GCSE Maths Practice: tree-diagrams

Tree Diagrams: Probability of Two Events Without Replacement

In GCSE Maths, tree diagrams are a clear way to organise two-step probability problems. They are especially useful when events are dependent, meaning the result of the first step affects the probabilities in the second step. This happens whenever you draw an item without replacement (you do not put it back).

Key Idea: The Numbers Change After the First Draw

At the start, you calculate the first probability using:

favourable outcomes ÷ total outcomes

After the first draw, the total number of items usually decreases by 1. If the first draw removed an item of the colour you care about, the number of favourable outcomes also decreases by 1. That is why the second probability is written as a conditional probability (for example, “second red given that the first was red”).

How to Set Up a Tree Diagram

• Step 1: Draw two branches for the first draw (e.g., Red / Not Red).
• Step 2: From the “Red” branch, draw two more branches for the second draw (Red / Not Red) using updated totals.
• Step 3: Multiply along the path you want (e.g., Red then Red).

Worked Example 1 (Different Numbers)

A bag has 4 red balls and 1 green ball. Two balls are drawn without replacement. Find the probability that both are red.

• P(first red) = 4/5
• After a red is taken, reds left = 3 and total left = 4, so P(second red | first red) = 3/4
• Multiply: 4/5 × 3/4

Worked Example 2 (A Common Wrong Method)

A frequent mistake is to assume the second probability stays the same as the first. For example, with 6 blue and 4 yellow (10 total), some students write P(two blues) as (6/10) × (6/10). This is incorrect because after taking a blue, there are only 5 blues left out of 9. Always reduce the total by 1 for the second draw when there is no replacement.

Common Mistakes to Avoid

• Forgetting the total changes: second denominator should be one less.
• Forgetting the favourable changes: if you removed a red, you now have one fewer red.
• Adding instead of multiplying: “and then” usually means multiply along a path.
• Over-simplifying answer options: remember equivalent fractions represent the same probability.

Real-Life Context

This idea appears in card games (drawing two hearts), quality checks (testing two items from a batch), and simple experiments (selecting two people from a group). Whenever you choose items one after another and don’t return them, you are using dependent probability.

Mini FAQ

• Do I always use a tree diagram? Not always, but it is the clearest method for two-stage GCSE questions.
• When do probabilities stay the same? Only with replacement, or if events are truly independent.
• Do I simplify the final fraction? Usually yes, unless the question requests an exact form shown in a specific way.

Study tip: When you see “without replacement”, immediately think: “second fraction uses a denominator one smaller.”