Probability Trees

Overview

In this lesson (lesson 6 of 8), students learn to represent combined events using probability trees and use the tree to calculate probabilities. They build from earlier work on outcomes and frequency/probability ideas, moving to structured theoretical models.

Learning intentions

• Students will understand what a probability tree shows for sequential events.
• Students will construct probability trees for events with equally likely outcomes.
• Students will calculate probabilities of combined outcomes using tree branches.
• Students will interpret probability trees to answer questions about events.

Success criteria

• I can draw a probability tree with correct branch probabilities.
• I can list the full set of outcomes shown by the tree.
• I can find the probability of a specific combined event by multiplying branch probabilities.
• I can state the probability as a number on the 0–1 scale and explain what it means.

Curriculum links

• Probability: generate theoretical sample spaces for single and combined events with equally likely outcomes and use these to calculate theoretical probabilities.
• Probability: understand that the probabilities of all possible outcomes sum to 1.
• Probability: record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, and equally/un-equally likely outcomes (linking tree results to expected behaviour).
• Working mathematically: begin to express arguments formally about what a probability means and how calculations follow from the tree.

Lesson structure (60 minutes)

1. 0–5 min · Do Now. Students face front during teaching time; in silence, they brainstorm events that could happen in a sequence (e.g., “flip a coin then roll a die”, “choose a card then choose again”) and write one short example on their mini-whiteboard.

• Teacher prompts: “What happens first? What can happen next?”

2. 5–15 min · I Do (modelling). Teacher demonstrates a simple tree on the board (e.g., coin then coin, or coin then spinner with 2 equal sections). Teacher speaks through: first event branches, then second event branches, and labels each branch with probability.

• Students copy the model, then answer one prompt: “What is the probability of the path ‘Heads then Tails’?” (quick call-and-response).

3. 15–28 min · We Do (guided construction). Whole class builds a probability tree together for a combined event with equally likely outcomes (teacher chooses one scenario). Example: “A bag has 2 red and 2 blue sweets. You draw one sweet, then replace it and draw another.”

• Teacher leads step-by-step: write first choice branches (each with probability 1/2), then second choice branches (each with probability 1/2).
• Students turn and talk for 2 minutes to decide: “How many outcomes does the tree show?” and “How do we get probability for one route?”

4. 28–38 min · We Do (using the tree). Teacher demonstrates how to calculate probability for a specific combined outcome (multiply along the route). Then demonstrates summing probabilities when there are multiple routes for “event” questions (e.g., “two different colours” includes two paths).

• Students complete 3 short questions on their sheet in pairs, then teacher checks: correct route, correct multiplication, and reasoning for events.

5. 38–53 min · You Do (independent practice). Students construct their own probability trees for two given scenarios printed on worksheets. Scenario A: equally likely outcomes in sequence (e.g., coin then die with two marked outcomes treated as equal parts). Scenario B: a slightly more complex tree still using equally likely outcomes (e.g., spinner with 3 equal sections then coin, or two fair dice but focusing on limited outcomes).

• Teacher circulates, using prompts: “Where is the first split?”, “What probability is each branch?”, “Which route matches your event?”
• Dyslexia-friendly reading option: provide a version with key words highlighted (FIRST, NEXT, PROBABILITY, EVENT) and allow students to have the scenario read aloud.

6. 53–60 min · Exit ticket (check understanding). Individually, students answer one final question: “Draw the probability tree for a fair coin then a fair coin. State the probability of ‘at least one Heads’.”

• Teacher collects for quick marking; students show at least one calculation and a statement linking to 0–1 probability.

Resources

• Probability tree worksheet (2 scenarios) with blank tree templates
• Mini-whiteboards and pens
• Teacher-made example tree (coin/coin or coin/spinner)
• Coloured pencils (optional: colour branches to show first vs second event)
• Dyslexia-friendly scenario cards (key terms highlighted; optional audio read-aloud by teacher)
• Prompt cards: “Multiply along a route”, “Sum routes for an event”, “Total = 1”
• Exit ticket slips

Assessment

• Formative checks during I Do: students identify the correct path probability verbally.
• Formative checks during We Do: teacher listens to turn-and-talk reasoning about number of outcomes and probability method.
• Summative-in-mini during You Do: teacher reviews one student’s tree for correct branch probabilities and one calculation using multiplication.
• Exit ticket: confirm ability to interpret “at least one” as a set of routes and compute using the tree.

Differentiation

• Support for learners needing scaffolding: provide a partially completed tree for Scenario A (first split already drawn) and sentence starters: “The route that matches ____ is ____.” / “So the probability is ____.”
• Extra visual support: allow students to use colour coding (same colour for each branch level) and provide a “route tracker” strip to follow paths.
• Challenge/extension for advanced learners: include one question requiring them to justify that probabilities of all outcomes sum to 1 by either counting routes or adding branch products from the tree.
• EAL/SEN: reduce reading load by using icons for FIRST/NEXT and provide a bilingual peer buddy for turn-and-talk; teacher checks understanding with short, specific questions rather than broad ones.