Conditional Probability Tables

Overview

Lesson 7 of 12 in Probability: From Chance to Models. Students use two-way tables to calculate and interpret conditional probabilities, focusing on how “given that” restricts the sample space and changes the denominator. They connect table calculations with probability notation, explanations, and a two-selection model with replacement.

Learning intentions

• WALT calculate conditional probabilities from two-way tables.
• WALT identify the restricted sample space in a conditional probability.
• WALT write conditional probabilities using correct notation and context.
• WALT explain how a condition changes the denominator.

Success criteria

• I can identify which row, column, or cell is the restricted sample space.
• I can write conditional probabilities correctly, including “given that”.
• I can calculate probabilities as simplified fractions or decimals.
• I can explain why the denominator changes when a condition is applied.

Curriculum links

• Interpret and apply mathematical and statistical information in context by making informed decisions from tables and other mathematical media.
• Demonstrate mathematical reasoning by using appropriate methods, notation, representations, and a logical sequence of steps.
• Use mathematical methods to explore problems related to life in Aotearoa New Zealand or the Pacific, communicating accurate mathematical information.
• Mathematics and Statistics refresh: students use mathematical and statistical thinking to reason, communicate, represent information, and make informed decisions.

Lesson structure (95 minutes)

1. 0–8 min · Retrieval hook. Open with the chocolate selection hook and display a two-way table about a box of 40 chocolates: 24 have fruit, 16 do not; 10 have hard centres, 30 have soft centres; 6 are fruit and hard-centred. Ask, “If I know the chocolate has a hard centre, should I still divide by 40?” Students complete a silent think, then explain their first response to a partner.

2. 8–20 min · Connect prior learning. Use the prior-learning review to revisit sample space, intersection, complementary events, and ordinary probability. Students label the table’s rows, columns, totals, and intersection, then answer: “What does the event fruit and hard mean?” and “What group remains possible if I am told the centre is hard?” Check responses using mini-whiteboards.

3. 20–38 min · Explicit teaching and modelling. Display the worked-example sequence and model (P(\text{fruit} \mid \text{hard})=\frac{\text{fruit and hard{\text{hard=\frac{6}{10}=\frac35). Highlight “given hard” in one colour and the restricted hard-centre column in another. Contrast this with (P(\text{hard} \mid \text{fruit})=\frac{6}{24}=\frac14). Students annotate the conditional probability notes and examples using fraction frames and complete the next example with the teacher. Emphasise that the condition determines the denominator.

4. 38–60 min · Guided table practice. Distribute the guided two-way table practice. Students work in pairs through questions such as (P(\text{soft}\mid\text{fruit})), (P(\text{not fruit}\mid\text{hard})), and a context-based interpretation. Require each answer to show: condition, restricted sample space, favourable cell or total, calculation, and a sentence in context. Teacher circulates, prompting students to point to the denominator before calculating. Pause halfway for a whole-class check and address common errors.

5. 60–78 min · Representations and reasoning. Return to the table-to-tree comparison. Students translate one conditional probability from the table into a two-stage tree representation and explain which branches would change if the first selection were not replaced. Invite selected pairs to explain why (P(A\mid B)) and (P(B\mid A)) usually differ. Use the sentence frame: “Because the condition is ___, the possible outcomes reduce from ___ to ___, so the denominator is ___.”

6. 78–90 min · Extension investigation. Students who are ready complete the two-selection challenge: select two chocolates with replacement and calculate probabilities involving fruit and hard centres. They explain assumptions, including that replacement restores the original proportions and makes the selections independent. Other students finish the core questions with teacher support. Ask advanced students to compare with selection without replacement and predict which probabilities increase or decrease.

7. 90–95 min · Exit check and reflection. Use the plenary and exit questions. Students answer independently: “A table has 12 fruit chocolates, 8 of which have soft centres. What is (P(\text{soft}\mid\text{fruit}))? What is the restricted sample space, and why is it the denominator?” Students add one sentence explaining how the condition changed their calculation, then submit the response.

Resources

• conditional probability teaching deck
• conditional probability notes and practice worksheet
• Whiteboards and pens
• Projector or interactive display
• Calculators
• Highlighters in two contrasting colours
• Optional headphones or text-to-speech/read-aloud support

Assessment

• Check mini-whiteboard responses for accurate identification of rows, columns, intersections, and restricted sample spaces.
• During pair work, require students to point to and justify the denominator before calculating; listen for correct use of “given that”.
• Use the independent exit response to identify whether students can calculate, represent, and explain a conditional probability.

Differentiation

• Support learners with “given that” highlighting, colour-coded tables, fraction frames, partially completed examples, and the prompt: “Given what?” before every calculation.
• Provide dyslexia-friendly notation: clear sans-serif font, generous spacing, one question per line, consistent colour coding, minimal visual clutter, and fractions written both horizontally and vertically where useful.
• Offer read-aloud instructions, partner reading, text-to-speech, and verbal response options. Allow students to trace from the condition to the correct row or column before recording.
• Use a table-to-tree cross-representation and step-by-step worked examples for students needing additional structure; extend confident students through replacement versus non-replacement comparisons and assumption checks.

Extension

• Solve the chocolate questions involving two selections with replacement, including compound events such as fruit then hard-centred and hard-centred on both selections.
• Explain why replacement supports the assumption of independence, and compare one result with the equivalent situation without replacement.