Theoretical Versus Experimental

Overview

In lesson 5 of Probability: From Chance to Models, students compare theoretical probability with experimental relative frequency using die-roll data. They analyse results after 20, 40, 60, 80 and 100 rolls, describe stabilisation, and use evidence to explain variation between groups.

Learning intentions

• WALT compare theoretical and experimental probabilities.
• WALT describe how relative frequency changes as the number of trials increases.
• WALT explain why experimental probability tends to approach theoretical probability over many trials.
• WALT communicate a conclusion using evidence from a table and graph.

Success criteria

• I can identify the closest and furthest experimental estimates from the theoretical probability.
• I can describe patterns of stabilisation in the data.
• I can use numerical and graphical evidence to explain why groups obtain different results.
• I can evaluate whether more trials always guarantee an exact result.

Curriculum links

• Interpret and apply mathematical and statistical information in context by making an informed judgement from tables and graphs.
• Explore data using a statistical enquiry process by presenting and describing data with an appropriate visualisation.
• Demonstrate mathematical reasoning by using probability concepts, relative frequency, accurate calculations and appropriate representations.
• Use mathematical methods to explore a problem related to life in Aotearoa New Zealand or the Pacific by communicating accurate mathematical information.

Lesson structure (95 minutes)

1. 0–8 min · Hook and retrieval. Teacher opens with the hook and retrieval slides and asks, “If a fair die is rolled 20 times, must each number appear exactly as often as the others?” Students make an individual prediction, then recall that the theoretical probability of rolling a particular number is (1/6), approximately 0.167.

2. 8–20 min · Explicit teaching and modelling. Teacher uses the comparison and modelling slides to distinguish theoretical probability from experimental probability, models relative frequency as [ \text{relative frequency}=\frac{\text{number of successful outcomes{\text{number of trials, ] and calculates an example at 20 rolls. Students annotate the die investigation analysis sheet and explain what the numerator and denominator represent.

3. 20–30 min · Reading the investigation. Teacher distributes the die investigation analysis sheet and checks that students understand the cumulative data for rolling a particular face at 20, 40, 60, 80 and 100 rolls. Students work in pairs to calculate any missing relative frequencies, compare each value with (1/6), and identify the closest and furthest estimates.

4. 30–52 min · Graphing and paired analysis. Teacher models how to plot number of rolls on the horizontal axis and relative frequency on the vertical axis, including a horizontal reference line at (1/6), using the graphing instructions and worked example. Students complete the graph on the worksheet, using the dyslexia-friendly graph labels or the graph template provided, then answer the paired-analysis questions about movement, variation and stabilisation.

5. 52–68 min · Comparing groups. Teacher displays two or three groups’ data in the group comparison slides and prompts: “Why are the results not identical?” and “What evidence suggests the estimates are becoming more stable?” Students compare graphs and data, discuss sampling variation, and write one evidence-based explanation using the sentence starters: “At ___ rolls…”, “This differs from (1/6) by…”, and “The results vary because…”.

6. 68–83 min · Class discussion and reasoning. Teacher facilitates a mini-whiteboard check and explicitly models a strong response: “The experimental probability does not have to equal (1/6) because each set of rolls is random. As the number of trials increases, unusually high or low results have less influence, so the relative frequency often becomes more stable near (1/6). However, this is a tendency, not a guarantee.” Students improve their own written explanation, adding at least two numerical or graphical references.

7. 83–95 min · Extension and exit assessment. Teacher introduces the extension through the pooling and evaluation slides. Students who are ready pool class totals, calculate combined relative frequencies at each trial size, and evaluate the claim “More trials always guarantee an exact result.” All students complete the final worksheet exit question: “Use evidence from the 20–100 roll data to explain why experimental probability tends to approach, but may not equal, theoretical probability.”

Resources

• the theoretical versus experimental probability slide deck
• the die investigation analysis sheet
• One fair six-sided die per pair, or pre-recorded die-roll data
• Calculators
• Rulers and pencils
• Mini-whiteboards and pens
• Projector or interactive display
• Optional pooled class data table

Assessment

• Formative questioning checks whether students distinguish (1/6) from relative frequency and understand cumulative trial totals.
• Teacher circulates during graphing, checking correct axes, scales, calculations, the (1/6) reference line and contextual descriptions of variation.
• Exit responses are assessed for a correct comparison, a description of stabilisation and evidence-based reasoning about random variation.

Differentiation

• Provide the worksheet with partially completed calculations, a graph template, clearly spaced dyslexia-friendly labels, high-contrast printing and reduced-copying tasks.
• Offer paired data analysis, a worked example and sentence starters; read instructions aloud and allow students to use a calculator or verbal rehearsal before writing.
• Support EAL and dyslexic learners with short instructions, uncluttered pages, a sans-serif font, increased line spacing and key terms explained in context.
• Challenge advanced learners to pool class data, calculate combined relative frequencies at each stage, compare pooled and individual graphs, and evaluate why a larger number of trials does not guarantee an exact result.

Extension

• Investigate whether the pooled data is consistently closer to (1/6) than every individual group’s data, using absolute differences from (1/6).
• Write a brief evaluation of the statement: “With enough trials, the experimental probability will definitely equal the theoretical probability.”