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📖 Lesson 19 ⏱ ~30 min Year 10 · Unit 4 ⚡ +50 XP

Conditional Probability

Calculate and interpret conditional probabilities in real-world contexts using two-way tables and tree diagrams.

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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 know someone is in the school choir, does that change the chance they are also in the band? Why might it?

Q2 · If a medical test is 95% accurate and you test positive, are you 95% likely to have the disease? What else matters?

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From the lesson
Intentions

Learning Intentions

Know

  • Conditional probability P(A|B) is the probability of A given that B has occurred. P(A|B) = P(A and B) / P(B).

Understand

  • How conditioning on an event changes the sample space and therefore the probability.

Can Do

  • Calculate conditional probabilities from two-way tables, Venn diagrams and tree diagrams.
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From the lesson
Key Terms

Key Terms

Conditional probability — The probability of an event occurring given that another event has already occurred.
P(A|B) — Probability of A given B; read as 'A given B'.
Reduced sample space — The subset of outcomes where the given condition is satisfied.
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From the lesson
Misconceptions

Misconceptions to Fix

✗

Wrong: P(A|B) means the probability of A and B both occurring.

✓

Right: P(A|B) means the probability of A given that B has already occurred. The sample space is restricted to outcomes where B is true.

✗

Wrong: If A and B are independent, then P(A|B) = 0.

✓

Right: If A and B are independent, then P(A|B) = P(A). Knowing B does not change the probability of A.

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From the lesson
Content

Conditional Probability

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

Remember Conditional probability formula: P(A|B) = P(A and B) / P(B). The denominator P(B) restricts the sample space to only those outcomes where B occurs.
HSC Note Conditional probability questions often involve two-way tables or Venn diagrams. Always identify what is "given" before calculating.
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From the lesson
Activity
✏ Activity 1 — Conditional Probability

Calculate each conditional probability:

  1. P(heart | red card) from a standard deck.
  2. P(even | greater than 3) when rolling a die.
  3. P(boy | plays sport) from a class of 30 where 18 are boys and 12 play sport, with 8 boys playing sport.
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From the lesson
Worked Example

Worked Example

Step-by-step
In a class of 30 students, 18 play sport and 12 play music. 8 students play both sport and music. Find P(music | sport).
  1. 1
    Step 1: Given that the student plays sport, we restrict to the 18 students who play sport.
  2. 2
    Step 2: Of these 18, 8 also play music.
  3. 3
    Step 3: P(music | sport) = 8/18 = 4/9 ≈ 0.444.
  4. 4
    Check: Using the formula: P(music and sport) = 8/30, P(sport) = 18/30. P(music | sport) = (8/30) / (18/30) = 8/18 = 4/9. ✓
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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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Multiple Choice

Multiple Choice

Select the best answer for each question.

1 mark P(A|B) means:

1 mark In a class of 30, 18 play sport and 12 of those play basketball. P(basketball | sport) =

1 mark If P(A and B) = 0.15 and P(B) = 0.3, then P(A|B) =

1 mark Conditional probability is useful when:

1 mark From a two-way table, conditional probability is found by:

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From the lesson
Short Answer

Short Answer

Show all working and justify your answers.

1. 4 marks In a school, 60% of students play sport, 40% play music, and 25% do both.
(a) Find P(music | sport).
(b) Find P(sport | music).
(c) Are playing sport and playing music independent? Show working.

2. 3 marks A two-way table shows the results of a survey on eye colour and hair colour. Explain how you would use the table to find P(blue eyes | blonde hair).

3. 3 marks If P(A) = 0.6, P(B) = 0.4, and P(A and B) = 0.24, show that A and B are independent.

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

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