Sample spaces and probability

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

• Slides Download Copy address for Prepare Present from the browser, or export to PowerPoint or PDF.
• Worksheet Download Copy address for Prepare Print the student copy; the Teacher edition has the answers.

Students will be able to

• I can list the sample space for a single event.
• I can work out a probability as a fraction, decimal or percentage.
• I can tell when outcomes are not equally likely, and use sizes or counts instead.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 12 min · Practise and Make 13 min

Before class

Do the Run board first. Students need fractions, decimals and percentages (Number Foundations).

This is a hinge: everything in Chance builds on a complete sample space and correct probabilities. The common mistake is treating every listed outcome as equally likely (three colours, so 1/3 each); it is planted in the Predict and in Kai's reasoning.

The board's random spins are different every time; with 1000 or more spins the relative frequencies are reliably close to the probabilities.

Running the lesson

1. Predict: take a show of hands; ask students who chose 1/3 how they decided.
2. Run: ask what relative frequency you would expect if each colour really were 1/3.
3. Investigate: write the sample space for each device on the board before any probability.
4. Modify: for repeated labels, draw the spinner with A, A, B.
5. Make: ask for the smallest bag with P(red) = 1/3.

When students get stuck

Answers to the checks

1. a third each?: A spinner is half red, a quarter blue and a quarter green. What is the probability of blue?
   1/4. There are three colours, but they are not equally likely: blue is a quarter of the spinner, so P(blue) = 1/4.
2. spin it: Run at least 1000 spins. Is the relative frequency of blue nearer $\tfrac{1}{4}$ or $\tfrac{1}{3}$? Was your prediction right?
   
3. List, then count: Roll a fair dice. What is the probability of an even number? Give a fraction in simplest form.
   1/2
   Sample space {1, 2, 3, 4, 5, 6}; even: 2, 4, 6. P = 3/6 = 1/2.
4. Find the mistake: Kai works out probabilities for the half-red spinner. Click the line where his reasoning goes wrong.
   line 3
   Line 3: favourable ÷ total only works when the outcomes are equally likely. Red is half the spinner: P(red) = 1/2, P(blue) = P(green) = 1/4.
5. Counters in a bag: A bag has 3 red, 5 blue and 2 green counters. You take one without looking. What is $P(\text{blue})$? Simplest form.
   1/2
   10 counters, 5 blue: 5/10 = 1/2.
6. A different event: Roll a fair dice. What is the probability of a number greater than 4? Simplest form.
   1/3
   {5, 6}: 2 of 6 = 1/3.
7. Repeated labels: A spinner has three equal sectors labelled A, A and B. What is the sample space, and $P(A)$?
   {A, B}, with P(A) = 2/3
   Two possible outcomes, A and B, but A covers 2 of 3 equal sectors: P(A) = 2/3.
8. The missing probability: A spinner has $P(\text{red}) = 0.5$ and $P(\text{blue}) = 0.3$. The only other colour is green. What is $P(\text{green})$?
   0.2
   0.5 + 0.3 + P(green) = 1, so P(green) = 0.2.
9. As a percentage: Write $P(\text{heads})$ for a fair coin as a percentage.
   50
   1/2 = 0.5 = 50%.
10. Impossible and certain: Edge cases on a fair dice: $P(7)$ and $P(\text{less than } 7)$?
    0 (impossible) and 1 (certain)
    No 7 on the dice: probability 0. Every roll is less than 7: probability 1.
11. A bag: A bag has 4 yellow and 6 purple counters. $P(\text{yellow})$? Simplest form.
    2/5
    4 of 10 = 2/5.
12. Letters: Pick a letter at random from the word MATHS. What is the probability it is a vowel? (A fraction.)
    1/5
    5 letters, 1 vowel (A): 1/5.
13. Sample space: Which list is the full sample space for spinning this spinner: four equal sectors labelled 1, 2, 2, 3?
    {1, 2, 3}
    The possible outcomes are 1, 2 and 3; 2 has probability 1/2, the others 1/4 each.
14. Review: comparing: Review: Group A's median is 12 (range 4); Group B's is 12 (range 15). Which is right?
    Same centre; B is much more spread
    Equal medians, but B varies far more.
15. design a bag: Design a bag of 10 counters with $P(\text{red}) = \tfrac{3}{10}$ and $P(\text{blue}) = \tfrac{1}{2}$; the rest are green. How many green counters?
    2
    3 red, 5 blue (1/2 of 10), so 10 − 8 = 2 green. P(green) = 2/10 = 1/5.

Stretch and support

Stretch: Design a spinner with four colours where one colour is three times as likely as each of the others.

Support: List the sample space in a table with a probability column; check the column adds to 1.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand the ideas in ONE lesson only, from the lesson "Sample spaces and probability".

The ideas:
- Identifying and listing sample spaces: I can Students identify the sample space for a single-stage event and list all possible outcomes systematically.
- Assigning probabilities to outcomes: I can Students assign theoretical probabilities to outcomes of a single-stage event using P = favourable/total and express these as fractions, decimals or percentages.

What understanding looks like:
- Lists a complete sample space for a given single-stage event.
- Explains what sample space means.
- Verifies completeness (all outcomes accounted for).
- Correctly assigns the probability to each outcome in a sample space.
- Verifies that all probabilities sum to 1.
- Expresses probability in the most appropriate form for the context.

Mistakes students often make here:
- A student gives each colour 1/3 (Kai's mistake).
- A student counts colours instead of counters.
- A student gives the count instead of the probability (3 instead of 1/2).

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Sample spaces and probability".

Exit ticket

A bag has 2 red, 3 green and 5 yellow counters. List the sample space and give each probability.

{red, green, yellow}: P(red) = 2/10 = 1/5, P(green) = 3/10, P(yellow) = 5/10 = 1/2. They add to 1.

What each step is for

• a third each?: Committing to P(blue) on a half-red spinner shows whether they assume the listed outcomes are equally likely.
• spin it: Spinning a thousand times shows blue settling near a quarter, not a third: the outcomes in a sample space need not be equally likely.
• list, then count: List the sample space completely, check whether the outcomes are equally likely, then favourable ÷ total, or the sizes; the probabilities add to 1. Kai's mistake treats every listed outcome as equally likely.
• one change at a time: One change at a time: a different event, a spinner with repeated labels, a missing probability, a percentage, then the edge cases of impossible and certain.
• probabilities: Sample spaces and probabilities for coins, dice, bags and spinners, mixed; the equally-likely slip planted again; the review keeps comparing data fresh.
• design a bag: Designing a bag to match given probabilities reverses the lesson: from probabilities back to counts, using equivalent fractions.

Where it leads

Next: Predict and test (AC9M7P01 node 3, AC9M7P02 node 1): expected counts and experiments.

Where it leads: complementary events and two-step experiments (Year 8).