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📖 Lesson 16 ⏱ ~30 min Year 10 · Unit 4 ⚡ +50 XP

Compound Events and Tree Diagrams

Calculate probabilities of compound events using tree diagrams and the addition and multiplication rules.

Today's hook:
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Core learning
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Worksheet

Worksheet

Use the worksheet to complete this lesson in your book or digitally.

Warm-up
Think First
+5 XP each

Q1 · If you roll a die and flip a coin, does the die result affect the coin result? Why or why not?

Q2 · How would you count all possible outfits when choosing one of three shirts AND one of four pairs of pants?

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From the lesson
Intentions

Learning Intentions

Know

  • For independent events: P(A and B) = P(A) × P(B). For mutually exclusive events: P(A or B) = P(A) + P(B).

Understand

  • Why tree diagrams help visualise all possible outcomes and their probabilities in multi-step experiments.

Can Do

  • Draw tree diagrams and use them to calculate probabilities of compound events.
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From the lesson
Key Terms

Key Terms

Compound event — An event made up of two or more simple events.
Independent events — Events where the outcome of one does not affect the other.
Mutually exclusive — Events that cannot occur at the same time.
Tree diagram — A branching diagram showing all possible outcomes and their probabilities.
With replacement — Returning an item to the sample space before the next selection.
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From the lesson
Misconceptions

Misconceptions to Fix

✗

Wrong: In a Venn diagram, the two circles must always overlap.

✓

Right: The circles only overlap if there are elements common to both sets. If A and B are mutually exclusive, the circles do not overlap.

✗

Wrong: n(A or B) = n(A) + n(B).

✓

Right: n(A or B) = n(A) + n(B) − n(A and B). The intersection is subtracted because it is counted twice.

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From the lesson
Content

Compound Events and Tree Diagrams

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

Remember Two-way tables show frequencies for two categorical variables. Venn diagrams show relationships between sets. Both are useful for calculating conditional probability.
Exam Tip When filling in a two-way table, always check that the row totals and column totals add up correctly.
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From the lesson
Activity
✏ Activity 1 — Complete Tables and Diagrams

Complete the following:

  1. A two-way table for 50 students: 20 play sport, 30 play music, 10 do both.
  2. A Venn diagram for the same data.
  3. Calculate P(sport or music) from the table.
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From the lesson
Worked Example

Worked Example

Step-by-step
In a class of 30 students, 18 play basketball, 15 play soccer, and 8 play both. Represent this in a two-way table and a Venn diagram.
  1. 1
    Step 1: Basketball only = 18 − 8 = 10. Soccer only = 15 − 8 = 7. Both = 8. Neither = 30 − (10 + 7 + 8) = 5.
  2. 2
    Step 2: Two-way table: Rows = Basketball (Yes/No), Columns = Soccer (Yes/No). Fill in frequencies.
  3. 3
    Step 3: Venn diagram: Left circle = Basketball (10 in only, 8 in overlap). Right circle = Soccer (7 in only, 8 in overlap). Outside = 5.
  4. 4
    Step 4: Check: 10 + 8 + 7 + 5 = 30. ✓
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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 Two independent events A and B have P(A) = 0.5 and P(B) = 0.4. P(A and B) =

1 mark Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.4. P(A or B) =

1 mark A coin is tossed and a die is rolled. The number of outcomes in the sample space is:

1 mark Drawing two cards without replacement means the events are:

1 mark On a tree diagram, probabilities along a branch are:

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From the lesson
Short Answer

Short Answer

Show all working and justify your answers.

1. 4 marks In a survey of 80 people, 45 own a dog, 35 own a cat, and 20 own both.
(a) Complete a two-way table.
(b) Draw a Venn diagram.
(c) Find the probability that a randomly chosen person owns a dog or a cat.

2. 3 marks Explain the difference between the intersection and the union of two sets. Use a Venn diagram to illustrate your answer.

3. 3 marks A two-way table shows that 60% of boys and 40% of girls play sport. Can you conclude that boys are more likely to play sport? What additional information do you need?

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

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