Math // Probability

Ordering Likelihoods

Compare the likelihood of different events and order them from least to most likely — including situations with unequal outcomes such as a bag with more of one colour than another, or a spinner with sections of different sizes — and explain reasoning using informal language

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

Quick check

If a bag has 8 red counters and 2 blue ones, does the learner understand that picking red is much more likely than picking blue — and can they put different events in order from least to most likely?

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.
  • Order four or more events from least likely to most likely and justify each placement
  • Compare likelihoods when outcomes are not equally likely — e.g. 'Drawing red from a bag with 7 red and 3 blue is more likely than drawing blue'
  • Explain why some events are closer to 'even chance' and others are closer to 'certain' or 'impossible'

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
  • Likelihood LanguageRequired

    Probability // Ages 9 to 10

    Comparing and ordering likelihoods requires first knowing the vocabulary of likelihood (certain, likely, unlikely, impossible)

You are here

Ordering Likelihoods

Ages 9 to 10 // Concept

Leads to

1 topic
  • Equally Likely OutcomesRequired

    Probability // Ages 9 to 10

    Understanding what 'equally likely' means is a specific case of comparing likelihoods that requires the general comparison skill 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 aged 9–10 rank several chance events from least to most likely, including unequal outcomes (a bag with more of one colour, a spinner with uneven sections). Same-total comparisons yield to simple more/less reasoning; comparing across different totals demands proportional reasoning, which research shows develops slowly. Teaching should secure confident additive comparison, then nudge attention from absolute counts toward the proportion of favourable outcomes. The signature task sets up a case where the bag with MORE red counters has a LOWER chance of red (3 red of 4 vs 6 red of 12), and uses repeated sampling to let a count-based guess be publicly corrected by a long-run tally.

Common misconceptions

  • Comparing by absolute count of favourable outcomes ('Bag B has 6 reds, Bag A only 3, so B is better') instead of proportion
  • The 'difference' strategy — comparing by the gap between the two colours (reds minus blues) rather than the fraction of the whole
  • Equiprobability bias — 'it's random, so every bag/colour is equally likely and you can't rank them'
  • Ranking by personal desire or by which outcome happened last, rather than by composition

Useful representations

  • Least-to-most-likely ordering line
  • Two (or more) bags of counters compared side by side, part-whole visible
  • Spinners with sections of different sizes
  • Fraction/area strips linking proportion of red to chance of red

Teaching practices

  • Build from same-total comparisons (which bag/section gives red the best chance when totals match) before introducing different totals
  • Make the proportional jump visible with a two-bag comparison where the bag with MORE red counters actually has a LOWER chance of red (e.g. 3 red of 4 vs 6 red of 12)
  • Have children physically order event cards along a least→most-likely line and defend each move
  • Press for reasons in terms of 'how many out of the whole' rather than 'how many reds', to shift attention from absolute number to proportion
  • Confront the 'it's random so you can't tell' response by sampling repeatedly and tallying, so order shows up in the long run

Activities to try

  1. 1

    Rank two bags, then test by sampling

    Why it works: A better chance of red depends on the proportion of red in the whole bag (three-quarters beats one-half), not on the raw number of red counters — so a bag with more reds can still be the worse bet.

    1. Look at Bag A (3 red, 1 blue) and Bag B (6 red, 6 blue), both fully visible, and rank which gives the better chance of pulling red before any draws.
    2. State a reason for the ranking out loud.
    3. Draw many times from each bag (e.g. 50 draws each), noting each drawn counter's colour and returning it to the bag before the next draw so the proportion stays fixed, and tally how often red comes up, comparing the two tallies.
    4. Revise the ranking if the tally contradicts the first guess, and say why — in terms of how big a share of each bag is red, not how many reds.
    5. Look at a shaded red-fraction strip beside each tally (three-quarters of A, one-half of B) and connect the share to the observed long-run frequency.
  2. 2

    Order four or more events on a least→most-likely line

    Why it works: Events can be placed in a consistent order of likelihood by comparing the proportion of favourable outcomes; same-total cases need only more/less, but different-total cases need the share of the whole.

    1. Physically place four or more event cards along a 'least likely → most likely' rail (e.g. different bags or spinners with varying red fractions).
    2. Defend each placement, and each time a card is moved, say why it should sit higher or lower.
    3. Respond to a challenge such as 'why is that one higher? what if I doubled the blues?' by giving a reason in terms of the share of the whole, not just the number of reds.
    4. Start with cards that share the same total (so simple more/less reasoning works) before mixing in cards with different totals.
  3. 3

    Spinner prediction: does section size matter?

    Why it works: When outcomes are not equally sized, 'it could land on either' does not make them equally likely — the size (proportion) of each section governs the long-run frequency, and a more-likely outcome can still fail on a single trial.

    1. Look at a spinner where red fills three-quarters and blue one-quarter, alongside a fair coin.
    2. Before spinning, predict: over 20 spins, will red and blue come up about the same number of times, or will one come up much more — and state a reason.
    3. Spin 20 times and tally red versus blue (expect red near 15, blue near 5).
    4. Compare the tally to the prediction and explain what the size of each section did to the outcome.
    5. Consider follow-ups: could the less-likely colour still come up on the very next spin? What does 'more likely' actually mean here?

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