Math // Probability

Equally Likely Outcomes

Understand that 'equally likely' means every outcome has exactly the same chance of occurring; identify whether a given situation has equally likely outcomes (a fair coin, a fair die, a spinner with equal sections) or unequally likely outcomes (a bag with more of one colour, a spinner with unequal sections)

Ages
9 to 10
Type
Concept
Builds on
1 topic
Leads to
2 topics

Quick check

Can the learner tell the difference between a fair situation — like rolling a normal die where every number has the same chance — and an unfair one, like a bag with many more of one colour than another?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Explain that a fair coin has equally likely outcomes because heads and tails each have the same chance
  • Identify whether a spinner with unequal sections has equally likely outcomes or not, and explain why
  • Give an example of a situation with equally likely outcomes and one without, explaining the difference

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

1 topic
  • Ordering LikelihoodsRequired

    Probability // Ages 9 to 10

    Understanding what 'equally likely' means is a specific case of comparing likelihoods that requires the general comparison skill first

You are here

Equally Likely Outcomes

Ages 9 to 10 // Concept

Leads to

2 topics
  • Probability as a FractionRequired

    Probability // Ages 9 to 10

    Using fractions to represent probability only makes sense for equally-likely outcomes, so the equally-likely concept must come first

  • Calculating Simple ProbabilityRequired

    Probability // Ages 10 to 11

    The probability formula only applies to situations with equally likely outcomes — this concept must be secure first

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Children learn that 'equally likely' means every outcome has exactly the same chance, decided by the SIZE/number of each outcome region rather than by how many different outcomes there are. They distinguish fair devices (balanced coin, balanced die, equal-section spinner) from unfair ones (bags with more of one colour, uneven spinners), confront the 'two outcomes means 50-50' trap, and separate 'equally likely' from 'random'.

Common misconceptions

  • Assuming that because there are two (or three) possible outcomes, they must be equally likely — counting the number of outcome types rather than their sizes
  • Equiprobability bias — believing any chance situation gives all outcomes an equal chance because the result is 'random/luck'
  • Judging a die or coin's fairness by how it is thrown or how well it is made, rather than by its identical faces (Watson & Moritz)
  • Confusing 'fair' as equal-chance with 'fair' as 'I have a good chance of winning'

Useful representations

  • Spinners: equal sections vs unequal sections (area = chance)
  • Fair coin and balanced die (identical, interchangeable faces)
  • Bags with equal vs unequal numbers of each colour
  • Area/region model in which equal chance corresponds to equal area

Teaching practices

  • Contrast a spinner with equal sections against one with the same colours but unequal sections, so children see that equal chance depends on section size, not on how many colours there are
  • Use identical-face devices (coin, balanced die) to define equal likelihood by symmetry — the faces are interchangeable
  • Directly challenge 'two outcomes means 50-50' with a 1-red / 5-blue cube: two outcomes, but chances of 1/6 and 5/6
  • Distinguish 'equally likely' from 'random': an uneven spinner is still random, but its outcomes are not equally likely — sample repeatedly to show the unequal long-run pattern
  • Have children both identify and CREATE fair and unfair devices, explaining what makes each fair or not

Activities to try

  1. 1

    Build a fair three-colour spinner and test it by sampling

    Why it works: Equal chance depends on each outcome region being the same size (or the same number of equal ways), not on how many different outcomes there are; sampling many times reveals equal-area regions producing roughly equal frequencies and unequal regions producing a lopsided tally.

    1. Try to divide a spinner into three colours (red, blue, green) so that each colour is equally likely to be landed on.
    2. Spin the spinner many times (e.g. 100 times) and keep a tally of how often each colour comes up.
    3. If one colour comes up clearly more often over many spins, examine the spinner: is one wedge bigger than the others? (Small differences in the tally can happen by chance even on a fair spinner.) Adjust the boundaries until the three wedges are the same size, checking the sizes directly, then spin and tally again.
    4. State in your own words what makes the three colours equally likely — the equal AREA of each section, not the fact that there are three colours.
    5. Then build the opposite: a cube with 1 red face and 5 blue faces (six equal faces), or a spinner split into six equal sections with 1 red and 5 blue; predict and then tally the long-run result.
  2. 2

    Sort and decide: fair or not fair?

    Why it works: Deciding fair vs not-fair is a categorisation grounded in comparing the size or number of each outcome; the sharpest traps are the three-colour uneven spinner ('three colours so equal') and the two-outcome cube ('two outcomes so 50-50').

    1. Look at a set of chance devices — spinners with equal and unequal sections, a balanced coin, a balanced die, bags with equal and unequal numbers of each colour.
    2. Sort each device into 'equally likely outcomes' or 'not equally likely'.
    3. For a spinner that has three colours but with red filling half and blue and green each a quarter, decide whether the three colours are equally likely and explain why or why not.
    4. For each device, give the reason that decided your answer — equal section size or equal counts, not the number of different colours.
    5. Answer the trap question: 'A cube has one red face and five blue faces — there are two colours, so does that make red and blue equally likely?' Explain why not.
  3. 3

    Argue down the carnival host (equally likely vs random)

    Why it works: Children can separate two ideas: an outcome being random (unpredictable on a single spin) versus outcomes being equally likely (same chance each). An uneven spinner is random but not equally likely.

    1. Listen to a carnival host who insists a spinner with a big red section is 'totally fair — three colours, three chances'.
    2. Explain to the host why having three colours does not make the colours equally likely.
    3. Point out that red's section is bigger than the others, so red is more likely.
    4. Address the host's fallback claim that 'it's random, so you can't say one colour is more likely' — explain that the spin is random but the colours are still NOT equally likely because the sections differ in size.
    5. Say what the host would have to change to make the spinner genuinely fair (three equal-sized sections).

Math // Probability

More in Probability

All 15 topics