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Maths · KS3 and GCSE · Interactive lesson

Probability: one draw and two draws

Find the chance of drawing a coloured counter. Start with one draw, then explore two draws with or without replacement.

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Explore an interactive example, then try the questions at your own pace.

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Start with the idea, then use the method when you are ready. Changing these settings changes the example. Your current practice question is kept.

Start with the idea explains the basics. Use the method builds on them with more detail.

KS3: count equally likely outcomes and recognise fractions. Method: single-event chances, then GCSE two-draw trees with fraction multiplication.

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KS3 / GCSE · start with the idea

By the end of this lesson: Connect the number of matching outcomes with the chance of an event.

The interactive diagram and controls are in Stage 2, “Explore one example”. Read the idea here, then try it next.

The idea in small steps
  • Each counter is equally likely to be chosen from the mixed bag.
  • Chance of red = number of red counters ÷ total counters.
  • A chance can be any value from 0 to 1, including 0 and 1. Zero means impossible; one means certain.
What do the words mean?
Probability
A measure of chance, from 0 (impossible) to 1 (certain).
Replacement
Putting the first counter back and mixing before the next draw.
Tree route
One sequence of outcomes, such as R then B.
Multiplying fractions
Multiply the top numbers together and the bottom numbers together: 3/5 × 2/4 = (3 × 2)/(5 × 4).

Change the example

Something to try

Keep the blue counters the same. Add a red counter. What happens to the chance of red? Does it always increase?

Notice what changes

3 red (R) + 2 blue (B) = 5 countersRRRBBChance of red3 out of 5Every counter is equally likely
3 out of 5 equally likely counters are red.
About this model

Imagine a well-mixed bag and a random draw. Counts determine chance, not a guaranteed outcome. The bag always has at least one counter; two draws without replacement need at least two. ≈ marks a rounded decimal.

The example you chose
3 red (R) + 2 blue (B) = 5 countersRRRBBChance of red3 out of 5Every counter is equally likely

3 out of 5 equally likely counters are red.

Still picture of the example from Stage 2. Use the back arrow to change it.

Work through it

Starting example: read the steps below

  1. Count the whole bag: 3 + 2 = 5.
  2. Count the counters we want: 3 red counters.
  3. Chance of red = 3/5.
  4. A chance is from 0 to 1, including both ends. Zero means impossible; one means certain.

Try on your own

Your turn. No rush.

Question 1

A bag has 3 red and 2 blue counters. What is the chance of red in one draw? Give a fraction or decimal.

Enter a number or fraction, such as −3, 0.75 or 3/4. You can include the matching unit shown above. If rounding is needed, the question tells you how many decimal places to use.

About these questions

Questions appear in a mixed order. Work at your own pace and stop whenever you like.

These questions use their own numbers, separate from the lesson example. You can stop whenever you like. Answers last for this visit to the page.

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Can you count all the outcomes? For two draws, can you explain whether the bag changes?

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Notes for parents and tutors

These lessons cover selected KS3 and GCSE ideas for England; they are not a complete course. The level notes explain where GCSE methods or optional extensions begin. Maurice can help you choose material for your course and exam board.

The content draws on the DfE Maths curriculum, DfE Science curriculum and AQA GCSE Physics specification. Level labels are guidance, not exam-board endorsements. For AQA Radioactivity, calculating net decline as a ratio is Higher-tier work. Net means the overall change; ask Maurice whether this extension is relevant to your course. Explanations and practice were reviewed during development on 21 October 2026; Maurice should check their fit before assigning work.

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