Math // Probability

Calculating Simple Probability

Calculate the probability of a simple event with equally likely outcomes using the formula: probability = number of favourable outcomes ÷ total number of possible outcomes; express the result as a fraction in its simplest form; apply to rolling dice, drawing from bags, and other simple chance situations

Ages
10 to 11
Type
Method
Builds on
2 topics
Leads to
3 topics

Quick check

If there are 3 red and 7 blue balls in a bag, can the learner work out the probability of picking a red one and express it as a fraction in its simplest form?

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.
  • Calculate the probability of drawing a blue marble from a bag of 3 blue and 7 red as 3/10
  • Use the formula P(event) = favourable outcomes ÷ total outcomes to solve at least three different problems
  • Explain why increasing the number of favourable outcomes increases the probability

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

2 topics
  • Equally Likely OutcomesRequired

    Probability // Ages 9 to 10

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

  • The 0-to-1 Probability ScaleRequired

    Probability // Ages 10 to 11

    Calculating probability using favourable/total outcomes requires understanding probability as a number on a 0-1 scale

You are here

Calculating Simple Probability

Ages 10 to 11 // Method

Leads to

3 topics
  • Experimental vs TheoreticalRequired

    Probability // Ages 10 to 11

    Comparing experimental results with theoretical predictions requires being able to calculate the theoretical probability first

  • The Probability ScaleRequired

    Probability // Ages 11 to 13

    The formal KS3 treatment of probability as a 0-1 scale with fairness and equally-likely outcomes formalises what was practised in the age-10-11 calculation node

  • Probabilities Sum to OneHelpful

    Probability // Ages 10 to 11

    Using the complement rule is easier once students can calculate basic probabilities and see that favourable + unfavourable outcomes cover all possibilities

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

Learners build fluency with P(event) = favourable outcomes / total outcomes for equally-likely situations, giving answers as fractions in simplest form. The core routine is: check the outcomes are equally likely, count favourable outcomes, count the TOTAL (which includes the favourable ones), form the fraction, and simplify. Practice should span varied contexts (dice, bags, spinners) and target the part-part error (favourable-to-unfavourable) and the misapplication of the formula to unequal outcomes.

Common misconceptions

  • Writing favourable-to-unfavourable (part-part) instead of favourable-over-total — e.g. 3/7 rather than 3/10 for 3 blue among 10
  • Applying the formula to situations whose outcomes are not equally likely (an uneven spinner, a weighted die)
  • Failing to simplify, or simplifying incorrectly
  • Miscounting the total — forgetting to include the favourable outcomes in the denominator

Useful representations

  • P(event) = favourable/total written under the situation
  • Bags, dice, spinners with countable equally-likely outcomes
  • Simplified-fraction equivalence (2/8 = 1/4) on the 0-to-1 scale
  • Part-whole diagram of the outcome set

Teaching practices

  • Rehearse the same three-step routine across varied contexts — count favourable, count total, write and simplify — so it transfers beyond dice
  • Make explicit that the total is the WHOLE set of outcomes (favourable included), heading off the part-part error
  • Require a check that outcomes are equally likely BEFORE applying the formula
  • Connect 'more favourable outcomes → larger probability' by varying a bag's composition and watching the fraction grow
  • Always express in simplest form and confirm the simplified fraction names the same point on the 0-to-1 scale

Activities to try

  1. 1

    Varied probability ladder with simplest form

    Why it works: The denominator is the whole set of equally-likely outcomes, favourable included \u2014 so P(blue) among 3 blue and 7 red is 3/10, not 3/7. The simple favourable/total formula only applies when outcomes are equally likely, and simplifying (2/8 = 1/4) names the same point on the 0-to-1 scale.

    1. Work a short series of problems, each a probability to be formed and then simplified: (a) a bag of 3 blue and 7 red marbles \u2014 find P(blue); (b) a fair six-sided die \u2014 find P(rolling a 4); (c) a spinner with 8 equal sections, 2 of them gold \u2014 find P(gold).
    2. For each item, first confirm the outcomes are equally likely, then count the favourable outcomes, then count the WHOLE set of possible outcomes (including the favourable ones) for the denominator.
    3. Write each probability as a fraction and simplify where possible (e.g. 2/8 \u2192 1/4).
    4. On the bag item, if you write 3/7, stop and recount: the denominator is all 10 marbles, including the 3 blue ones, so the answer is 3/10.
    5. Encounter at least one unequal-outcome situation (e.g. an uneven spinner with different-sized sections) and decide whether the simple formula applies before using it.
  2. 2

    Editable bag: watch the fraction form and simplify

    Why it works: More favourable outcomes relative to the whole set raise the probability; the denominator is always the whole outcome set. Seeing the fraction and its position on the scale update as the bag changes connects composition to the value of the probability.

    1. Edit the composition of a bag (e.g. add or remove blue and red counters) and read the probability fraction as it forms.
    2. Identify the WHOLE set as the denominator each time the bag changes, confirming favourable counters are part of that total.
    3. Watch the simplified form of the fraction and a marker on a 0-to-1 line move as the composition changes.
    4. Predict: 'If I add 5 more red marbles, which number in the fraction changes, and does P(blue) go up or down?' then check against the live display.
  3. 3

    Worked example and Socratic 'out of how many?' check

    Why it works: Articulating the total as the whole set (favourable included) catches the part-part slip before it becomes a habit and makes the structure of the formula explicit.

    1. Study a worked die or bag example: identify the favourable outcomes, then the whole set, then reveal the simplified fraction.
    2. When asked 'and out of how many altogether?', state the full count of possible outcomes, not just the non-favourable ones.
    3. Explain in your own words why the total includes the favourable outcomes.

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