Experimental Probability

Overview

In lesson 4 of 12 in Probability: From Chance to Models, students investigate how experimental probability is estimated from repeated trials. They roll a die 100 times, record results in blocks of 20, calculate the relative frequency of rolling a 6, and compare group and class results with the theoretical probability of (1/6).

Learning intentions

• WALT collect and organise experimental data accurately.
• WALT calculate experimental probability as relative frequency.
• WALT compare experimental results with the theoretical probability of (1/6).
• WALT describe variation and use evidence to discuss whether a die appears fair.

Success criteria

• I can tally die results and record cumulative totals after every 20 rolls.
• I can calculate the relative frequency of rolling a 6.
• I can compare my result and my group’s result with (1/6).
• I can describe variation between groups and support my ideas with evidence.

Curriculum links

• Explore data using a statistical enquiry process: explain the data source, present data appropriately, and describe features in context.
• Interpret and apply mathematical and statistical information in context by making an informed judgement about fairness using evidence.
• Demonstrate mathematical reasoning by using appropriate probability methods, representations, and mathematical statements.
• New Zealand Curriculum Refresh: develop statistical thinking, critical thinking, communication, collaboration, and the ability to make evidence-informed decisions.

Lesson structure (95 minutes)

1. 0–8 min · Hook and prior knowledge. Display a die image and ask, “If a fair die is rolled 100 times, must a 6 appear exactly 16 or 17 times?” Open with the hook and prediction slide. Students make an individual prediction, then discuss what “fair” and “variation” might mean. Take several responses without confirming an answer.

2. 8–20 min · Model the mathematics. Use the modelling slides to revisit theoretical probability (P(6)=1/6), tally marks, cumulative frequency, and relative frequency: [ \text{relative frequency}=\frac{\text{number of sixes{\text{number of rolls. ] Model a short example, including conversion to a decimal or percentage and comparison with (1/6\approx0.167). Students complete two quick checks on the probability investigation worksheet and explain their calculation to a partner.

3. 20–27 min · Set up the investigation. Demonstrate safe, consistent rolling and recording procedures using the investigation instructions. Organise students into five groups of five and assign role cards: roller, tally recorder, checker, calculator/data analyst, and reporter. Students collect one die, a recording table, and a calculator for their group.

4. 27–52 min · Collect experimental data. Students roll the die 100 times, recording the result after each roll and pausing at rolls 20, 40, 60, 80, and 100. At each checkpoint, they record the cumulative number of sixes and calculate the current relative frequency on the probability investigation worksheet. The teacher circulates, checks tallies, prompts accurate counting, and models again for groups needing support.

5. 52–70 min · Represent and analyse results. Students enter their five checkpoint results into the table and create an appropriate visualisation, such as a line graph of cumulative relative frequency against number of rolls, on the worksheet. They answer: When was the result closest to (1/6)? When was variation greatest? Did the estimate become more stable? Groups compare their final relative frequencies and calculate the difference from (1/6).

6. 70–87 min · Class discussion: fairness and evidence. Display the discussion prompts in the analysis and fairness discussion slides. Each group reports its final number of sixes and relative frequency. Collate results on the board and discuss why groups differ, whether the class evidence suggests the die is fair, and why one experiment cannot prove fairness. Students must refer to at least one numerical result and distinguish between “close to expected” and “exactly expected”.

7. 87–95 min · Plenary and exit response. Students complete the final reflection on the probability investigation worksheet: “Our group’s result was ___; this compares with (1/6) because ___; variation occurred because ___.” Invite two students to share. Collect worksheets to check calculation accuracy, contextual descriptions, and evidence-based reasoning.

Resources

• the complete probability investigation slide deck
• the probability investigation worksheet
• Five standard six-sided dice
• Role cards for roller, recorder, checker, calculator/data analyst, and reporter
• Calculators or spreadsheet software
• Board or shared class data table
• Rulers, pencils, and graphing equipment
• Optional speech-to-text or audio recording tools

Assessment

• During modelling, check that students can distinguish theoretical probability from experimental relative frequency and calculate (1/6).
• During data collection, check tally accuracy, cumulative totals, correct denominators, and appropriate graph scales.
• Use the completed reflection as an exit assessment: students should compare with (1/6), describe variation, and make a cautious judgement about fairness using evidence.

Differentiation

• Provide shared recording tables with pre-labelled checkpoints, a worked example, and the formula for relative frequency. Offer calculator support and teacher modelling in a small group.
• Use role cards so students can contribute through practical, checking, speaking, or recording tasks. Pair students strategically and allow oral explanations before written responses.
• Offer dyslexia-friendly options: larger data tables, uncluttered sans-serif text, increased spacing, audio prompts, read-aloud instructions, and speech-to-text for the reflection.
• For advanced learners, predict the likely number of sixes and the relative frequency in 1,000 rolls. Explain why estimates generally become more stable as the number of trials increases, while recognising that variation does not disappear.