Practice Maths

Probability Language and Single Events

Key Ideas

Key Terms

Probability
A measure of how likely an event is to occur, always between 0 (impossible) and 1 (certain).
Event
A specific outcome or set of outcomes from an experiment (e.g. rolling an even number).
Outcome
A single possible result of an experiment (e.g. rolling a 4).
Sample space
The complete set of all possible outcomes, written in braces: e.g. {H, T} for a coin flip.
Theoretical probability
P(event) = number of favourable outcomes ÷ total outcomes, assuming all outcomes are equally likely.
Equally likely outcomes
Outcomes that each have the same probability of occurring (e.g. each face of a fair die).
Certain
An event with probability 1 — it will definitely happen.
Impossible
An event with probability 0 — it can never happen.
Unlikely
An event with probability closer to 0 than to 1 (less than 0.5).
Likely
An event with probability closer to 1 than to 0 (greater than 0.5).
Even chance
An event with probability exactly 0.5 — equally likely to happen or not happen.

The Probability Scale

  Impossible   Unlikely   Even chance   Likely    Certain
      |            |           |           |          |
      0           0.25        0.5         0.75        1
Hot Tip Always write out the sample space before calculating probability. Count carefully — every outcome must be equally likely. If outcomes are not equally likely (e.g. a loaded die), theoretical probability based on equal outcomes does not apply.

Worked Example

Question: A bag contains 3 red, 5 blue, and 2 green marbles. A marble is drawn at random. Find: (a) P(red)   (b) P(blue)   (c) P(green)   (d) P(yellow)

Total outcomes = 3 + 5 + 2 = 10

(a) P(red) = 3/10

(b) P(blue) = 5/10 = 1/2

(c) P(green) = 2/10 = 1/5

(d) P(yellow) = 0/10 = 0 (impossible — no yellow marbles)

Check: 3/10 + 5/10 + 2/10 = 10/10 = 1 ✓

The Language of Probability

Probability is the measure of how likely an event is to happen. We express it as a number between 0 and 1 (or equivalently, as a percentage between 0% and 100%). An event with probability 0 is impossible — it can never happen (like rolling a 7 on a standard die). An event with probability 1 is certain — it will definitely happen (like rolling a number less than 7 on a standard die).

In between, we use language like: unlikely (close to 0), even chance (exactly 0.5 — equally likely to happen or not happen), and likely (close to 1). When you flip a fair coin, the chance of getting heads is 0.5 — an even chance.

Theoretical Probability

Theoretical probability is calculated using logic and symmetry, assuming all outcomes are equally likely. The formula is:

P(event) = number of favourable outcomes ÷ total number of possible outcomes

Example: Rolling a standard six-sided die. The sample space is {1, 2, 3, 4, 5, 6} — 6 equally likely outcomes. The probability of rolling a 4 is 1/6. The probability of rolling an even number is 3/6 = 1/2 (because 2, 4, and 6 are even).

Example: Drawing a card from a standard deck of 52 cards. P(Ace) = 4/52 = 1/13 (there are 4 aces in the deck). P(red card) = 26/52 = 1/2.

Sample Spaces

The sample space is the complete list of all possible outcomes of an experiment. Listing the sample space carefully is an essential first step — if you miss an outcome, your probability will be wrong.

For a coin flip: sample space = {Head, Tail}. For a die: {1, 2, 3, 4, 5, 6}. For a spinner divided into 4 equal sections coloured Red, Blue, Green, Yellow: {Red, Blue, Green, Yellow}.

Make sure you only include outcomes that are actually possible, and count each outcome once. If a spinner has two red sections and one blue section (out of 3 equal sections), the sample space listed by section is {Red, Red, Blue}, giving P(Red) = 2/3, not 1/2.

Complementary Events

The complement of an event A (written A' or "not A") is everything that is NOT in event A. Since either A happens or it doesn't, the probability of A and the probability of A' must add up to exactly 1:

P(A) + P(A') = 1, so P(A') = 1 − P(A)

Example: The probability of rolling a 6 on a die is 1/6. So the probability of NOT rolling a 6 is 1 − 1/6 = 5/6. This matches the fact that {1, 2, 3, 4, 5} are the 5 outcomes that are not a 6.

Expressing Probability as a Fraction, Decimal, or Percentage

Probability can be written in any of these equivalent forms: 1/4 = 0.25 = 25%. They all mean the same thing. In school maths, fractions are usually preferred because they are exact, but decimals and percentages are also acceptable. Always simplify fractions where possible (e.g. 3/6 = 1/2).

When asked "how likely" something is, you can use the probability value on a 0-to-1 number line: 0 means impossible, 0.5 means an even chance, and 1 means certain.

Key tip: Always check that your probabilities make sense. P(event) must be between 0 and 1 — if you get an answer like 7/5 or −0.3, something has gone wrong. Also check that all the probabilities in your sample space add up to 1. If they don't, you have missed an outcome or counted one twice.

Mastery Practice

  1. Place each event on the probability scale by writing impossible, unlikely, even chance, likely, or certain. Fluency

    1. Rolling a 7 on a standard six-sided die.
    2. Flipping a fair coin and getting heads.
    3. Drawing any card from a standard deck of 52 cards.
    4. It will rain somewhere in Queensland this year.
    5. Rolling an even number on a standard die.
    6. Drawing a heart from a standard deck of cards.
    7. Rolling a number less than 3 on a standard die.
    8. A newborn baby will be male or female.

  2. List the sample space for each experiment. Fluency

    1. Flipping a fair coin once.
    2. Rolling a standard six-sided die once.
    3. Selecting a day of the week at random.
    4. Randomly picking a letter from the word “SCHOOL”.
    5. Randomly picking a number from 1 to 5.
    6. Selecting a month of the year at random.
    7. Randomly picking a card suit from a standard deck.
    8. Rolling a four-sided die (faces 1, 2, 3, 4).

  3. Calculate the probability of each event. Express your answer as a fraction in simplest form. Fluency

    A standard six-sided die is rolled once.

    1. P(rolling a 4)
    2. P(rolling an even number)
    3. P(rolling a number greater than 4)
    4. P(rolling a factor of 6)
    5. P(rolling a number less than 1)
    6. P(rolling a prime number)
    7. P(rolling a number that is a multiple of 3)
    8. P(rolling a 1 or a 6)

  4. Calculate each probability. Give answers as fractions in simplest form. Fluency

    A bag contains 4 red, 6 blue, 2 yellow, and 8 green marbles.

    1. P(red)
    2. P(blue)
    3. P(yellow)
    4. P(green)
    5. P(not green)
    6. P(red or yellow)
    7. P(purple)
    8. P(not red)

  5. Express each probability as a fraction, a decimal, and a percentage. Understanding

    1. A spinner has 8 equal sections numbered 1 to 8. P(spinning a number greater than 5).
    2. A jar contains 15 red jellybeans and 5 blue jellybeans. P(drawing a blue jelly bean).
    3. A bag of 24 mixed lollies has 6 chocolate, 8 caramel, 6 strawberry, and 4 lemon. P(caramel or lemon).
    4. The letters of the word “PROBABILITY” are written on cards and one card is drawn. P(drawing a letter B).
    5. A box contains 3 faulty and 27 working light bulbs. P(drawing a faulty bulb).
    6. A standard die is rolled. P(rolling a number less than 5).

  6. Single event probability problem solving. Problem Solving

    1. A bag contains red and blue marbles only. P(red) = 3/8. How many blue marbles are there if there are 24 marbles in total?
    2. A spinner has sections coloured red, blue, green, and yellow. P(red) = 0.3, P(blue) = 0.25, P(green) = 0.15. What is P(yellow)? Explain your reasoning.
    3. An exam has 40 multiple-choice questions. Each question has 4 options (A, B, C, D). If a student guesses randomly on every question:
      1. What is the probability of guessing any single question correctly?
      2. How many questions would you expect the student to get right by guessing?
    4. A class of 30 students has 12 who play sport, 8 who play a musical instrument, and 10 who do neither. If a student is chosen at random:
      1. What is P(plays sport)?
      2. What is P(plays an instrument)?
      3. What is P(does neither)?
      4. Verify that your probabilities are consistent (explain what “consistent” means in this context).
  7. Design a spinner. You are designing a spinner with sections coloured Red, Blue, Green, and Yellow. The spinner must satisfy all of the following conditions:
    • P(Red) = 1/4
    • P(Blue) = 2 × P(Green)
    • P(Yellow) = 3/10
    1. Find P(Green) and P(Blue).
    2. Verify that all four probabilities sum to 1.
    3. If the spinner has 20 equal sections, how many sections should be each colour?
    Problem Solving
  8. Letter tiles. The letters of the word QUEENSLAND are written on individual tiles and placed in a bag. A tile is chosen at random.
    1. List the sample space (count each letter as it appears).
    2. Find P(vowel).
    3. Find P(letter N).
    4. Find P(a letter that appears more than once in the word).
    Problem Solving
  9. Working backwards. A bag contains only red, blue, and green marbles. P(red) = 2/7 and P(blue) = 3/7.
    1. Find P(green).
    2. If there are 14 marbles in total, how many of each colour are there?
    3. Five more green marbles are added to the bag. What is P(green) now? Express as a fraction in simplest form.
    Problem Solving
  10. School survey. In a survey of 40 Year 8 students, 18 preferred soccer, 14 preferred basketball, and 8 preferred neither sport. A student is selected at random.
    1. Find P(prefers soccer).
    2. Find P(prefers basketball).
    3. Find P(prefers neither).
    4. The probabilities from (i), (ii), and (iii) sum to 1. What assumption about the students does this require? Is that assumption reasonable in this situation?
    Problem Solving
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