GCSE Maths Practice: tree-diagrams

Tree Diagrams: “Both the Same Colour” Without Replacement

This GCSE Maths question is about selecting two items one after another without replacement. That phrase is crucial: it means the first item is not put back before the second selection, so the probabilities in the second step can change. These are called dependent events because the second event depends on what happened first.

Why the Second Fraction Changes

Probability is always:

Probability = favourable outcomes ÷ total outcomes

After the first selection, the total number of items decreases by 1. If the first selection was a blue ticket (or card, sweet, sticker, etc.), then the number of blue items also decreases by 1. That is why the second probability must be updated.

Tree Diagram Method

A tree diagram helps you structure the information:

• Step 1: Split into Blue and Not Blue for the first pick.
• Step 2: From the Blue branch, update the numbers (one item has been removed).
• Step 3: To find “Blue then Blue”, multiply the two probabilities on that path.

Worked Example 1 (Different Numbers)

A roll of raffle tickets has 5 blue tickets and 7 red tickets. Two tickets are torn off at random without replacement. Find the probability that both tickets are blue.

• P(blue first) = 5/12
• After a blue is removed: blue left = 4, total left = 11, so P(blue second | blue first) = 4/11
• Multiply: 5/12 × 4/11

Worked Example 2 (Same Skill, Different Context)

A box contains 6 pens: 2 blue and 4 black. Two pens are taken without replacement. Find the probability both pens are blue.

• P(blue first) = 2/6
• Then P(blue second | blue first) = 1/5
• Multiply: 2/6 × 1/5

Common Mistakes to Avoid

• Forgetting “without replacement”: the second denominator should be one less than the first.
• Not reducing the favourable number: if you took a blue first, there is one fewer blue available next.
• Adding instead of multiplying: “both” and “and then” usually mean multiply along a path.
• Arithmetic slips: after multiplying fractions, check simplification carefully (e.g., simplify 6/90 correctly).

Real-Life Link

These questions model real situations: selecting two winners from a set of tickets, choosing two items to test from a batch, or picking two cards from a deck. The idea is always the same: once something has been taken out, what remains has changed.

Mini FAQ

• Do I have to draw the full tree? Not always. For Foundation questions, it is often enough to expand only the branch you need for the required outcome.
• Should I simplify the final answer? Usually yes, but equivalent fractions are acceptable if they represent the same probability.
• When would probabilities stay the same? Only if items are replaced, or if events are independent.

Study tip: When you see “both are blue”, think “Blue then Blue”, then multiply the two fractions after updating the second one.