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📖 Lesson 15 ⏱ ~30 min Year 10 · Unit 4 ⚡ +50 XP

Introduction to Probability

Understand probability as a measure of likelihood, calculate simple probabilities, and use sample spaces.

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 flip a fair coin 10 times and get 8 heads, is the coin broken? Or is that just how luck sometimes works?

Q2 · What does "50% chance of rain" actually mean to a weather forecaster? Does it mean it will rain for half the day?

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From the lesson
Intentions

Learning Intentions

Know

  • Probability = (number of favourable outcomes) / (number of possible outcomes). All probabilities lie between 0 and 1.

Understand

  • Why the sum of probabilities of all possible outcomes in a sample space equals 1.

Can Do

  • Calculate simple probabilities from sample spaces and express them as fractions, decimals and percentages.
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From the lesson
Key Terms

Key Terms

Probability — A measure of how likely an event is to occur, from 0 (impossible) to 1 (certain).
Sample space — The set of all possible outcomes of an experiment.
Event — A set of one or more outcomes from a sample space.
Favourable outcome — An outcome that satisfies the conditions of the event.
Complementary event — The event 'not A'; P(not A) = 1 − P(A).
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From the lesson
Misconceptions

Misconceptions to Fix

✗

Wrong: When flipping a coin twice, there are 3 outcomes: HH, HT, TT.

✓

Right: When flipping a coin twice, there are 4 outcomes: HH, HT, TH, TT. TH and HT are different outcomes.

✗

Wrong: Tree diagrams are only useful for two-stage experiments.

✓

Right: Tree diagrams can be used for any number of stages. They become large for multi-stage experiments but remain valid.

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From the lesson
Content

Introduction to Probability

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

Remember To find the probability of a combined outcome, multiply probabilities along the branches. The sum of probabilities from any branch point equals 1.
HSC Note When constructing a tree diagram, always label each branch with its probability. Check that the probabilities from each node sum to 1.
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From the lesson
Activity
✏ Activity 1 — Build Tree Diagrams

Construct a tree diagram and find the probabilities for each experiment:

  1. Flip a coin and roll a die.
  2. Draw two cards from a deck with replacement.
  3. Spin a spinner (R, B, G) twice.
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From the lesson
Worked Example

Worked Example

Step-by-step
A coin is flipped and a die is rolled. Draw a tree diagram and find P(head and even number).
  1. 1
    Step 1: First branch: Head (P = 1/2) and Tail (P = 1/2).
  2. 2
    Step 2: From each coin outcome, branch into die outcomes: 1, 2, 3, 4, 5, 6 (each P = 1/6).
  3. 3
    Step 3: P(head and even) = P(head) × P(even) = 1/2 × 3/6 = 1/2 × 1/2 = 1/4.
  4. 4
    Step 4: Check: There are 12 combined outcomes, each with P = 1/12. Three outcomes give head and even (H2, H4, H6). P = 3/12 = 1/4. ✓
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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 The probability of an impossible event is:

1 mark If P(A) = 0.3, then P(not A) =

1 mark A fair die is rolled. The probability of rolling a number greater than 4 is:

1 mark The sum of all probabilities in a sample space is:

1 mark A bag has 3 red and 7 blue marbles. P(red) =

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From the lesson
Short Answer

Short Answer

Show all working and justify your answers.

1. 4 marks A bag contains 3 red and 2 blue marbles. Two marbles are drawn without replacement.
(a) Draw a tree diagram showing all probabilities.
(b) Find the probability of drawing two red marbles.
(c) Find the probability of drawing one red and one blue marble (in any order).

2. 3 marks Explain why the probabilities on the second draw change when drawing without replacement, but stay the same when drawing with replacement.

3. 2 marks A coin is flipped three times. How many outcomes are in the sample space? List all outcomes.

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

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