Independent Event Products

Overview

Lesson 9 of 12 in Probability: From Chance to Models. Students build on single-event probability and sample spaces by investigating two-stage experiments, identifying independence, and using multiplication to calculate “A and B” probabilities. They move from concrete dice-and-coin trials to tables, outcome lists and symbolic notation, then evaluate whether repeated results support their model.

Learning intentions

• WALT identify when two events are independent.
• WALT calculate the probability of two independent events using (P(A\text{ and }B)=P(A)\times P(B)).
• WALT represent combined outcomes using tables and outcome lists.
• WALT compare theoretical probabilities with results from repeated trials.

Success criteria

• I can explain why the outcome of one event does not change the probability of the other.
• I can multiply two probabilities accurately and simplify my answer where appropriate.
• I can list or organise all combined outcomes without omissions or repetitions.
• I can compare experimental results with a theoretical probability and describe possible variation.

Curriculum links

• Interpret and apply mathematical and statistical information in context: use experimental results, tables and probability information to make an informed judgement.
• Demonstrate mathematical reasoning: use an appropriate sequence of steps, mathematical notation and representations to justify a probability calculation.
• Use mathematical methods to explore problems that relate to life in Aotearoa New Zealand or the Pacific: apply number methods accurately in a familiar game and chance context.
• NZ Curriculum Refresh: develops mathematical and statistical literacy, reasoning, problem solving and communication, alongside the competencies of thinking, managing self, and participating and contributing.

Lesson structure (95 minutes)

1. 0–8 min · Hook and retrieval. Open with the hook and retrieval slides showing the question, “Is getting a six and then heads twice as unlikely as getting a six or heads?” Students complete three quick retrieval questions independently: probability of a six, probability of heads, and the meaning of “and”; they then share answers and predictions.

2. 8–23 min · Concrete investigation. Give each pair one die and one coin, and display the experiment instructions. Students conduct 30 trials of “roll a six and toss heads”, recording successes in a simple frequency table; pause to ask whether the coin probability changes after the die result is known. Emphasise that the die and coin do not affect each other, so the events are independent.

3. 23–38 min · Model the representation. Model a product grid using the board and the worked example slides: list die outcomes across one side and coin outcomes down the other, then identify the single successful combination (6, heads). Students create a second grid for “roll an even number and toss tails”, shade successful cells, and calculate (P(\text{even and tails})=\frac{3}{6}\times\frac{1}{2}=\frac{1}{4}). Check that students can distinguish “and” from “or”.

4. 38–58 min · Guided-to-independent practice. Distribute the independent events practice worksheet and complete the first question together. Students then solve progressively less scaffolded questions involving dice, coins and two-stage outcome lists, explaining each multiplication. Circulate for a targeted check: ask, “What is event A? What is event B? Does either event change the other probability?” Students who need support use colour coding and the product grid before moving to notation.

5. 58–73 min · Repeated trials and variation. Students pool pair results from the first experiment, calculate the experimental proportion of successes, and compare it with the theoretical value (\frac{1}{12}). Use the variation discussion slides to prompt: “Why might our answer not be exactly (\frac{1}{12})?” Students write one statement about random variation and one limitation of using only 30 trials.

6. 73–88 min · Challenge: design a fair game. In pairs, students design a simple game using two independent events, such as a coin and die. They must state the winning condition, show the outcome list or grid, calculate the winning probability, and decide whether the game is fair for a player winning a one-point prize. Students use the challenge section of the game design task and give a one-minute explanation to another pair.

7. 88–95 min · Plenary and exit check. Return to the summary and exit prompt. Students complete an exit response: “A fair coin is tossed and a die is rolled. Find (P(\text{tails and a number greater than 4})), show a representation, and explain why multiplication is appropriate.” Take responses at the door and address common errors in the next lesson.

Resources

• the complete probability slide deck
• the independent events practice worksheet
• One die and one coin per pair
• Recording paper or exercise books
• Calculators
• Board and coloured pens
• Headphones or text-to-speech/audio-supported instructions
• Mini whiteboards or response cards

Assessment

• Listen for correct use of independent, and, outcome, theoretical probability and experimental probability during pair work.
• Use the guided practice questions to check accurate event identification, multiplication, simplification and grid construction; provide immediate verbal feedback.
• Mark the exit response for method, representation and explanation. Note whether errors arise from confusing “and” with “or”, incomplete outcome lists, or incorrect multiplication.

Differentiation

• Support: provide a two-stage experiment routine, colour-code event A and event B, and require students to build a product grid before using the formula. Keep a worked example visible and provide sentence starters: “The events are independent because…” and “I multiply because…”.
• Dyslexia-friendly access: offer audio-supported instructions, read questions aloud without requiring public reading, use a clear sans-serif font, generous spacing, short numbered instructions and uncluttered grids. Accept verbal explanations alongside written working where appropriate.
• EAL and diverse learners: pre-teach event, outcome, independent, and, multiply and theoretical. Pair students strategically and use symbols, gestures, coins and dice before abstract notation.
• Extension: ask advanced learners to compare multiplication for “and” with addition for mutually exclusive “or” events, then alter their game so it has a specified winning probability. They must justify whether the revised game is fair and identify assumptions or limitations in their model.

Extension

• Investigate how many trials are needed before the experimental proportion is usually close to (\frac{1}{12}), recording results in batches.
• Create two different independent-event games with the same winning probability and explain why their outcome structures are equivalent.
• Write a short advice statement for a player explaining why a long run of unsuccessful trials does not change the probability on the next trial.