Conditional Probability Models

Overview

Lesson 11 of 12 in Probability: From Chance to Models. Students connect conditional probability with restricted sample spaces, two-way tables, tree diagrams and independence, using a standard deck of cards. They build from earlier work on probability models and prepare to justify decisions using mathematical representations and clear reasoning.

Learning intentions

• WALT identify and use a condition to restrict a sample space.
• WALT represent conditional probability using a table, tree diagram and formula.
• WALT connect conditional probability with independence.
• WALT communicate mathematical reasoning clearly in a real-world-style context.

Success criteria

• I can identify the condition in a probability question.
• I can restrict the sample space to the 12 face cards.
• I can calculate (P(\text{King} \mid \text{face card})=\frac{4}{12}=\frac13).
• I can explain, using evidence, whether two events are independent.

Curriculum links

• Interpret and apply mathematical and statistical information in context by making an informed judgement from tables, diagrams and written information.
• Demonstrate mathematical reasoning by using appropriate methods, representations, mathematical statements 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.
• New Zealand Curriculum Refresh: students use mathematical and statistical thinking to investigate, communicate, justify and evaluate ideas; learning is connected to meaningful contexts and supports confident, critical participation.

Lesson structure (95 minutes)

1. 0–8 min · Hook and retrieval. Display the question, “If I tell you a card is a face card, has the chance of it being a King changed?” using the opening question slide. Students independently write a prediction, then recall the meaning of sample space, event and independence before sharing with a partner.

2. 8–23 min · Teacher modelling. Use the worked standard-deck example and a teacher think-aloud: the condition “given that the card is a face card” removes the 40 non-face cards, leaving 12 equally likely outcomes—four Jacks, four Queens and four Kings. Model the table, a restricted sample-space list and the calculation (4/12=1/3), stressing that the denominator is the condition, not the whole deck. Students annotate the conditional probability practice sheet and complete the first worked example.

3. 23–40 min · Representations link. Demonstrate how the same information appears in a two-way table and tree diagram with the table-and-tree comparison. Students complete the matching table and tree on the worksheet, label the condition, and explain to a partner why the relevant branch or row has only 12 outcomes. Check responses through mini-whiteboards and address the common error (4/52).

4. 40–58 min · Guided practice. In pairs, students solve progressively less-scaffolded deck questions on the guided practice questions: (P(\text{Queen}\mid\text{face card})), (P(\text{red}\mid\text{face card})), and (P(\text{face card}\mid\text{red card})). Students must show the restricted sample space and one representation. Pause after each question for a brief class check, asking, “What changed, and what stayed the same?”

5. 58–73 min · Independence investigation. Model with the independence comparison how to test whether events (A) and (B) are independent: compare (P(A\mid B)) with (P(A)), or compare (P(A\cap B)) with (P(A)P(B)). Students use the worksheet to test examples such as “King” and “face card”, then “red” and “face card”, recording a conclusion supported by numerical evidence. Emphasise that events can be related even when the wording does not say “without replacement”.

6. 73–87 min · Independent application and extension. Students complete the final integrated problem on the independent application task, choosing and linking a table, tree diagram and formula. Advanced learners create two related conditional questions, solve each in all three representations, and compare why the answers differ. Confer with students, prompting them to name assumptions and check that probabilities are between 0 and 1.

7. 87–95 min · Plenary and exit check. Return to the summary and exit prompt. Students complete an exit response: “Given that a card is a face card, find the probability it is a King. Identify the restricted sample space and state whether ‘King’ and ‘face card’ are independent, with evidence.” Collect responses to plan the final lesson.

Resources

• the conditional probability teaching deck
• the conditional probability practice sheet
• Standard decks of playing cards, one per pair
• Mini-whiteboards and pens
• Calculators, if routinely used
• Dyslexia-friendly cards with suit and face labels clearly separated
• Projector or interactive board

Assessment

• Check predictions, annotated models and mini-whiteboard answers during retrieval and modelling.
• Listen for correct use of “given that”, restricted sample space, numerator and denominator during pair explanations.
• Use the exit response to assess calculation, representation, independence reasoning and the next lesson’s support needs.

Differentiation

• Provide deck diagrams, partially completed tables and tree diagrams, a comparison table, and sentence frames such as “The condition is ___, so I restrict the sample space to ___” and “The events are/are not independent because ___.”
• Use teacher think-alouds, worked examples and frequent verbal checks before removing scaffolds; pair students deliberately and allow oral rehearsal before written explanations.
• Offer dyslexia-friendly cards with suit and face labels separated clearly, uncluttered worksheets, a clear sans-serif font, increased spacing, short instructions and the option to read questions aloud or use text-to-speech.
• Extend advanced learners by requiring three equivalent representations, creating related conditional questions, comparing solutions in a table, tree diagram and formula, and evaluating assumptions such as equally likely cards and drawing with or without replacement.

Extension

• Create a conditional question involving colour, suit or card value, then solve it using a two-way table, tree diagram and formula.
• Compare two questions with reversed conditions and explain why (P(A\mid B)) and (P(B\mid A)) need not be equal.
• Investigate how the answer changes when two cards are drawn without replacement.