half. These two possibilities are exhaustive: one toss cannot land as both heads and tails. In the diagram, the two first branches leave the starting point and their probabilities add to one.
Second stage
After either first result, the second fair toss again has heads and tails as possibilities. The first toss does not change the probabilities for the second toss. The complete outcomes are HH, HT, TH, and TT. The order matters: HT and TH are different sequences.
Multiply along a path
Each complete path has probability one half times one half, or one quarter. To find exactly one head, add the disjoint HT and TH paths: one quarter plus one quarter equals one half. Do not add probabilities along one path; multiplication is used across independent stages.
Worked example
Two fair coin tosses
What is the probability that exactly one of two fair coin tosses is heads?
The four terminal outcomes HH, HT, TH, and TT are disjoint and each has probability 0.25. This compact tree summarizes the complete paths.
- List outcomesThe ordered results are HH, HT, TH, and TT.
- Find one pathEach pair has probability 1/2 × 1/2 = 1/4.
- Combine casesExactly one head occurs in HT or TH, so 1/4 + 1/4 = 1/2.
Answer: The probability is 1/2, or 50%.
Check: Two of the four equally likely ordered outcomes qualify.
Common mistakes and how to avoid them
Counting HT and TH as the same ordered outcome.
Correction: They occur in different orders and are two separate paths in a two-toss experiment.
Lesson summary
- Branches leaving one point have probabilities that total one.
- Multiply probabilities along independent stages.
- Add probabilities of disjoint qualifying paths.
Check your understanding
Question 1
What is the probability of two tails?
- 1/2
- 1/4
- 3/4
- 1
Show answer and explanation
1/4
Only TT qualifies, and its path probability is 1/2 × 1/2 = 1/4.
Key terms
- Outcome
- One possible result of an experiment.
- Independent
- One result does not change the probability of the next result.
Continue through Grade 8 Math
View the complete Grade 8 Math Ontario Grade 8 Mathematics curriculum and lessons
- D1.6 · Critique scatter plots and misleading graphs before drawing conclusions
- E1.1 · Identify shape properties and transformations in tessellations
- D1.1 · Choose one-variable or two-variable data for a question
- D1.2 · Collect and tabulate continuous two-variable data
- D1.3 · Choose and label suitable graphs including scatter plots
- D1.4 · Create a data infographic with appropriate representations
About this lesson and its review
Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation D2.1. It is a study resource, not an official curriculum publication.
Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.