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📖 Lesson 17 ⏱ ~30 min Year 10 · Unit 4 ⚡ +50 XP

Venn Diagrams and Two-Way Tables

Use Venn diagrams and two-way tables to organise data and calculate probabilities involving overlapping events.

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Core learning
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Worksheet

Worksheet

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Warm-up
Think First
+5 XP each

Q1 · In your class, some students play basketball, some play netball, some play both. How would you draw a diagram showing who plays what?

Q2 · If 20 students like chocolate, 15 like vanilla, and 10 like both, how many students like at least one of them? How do you avoid double-counting?

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From the lesson
Intentions

Learning Intentions

Know

  • Venn diagrams show relationships between sets. Two-way tables organise data by two categorical variables.

Understand

  • How Venn diagrams and two-way tables help visualise intersections, unions and complements of events.

Can Do

  • Construct and interpret Venn diagrams and two-way tables to find probabilities.
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From the lesson
Key Terms

Key Terms

Venn diagram — A diagram using overlapping circles to show relationships between sets.
Intersection — The overlap of two sets; elements belonging to both A and B.
Union — All elements in either A or B or both.
Two-way table — A table showing frequencies for two categorical variables.
Conditional probability — The probability of an event given that another event has occurred.
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From the lesson
Misconceptions

Misconceptions to Fix

✗

Wrong: Complementary events are the same as mutually exclusive events.

✓

Right: Complementary events are a special case of mutually exclusive events where one must occur. Mutually exclusive events cannot both occur, but neither might occur.

✗

Wrong: If P(A or B) = P(A) + P(B), then A and B are always independent.

✓

Right: P(A or B) = P(A) + P(B) means A and B are mutually exclusive, not independent. Independence means P(A and B) = P(A) × P(B).

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From the lesson
Content

Venn Diagrams and Two-Way Tables

Work through the content, activities and worked examples below. Test your understanding with the questions in the Questions phase.

Remember Complementary: P(A) + P(A′) = 1. Mutually exclusive: P(A and B) = 0. Addition rule: P(A or B) = P(A) + P(B) − P(A and B).
HSC Note For mutually exclusive events, the addition rule simplifies to P(A or B) = P(A) + P(B). Always check whether events are mutually exclusive before using this simplification.
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From the lesson
Activity
✏ Activity 1 — Identify Event Types

Classify each pair of events as complementary, mutually exclusive, or independent:

  1. Rolling a 2 and rolling a 5 on a die.
  2. Rolling an even number and rolling an odd number.
  3. Drawing a red card and drawing a heart.
  4. Rain today and rain tomorrow.
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From the lesson
Worked Example

Worked Example

Step-by-step
A die is rolled. Let A = rolling an even number, B = rolling a 3, C = rolling an odd number. Identify which pairs are complementary, mutually exclusive, or independent.
  1. 1
    A and B: Even numbers are 2, 4, 6. B is 3. They cannot both occur. Mutually exclusive.
  2. 2
    A and C: Even and odd numbers cover all outcomes with no overlap. Complementary (and mutually exclusive).
  3. 3
    B and C: 3 is odd. B and C can both occur. Neither mutually exclusive nor complementary.
  4. 4
    Check: P(A) = 3/6 = 0.5, P(C) = 3/6 = 0.5, P(A and C) = 0. P(A or C) = 1. Complementary ✓
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From the lesson
Revisit

Revisit Your Thinking

Look back at your Think First response. What new understanding do you have now?

Reflect
Revisit your thinking
reflect

Earlier you were asked: What was your first thought on this topic?

Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?

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From the lesson
Multiple Choice

Multiple Choice

Select the best answer for each question.

1 mark In a Venn diagram, the overlapping region represents:

1 mark P(A or B) for non-mutually exclusive events is:

1 mark A two-way table shows:

1 mark In a two-way table, row totals give:

1 mark If P(A and B) = 0.2, P(A) = 0.5, P(B) = 0.4, then P(A or B) =

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From the lesson
Short Answer

Short Answer

Show all working and justify your answers.

1. 4 marks A card is drawn from a standard deck. Let A = drawing a heart, B = drawing a king, C = drawing a black card.
(a) Are A and C mutually exclusive? Explain.
(b) Are A and B complementary? Explain.
(c) Find P(A or B).

2. 3 marks Explain the difference between mutually exclusive events and complementary events. Give an example of events that are mutually exclusive but not complementary.

3. 2 marks If P(A) = 0.4, P(B) = 0.5, and A and B are mutually exclusive, find P(A or B) and P(A and B).

Marking guidance: 1 mark each for MCQs. See mark allocations for each short answer question.

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