Probability In Context

Overview

This 12-lesson Year 11 unit develops probability from simple events to distributions and histograms. Students use mathematical and statistical information to make informed judgements, explain variation, and communicate conclusions in familiar Aotearoa New Zealand contexts. Each lesson is designed for 90 minutes and a class of 25 students.

Learning intentions

• WALT calculate, represent and interpret probabilities.
• WALT use tables, tree diagrams, Venn diagrams and graphs to solve probability problems.
• WALT compare theoretical and experimental probability.
• WALT explain assumptions, variation and limitations when making probability-based decisions.

Success criteria

Students can:

• represent probabilities as fractions, decimals and percentages between 0 and 1;
• select and use an appropriate probability representation;
• show a logical sequence of calculations;
• explain conclusions using mathematical evidence and context.

Curriculum links

• Interpret and apply mathematical and statistical information in context, including explaining variation and evaluating assumptions and limitations.
• Explore data using a statistical enquiry process by collecting, representing and describing experimental probability data.
• Demonstrate mathematical reasoning through accurate methods, representations and logical statements.
• Use mathematical methods to explore problems relating to life in Aotearoa New Zealand or the Pacific, communicating results with correct units and context.
• Key competencies: thinking; using language, symbols and texts; managing self; participating and contributing; relating to others.

Lesson structure (12 × 90 minutes)

1. Simple probability

WALT define outcome, event and sample space, and calculate simple probabilities. Success criteria: I can place events on a probability scale; identify favourable outcomes; calculate and simplify a simple probability.

1. 0–10 min · Hook. Open with the probability unit introduction and impossible-to-certain scenarios; students rank events from impossible to certain and justify one choice.
2. 10–30 min · Explicit teaching. Model the probability scale, key vocabulary and (P(E)=\frac{\text{favourable outcomes{\text{total outcomes).
3. 30–55 min · Guided practice. Solve dice, cards, marbles and word-based examples together using the simple probability practice sheet.
4. 55–80 min · Independent practice. Students complete questions involving spinners, LOTTO-style numbers and coloured objects.
5. 80–90 min · Plenary. Students explain why every probability lies between 0 and 1 and complete an exit response.

2. Complementary and compound events

WALT calculate probabilities of complementary and combined events. Success criteria: I can use (P(\text{not }E)=1-P(E)); identify “and” and “or”; explain my method.

1. 0–10 min · Retrieval. Revisit the probability scale and correct common errors.
2. 10–30 min · Explicit teaching. Model complements, mutually exclusive events and simple compound events with dice and cards.
3. 30–55 min · Guided practice. Students annotate worked examples and complete “and/or” questions in pairs.
4. 55–80 min · Independent practice. Students solve contextual problems using the compound probability practice sheet.
5. 80–90 min · Plenary. Students write one example where using the complement is more efficient.

3. Experimental probability

WALT collect data and compare experimental and theoretical probability. Success criteria: I can calculate relative frequency; describe variation; explain why results may differ.

1. 0–10 min · Prediction. Students predict the number of heads in 100 coin tosses.
2. 10–25 min · Investigation setup. Model a fair trial, recording table and relative frequency.
3. 25–55 min · Experiment. Groups complete repeated coin, dice or spinner trials and record results on the experimental probability recording sheet.
4. 55–75 min · Analysis. Students compare results with theoretical probabilities and describe variation.
5. 75–90 min · Plenary. Groups share one limitation and one improvement to the investigation.

4. Probability from two-way tables

WALT read, complete and interpret two-way tables. Success criteria: I can find totals; calculate joint and conditional probabilities; support a conclusion with table evidence.

1. 0–10 min · Hook. Present a fictional student transport survey in the two-way table introduction.
2. 10–30 min · Explicit teaching. Model row totals, column totals, joint events and conditional probabilities.
3. 30–55 min · Guided practice. Complete a table together and discuss the meaning of the denominator.
4. 55–80 min · Independent practice. Students answer contextual questions using the two-way table practice sheet.
5. 80–90 min · Plenary. Students explain how changing the denominator changes the probability.

5. Experimental versus theoretical probability

WALT evaluate differences between predicted and observed probabilities. Success criteria: I can calculate both probabilities; describe variation; judge whether results support a model.

1. 0–10 min · Retrieval. Review relative frequency and expected outcomes.
2. 10–30 min · Explicit teaching. Model comparison using fractions, percentages and difference.
3. 30–55 min · Group analysis. Students examine class data and identify patterns, clusters and unusual results.
4. 55–80 min · Written response. Students complete the comparison and evaluation worksheet.
5. 80–90 min · Plenary. Students state whether more trials would improve reliability and why.

6. Tree diagrams with replacement

WALT use tree diagrams to represent multi-stage events with replacement. Success criteria: I can label branches; multiply along branches; add suitable paths.

1. 0–10 min · Hook. Display a two-stage coloured-ball problem.
2. 10–30 min · Explicit teaching. Model branch labels, multiplication along paths and addition of outcomes.
3. 30–55 min · Guided practice. Build tree diagrams collaboratively using cards or board diagrams.
4. 55–80 min · Independent practice. Students solve multi-stage problems with the tree diagram practice sheet.
5. 80–90 min · Plenary. Students identify where independence appears in a with-replacement problem.

7. Tree diagrams without replacement

WALT use changing probabilities in tree diagrams without replacement. Success criteria: I can update totals after each selection; calculate path probabilities; explain dependence.

1. 0–10 min · Starter. Compare with-replacement and without-replacement scenarios.
2. 10–30 min · Explicit teaching. Model changing numerators and denominators at each branch.
3. 30–55 min · Guided practice. Solve a two-colour bag problem step by step.
4. 55–80 min · Independent practice. Students complete the without-replacement worksheet.
5. 80–90 min · Plenary. Students explain why the second draw is not independent.

8. Venn diagrams

WALT represent and interpret unions, intersections and complements. Success criteria: I can place data correctly; use set notation or words; calculate probabilities from a Venn diagram.

1. 0–10 min · Hook. Use a class-interest example involving sport, music or transport.
2. 10–30 min · Explicit teaching. Model universal set, intersection, union and complement.
3. 30–55 min · Guided practice. Construct a Venn diagram from a two-way table.
4. 55–80 min · Independent practice. Students solve contextual questions using the Venn diagram practice sheet.
5. 80–90 min · Plenary. Students explain the difference between “A and B” and “A or B”.

9. Discrete probability distributions

WALT construct and interpret a discrete probability distribution. Success criteria: I can check probabilities sum to 1; calculate expected value; interpret results in context.

1. 0–10 min · Retrieval. Review possible outcomes and probability totals.
2. 10–30 min · Explicit teaching. Model a distribution table and expected value.
3. 30–55 min · Guided practice. Complete a distribution from a game or weather-related context.
4. 55–80 min · Independent practice. Students use the probability distribution worksheet.
5. 80–90 min · Plenary. Students interpret an expected value without claiming it must occur.

10. Probability histograms

WALT represent and interpret discrete probability data graphically. Success criteria: I can select suitable axes; plot probabilities accurately; describe the shape and most likely outcomes.

1. 0–10 min · Hook. Compare a table with its probability histogram.
2. 10–30 min · Explicit teaching. Model scale, labels, bars and interpretation.
3. 30–55 min · Guided practice. Build a histogram from a distribution table.
4. 55–80 min · Independent practice. Students complete the probability histogram worksheet.
5. 80–90 min · Plenary. Students write two accurate observations from a histogram.

11. Probability in context

WALT interpret mathematical and statistical information to make an informed judgement. Success criteria: I can extract relevant information; explain variation; identify an assumption or limitation.

1. 0–10 min · Hook. Present a probability-based claim about a game, forecast or sporting result.
2. 10–25 min · Explicit teaching. Model a response structure: evidence, calculation, interpretation and limitation.
3. 25–60 min · Collaborative task. Groups investigate a mixed representation using the probability decision-making task.
4. 60–80 min · Individual response. Students justify whether the claim is valid.
5. 80–90 min · Plenary. Peer-check responses using the exam command words reference mat.

12. Consolidation and assessment

WALT select appropriate probability methods and communicate a justified conclusion. Success criteria: I can choose a suitable representation; show accurate working; evaluate the reasonableness of my answer.

1. 0–10 min · Retrieval quiz. Students answer short questions covering the unit.
2. 10–25 min · Review. Address misconceptions identified in the quiz.
3. 25–70 min · Assessment task. Students complete a mixed-context task using the probability consolidation assessment.
4. 70–82 min · Self-assessment. Students review their work with the exam command words reference mat.
5. 82–90 min · Reflection. Students identify one secure skill, one next step and one strategy that helped them learn.

Resources

• the Year 11 probability slide deck
• the relevant probability worksheet
• Dice, coins, cards, spinners and coloured counters
• Whiteboard, markers and calculators
• Graph paper or digital graphing tool
• the exam command words reference mat

Assessment

• Use questioning, mini-whiteboard responses, worked examples and circulating conferences to check vocabulary, denominators, representations and reasoning.
• Collect experimental data and written explanations to assess calculation accuracy, interpretation of variation and awareness of limitations.
• Use Lesson 12 as a formative consolidation assessment, with feedback focused on method selection, mathematical communication and contextual judgement.

Differentiation

• Provide worked examples, labelled diagrams, formula reminders, reduced question sets and structured sentence starters such as “The probability is … because …”.
• Offer reading of instructions aloud, uncluttered layouts, clear sans-serif fonts, increased spacing, colour-coded representations and audio or teacher-recorded explanations.
• Pair students strategically and allow calculators, manipulatives and verbal rehearsal before written responses.
• Advanced learners can compare multiple methods, create a probability game, investigate fairness, calculate expected value, or evaluate how assumptions affect a conclusion.