Chocolate Table Probabilities

Overview

Lesson 6 of 12 in Probability: From Chance to Models. Students use chocolate-centre data to complete and interpret a two-way frequency table, then calculate probabilities from joint and marginal frequencies. The lesson develops statistical reasoning by requiring students to justify conclusions from data and consider what the table does—and does not—tell us.

Learning intentions

• WALT complete and interpret a two-way frequency table.
• WALT identify joint frequencies, marginal frequencies and the total frequency.
• WALT calculate probabilities from entries in a two-way table.
• WALT justify conclusions using evidence from data.

Success criteria

• I can fill missing cell values, row totals, column totals and the grand total.
• I can identify a joint frequency and a marginal frequency.
• I can calculate probabilities such as (P(\text{soft})), (P(\text{fruit})) and (P(\text{soft and fruit})).
• I can explain a conclusion using values from the table and appropriate probability language.

Curriculum links

• Interpret and apply mathematical and statistical information in context by making an informed judgement from a table.
• Explore data using a statistical enquiry process by presenting, describing and interpreting data in context.
• Demonstrate mathematical reasoning by using accurate methods, mathematical terms and appropriate representations.
• Develop mathematical thinking, communication and statistical literacy through explaining variation and justifying decisions from data.

Lesson structure (95 minutes)

1. 0–8 min · Hook and retrieval. Teacher displays a chocolate survey showing centres as soft or hard and fillings as fruit or caramel using the opening chocolate-data slide; students make a quick prediction about which combination is most common and recall the meaning of frequency and probability.

2. 8–20 min · Build the table. Teacher introduces the two-way table and models how the row and column headings organise two categorical variables. Use colour-coded rows and columns on the table vocabulary slides to distinguish a cell, row total, column total and grand total. Students annotate a blank table and explain what each position represents.

Use this example throughout the lesson:

Centre / Filling	Fruit	Caramel	Total
Soft	18	12	30
Hard	10	20	30
Total	28	32	60

Ask: “What does 18 represent?” “How can we check that the totals are consistent?” Emphasise that 18 is a joint frequency, while 30 and 28 are marginal frequencies.

3. 20–38 min · Guided completion. Teacher distributes the two-way table practice worksheet and works through the first incomplete table, modelling one step at a time: add across rows, add down columns, then check that both routes give the same grand total. Students complete the example with a partner, using coloured pencils or highlighters for rows and columns. Pause for mini-whiteboard checks: “What is missing?” “Which total can be found first?” “How do you know your answer is reasonable?”

4. 38–58 min · Probability from entries. Teacher models how to write probabilities as fractions, decimals and percentages where appropriate, using the completed chocolate table:

• (P(\text{soft})=30/60=1/2)
• (P(\text{fruit})=28/60=7/15)
• (P(\text{soft and fruit})=18/60=3/10) Students complete progressively challenging questions on the worksheet, including probabilities of a centre type, a filling type and an intersection. They must label the numerator in words before calculating.

5. 58–73 min · Interpret and justify. Teacher presents statements on the reasoning and discussion slides, such as “A randomly selected chocolate is more likely to have a caramel filling than a fruit filling” and “Soft chocolates are more likely to contain fruit than caramel.” Students decide whether each statement is supported, then justify their decision with table values or probabilities. Pairs compare explanations and improve one response using the structure: “The data show… because… Therefore…”

6. 73–87 min · Independent application and extension. Students complete the final worksheet questions independently, including a question requiring comparison of two probabilities. Advanced students design a new two-way table about two categorical features of a familiar product or activity, choose realistic frequencies, and write three questions involving intersections, row or column totals, then provide answers and justifications. Teacher conferences with students and checks accurate use of “and”, “or”, “total” and “out of”.

7. 87–95 min · Plenary and exit check. Teacher returns to the closing reflection slides and asks students to explain one way a two-way table supports a probability conclusion. Students complete the worksheet exit ticket: identify one joint frequency and one marginal frequency, calculate (P(\text{hard and caramel})), and write one evidence-based conclusion. Collect responses to plan the next lesson.

Resources

• the two-way table lesson slide deck
• the two-way table practice worksheet
• Mini-whiteboards, pens and erasers
• Coloured pencils or highlighters
• Enlarged blank two-way tables
• Calculators
• Projector or interactive display
• Board space for the worked example

Assessment

• Listen to partner explanations and use mini-whiteboard responses to check whether students distinguish cells, joint frequencies and marginal frequencies.
• Check completed tables for accurate totals and a consistent grand total; question students about how they know their answers are reasonable.
• Use the exit ticket to identify students needing further practice with table structure, intersections or probability notation.

Differentiation

• Support learners with colour-coded rows and columns, enlarged cells, a partially completed table, a worked example and a checklist: “across, down, check”.
• Provide table manipulatives or cut-out labels for students who benefit from physically matching categories and totals. Offer verbal descriptions of each table entry before requiring written notation.
• Use dyslexia-friendly copies: clear sans-serif font, enlarged spacing, minimal text, strong contrast, short instructions and one question at a time. Read instructions aloud and allow students to explain answers verbally before recording them.
• Pair students strategically and use guided questions rather than supplying answers. Extension students design a new table and create intersection and total questions, including written justifications.

Extension

• Create a realistic two-way table for another context, such as preferred transport and year level, ensuring all totals agree.
• Write and solve three questions: one joint probability, one marginal probability and one comparison requiring a justified conclusion.
• Explain one limitation of using the table to make a broader claim about all chocolates or all students.