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WillTeachMaths

Thinking-first maths tools

GCSE probability

Tree diagrams show how multi-step probability is built.

This prototype helps students explore two-stage probability trees, replacement, combined events and why we multiply along branches before adding final outcomes.

It is designed for active revision: predict what will change, adjust the bag, and explain the probability calculation.

GCSE probability lab

Follow the branches, multiply along them, then add outcomes.

This tree diagram models two draws from a bag. Choose an event and watch the relevant branches and final outcomes light up.

Two draws from a bag with 4 red and 3 blue countersThe first counter is not replaced, so the second draw depends on the first.Total counters at start: 74/73/71/21/22/31/3StartRBRBRBRR: Red, RedP = 2/7RB: Red, BlueP = 2/7BR: Blue, RedP = 2/7BB: Blue, BlueP = 1/7

Event

Same colour

Outcomes

RR, BB

Fraction

3/7

Decimal

0.429

Calculation

P(Same colour) = 3/7

RR: 2/7
BB: 1/7
Add them: 2/7 + 1/7 = 3/7

Guided tasks

Predict whether replacing the first counter will make same colour more or less likely. Then test it.
Choose at least one red. Which final outcome is excluded?
Change the bag so there are many more red counters than blue. Which branches become more likely?
Explain why the probabilities after a red first draw and a blue first draw are different when there is no replacement.

Joy in the process

The point is not just to finish. It is to notice, test and return.

These tools are invitations to explore. A good mistake, a surprising pattern or a question you cannot yet answer is part of the work, not a failure of it.

The challenge is deliberate: the site should support thinking, not remove the need for it.

Before changing a setting, pause and predict what you think will happen.
Change one thing at a time. What stayed the same, and what changed?
Try to create a surprising case, a broken case, or a beautiful pattern.
Ask what this connects to outside the page: maps, movement, nature, systems or decisions.
Reset, then try again with a new question in mind.

Future extensions

This can become a broader probability revision toolkit.

Add three-stage tree diagrams for more challenging GCSE and A Level questions.
Add exam-style questions where students choose which branches to include.
Add independent/dependent event comparisons with diagnostic feedback.
Add contexts beyond counters, such as spinners, medical tests and conditional probability.