Practice Maths

L17 — Sample Spaces

Probability Basics

In Maths, we use a specific formula to calculate the probability of an event occurring:

Probability = Number of favourable outcomes Number of possible outcomes

Key Terms

probability
a number between 0 and 1 that measures how likely an event is to occur (0 = impossible; 1 = certain)
sample space
the complete set of all possible outcomes of an experiment (e.g. the sample space for a coin toss is {Heads, Tails})
outcome
a single result of one trial (e.g. rolling a 4)
event
a specific result or set of results we are interested in (e.g. rolling an even number)
complementary event
the event of something NOT happening; P(event) + P(not event) = 1
Sample Space A sample space is a list of all possible results of an experiment. For example, the sample space for a coin toss is {Heads, Tails}.

Representing Chance:

  • Probability is always between 0 (Impossible) and 1 (Certain).
  • It can be written as a fraction, decimal, or percentage.

Worked Example

Question: A bag contains 3 red, 2 blue, 1 green marble. List the sample space and find P(red).

Step 1 — List all outcomes: Sample space = {R, R, R, B, B, G} — 6 outcomes total.

Step 2 — Count favourable outcomes: Favourable outcomes for red = 3.

Step 3 — Calculate probability: P(red) = 36 = 12

What Is a Sample Space?

A sample space is the complete list of all possible outcomes of an experiment or activity. For example, if you flip a coin, the sample space is {Heads, Tails} — those are the only two things that can happen. If you roll a standard die, the sample space is {1, 2, 3, 4, 5, 6}.

The sample space must include every possible outcome — nothing extra and nothing left out. Listing it carefully is the foundation of all probability calculations.

Listing Sample Spaces for Two-Step Experiments

When an experiment has two steps (like flipping two coins, or rolling a die and spinning a spinner), you need to list all the combinations. A systematic approach helps you avoid missing any outcomes.

Example: Flip two coins. The sample space is:

  • HH (Heads, Heads)
  • HT (Heads, Tails)
  • TH (Tails, Heads)
  • TT (Tails, Tails)

Notice that HT and TH are different outcomes, even though both have one head and one tail. The first coin result and the second coin result are separate events.

Using Tree Diagrams

A tree diagram is a branching diagram that shows all the paths through a multi-step experiment. For flipping two coins:

  • First branch: H or T (2 branches)
  • Each branch splits again: H or T (2 more branches each)
  • Total outcomes: 2 × 2 = 4 branches at the end

Tree diagrams are especially useful when each step has a different number of outcomes. To count the total number of outcomes, multiply the number of choices at each step.

Counting Outcomes

For independent steps, total outcomes = (outcomes at step 1) × (outcomes at step 2) × …

  • Roll a die AND flip a coin: 6 × 2 = 12 outcomes
  • Spin a 4-section spinner twice: 4 × 4 = 16 outcomes
  • Choose from 3 flavours AND 2 cup sizes: 3 × 2 = 6 outcomes
Key tip: When listing a sample space, work systematically to avoid missing outcomes. Fix the first result and list all second results, then move to the next first result. For example, rolling a die then flipping a coin: (1,H), (1,T), (2,H), (2,T), (3,H), (3,T)… This “fix one, vary the other” method ensures you capture every outcome.

Practice Questions

  1. The Likelihood Scale Fluency

    1. Order these terms from least likely to most likely: Equally Likely, Impossible, Certain, Unlikely, Highly Likely, Likely.
    2. Describe the likelihood of rolling a 7 on a standard 6-sided die.
    3. Describe the likelihood of a fair coin landing on Heads.
    4. Describe the likelihood of drawing a red card from a standard deck (26 red cards out of 52).
    5. Describe the likelihood of rolling a number less than 7 on a standard 6-sided die.

  2. Sample Spaces Fluency

    List all possible outcomes (the sample space) for each experiment:

    1. Tossing a standard coin.
    2. Choosing a day of the week that starts with the letter ‘S’.
    3. Rolling a standard 6-sided die.
    4. Choosing a vowel from the letters A, B, C, D, E.
    5. Spinning a fair 4-colour spinner with sections Red, Blue, Green, Yellow.

  3. The Marble Bag Fluency

    A bag contains 4 red marbles, 4 green marbles, and 2 yellow marbles.

    1. How many marbles are there in total?
    2. What is P(red)? Write as a fraction in simplest form.
    3. What is P(green)? Write as a fraction in simplest form.
    4. What is P(yellow)? Write as a fraction in simplest form.
    5. What is P(blue)?

  4. Fractions, Decimals, Percentages Fluency

    A spinner has 10 equal sections: 3 are blue, 5 are red, and 2 are green.

    R R R R R B B B G G Red (5) Blue (3) Green (2)
    1. Find P(red). Give your answer as a fraction, decimal, and percentage.
    2. Find P(blue). Give your answer as a fraction, decimal, and percentage.
    3. Find P(green). Give your answer as a fraction, decimal, and percentage.
    4. Add P(red) + P(blue) + P(green). What do you notice?
    5. What is P(not blue)?

  5. Complementary Events Understanding

    A bag contains 7 white balls and 3 black balls.

    1. What is P(black)?
    2. What is P(not black)?
    3. What is P(black) + P(not black)? What does this tell us about complementary events?
    4. If the probability of winning a game is 0.35, what is the probability of not winning?

  6. Equally Likely Outcomes Understanding

    1. Does rolling a standard 6-sided die give equally likely outcomes? Explain.
    2. A bag contains 3 red balls and 8 blue balls. Is picking each colour equally likely? Explain.
    3. A spinner has 4 sections but two sections are twice as large as the others. Are all outcomes equally likely? Explain.
    4. Give one example of an experiment with equally likely outcomes and one with non-equally likely outcomes.

  7. Spinner Logic Understanding

    A spinner has 8 equal sections and 2 are shaded purple.

    P P Purple (2) Other (6)
    1. What is P(purple)? Write as a simplified fraction.
    2. Convert P(purple) to a decimal and a percentage.
    3. What is P(not purple)?
    4. If you spin the spinner 80 times, how many times would you theoretically expect to land on purple?

  8. Real-World Probability Understanding

    1. A weather forecast gives the chance of rain as 0.75. Write this as a simplified fraction and a percentage.
    2. A bus company says there is a 1-in-5 chance a bus is late. Write this probability as a fraction, decimal, and percentage.
    3. Order these three probabilities from least to most likely: 0.75, 13, 45%.
    4. A student calculates a probability of 1.3 for an event. Explain why this is impossible.

  9. Trials, Outcomes and Events Problem Solving

    A student rolls a 6-sided die 30 times to investigate the probability of rolling a 5.

    1. What is the sample space for rolling this die?
    2. What is the theoretical probability of rolling a 5?
    3. After 30 rolls, the student rolled a 5 exactly 8 times. Calculate the experimental probability of rolling a 5. (Write as a fraction.)
    4. The student says: “I rolled 5 more than expected — this die must be unfair.” Do you agree? Explain your reasoning using both theoretical and experimental probability.

  10. The Spinner Game Problem Solving

    A game uses a spinner with 20 equal sections: 8 red, 5 blue, 4 green, 3 yellow.

    1. Write the theoretical probability of landing on each colour as a fraction in simplest form.
    2. After 40 spins, a player lands on red 14 times. Calculate the experimental probability of red and compare it to the theoretical probability.
    3. If you play the game 20 times, how many times would you theoretically expect to land on each colour? Show working for all four colours.
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