The Mathematics of Maybe: Introduction to Probability · Lesson 6 of 7
Tree Diagrams
“When outcomes start branching, a tree diagram keeps the family organised.”
• Understand why tree diagrams are useful. • Build tree diagrams for multi-step experiments. • Read complete outcomes from paths. • Use tree diagrams to list sample spaces. • Calculate simple probabilities from equally likely paths.
When an experiment happens in two or more steps, writing all possible outcomes in a single list can quickly become difficult to follow. A tree diagram makes the process easier by showing the choices step by step. At each stage, every possible choice is drawn as a separate branch, so we can clearly see how one decision or event leads to the next.
To find a complete outcome, we follow a path from the starting point of the tree all the way to an endpoint. Each complete path represents one possible result of the entire experiment. This makes tree diagrams especially useful for checking that no outcome has been missed and for understanding how the total number of possible outcomes is built from the choices available at each stage.
A branching diagram used to display all possible outcomes of a multi-step experiment.
Tossing a Coin Twice
On the first toss, the tree splits into H and T. From each of these branches, the second toss again splits into H and T. Reading complete paths gives HH, HT, TH and TT.
Problem
For two fair coin tosses, what is the probability of HH?
- 1.There are 4 equally likely complete outcomes.
- 2.Only one outcome is HH.
- 3.P(HH)=1/4=0.25=25%.
Tree Diagrams for Daily Choices
Tree diagrams are not only for coins. Suppose a school canteen offers three snacks and two drinks. Start with one branch for each snack, then split each snack branch into the two drink choices. The endpoints show every possible meal combination.
Problem
A child has 2 shirts and 3 pairs of pants. How many outfits are possible if one shirt and one pair of pants are chosen?
- 1.From each of the 2 shirt branches, draw 3 pants branches.
- 2.Each complete path is one outfit.
- 3.Total outcomes=2×3=6.
- 4.A tree diagram guarantees that no combination is accidentally missed.
Replacement Changes the Tree
If an item is selected and then replaced, the same possibilities are available on the next selection. If it is not replaced, the second-step possibilities can change. A correct tree must reflect what is actually available after the first step.
Problem
A box contains red, black and green pens. One pen is selected, replaced, and then another person selects a pen. What colour-pair outcomes are possible?
- 1.First choice can be R, B or G.
- 2.Because the pen is replaced, the second choice can again be R, B or G after every first branch.
- 3.The sample space has 3×3=9 colour-pair outcomes: RR,RB,RG,BR,BB,BG,GR,GB,GG.
Practice Problems
- Draw a tree diagram for tossing a fair coin twice and list the sample space.
- A basket A contains Apple and Orange. Basket B contains Banana and Mango. One fruit is picked from each basket. Draw a tree and list all pairs.
- A child has 2 shirts and 3 types of pants. Draw or describe a tree diagram for all outfits.
- A box has red, blue and green pens. One pen is drawn, replaced, then another is drawn. List the 9 colour-pair outcomes.
- Explain how a tree diagram would change if an object is not replaced before the second draw.
Key Takeaways
• Tree diagrams organise multi-step experiments. • Every complete path represents one outcome. • They help ensure the sample space is complete. • Branches can also display probabilities. • Replacement and non-replacement lead to different later branches.
Next, we bring all the ideas together in the chapter summary and mixed practice.