GCSE Maths Practice: tree-diagrams

Two-Step Probability Using Tree Diagrams

This GCSE Foundation question involves choosing two items one after another from the same group, where the first item is not replaced before the second is chosen. Because the first item is removed, the probabilities in the second step depend on what happened in the first step. These are known as dependent events.

Understanding “Then” and “Without Replacement”

The word “then” tells you that the events happen in order. The phrase “without replacement” tells you that the total number of items changes after the first selection. Both of these clues mean that you should multiply two fractions and adjust the second one.

Step-by-Step Method

• Step 1: Find the probability of the first colour.
• Step 2: Reduce the total number by 1 for the second selection.
• Step 3: Keep the number of favourable outcomes the same unless the first draw removed one of them.
• Step 4: Multiply the two probabilities.

Worked Example 1 (Different Context)

A pack contains 5 football cards and 3 basketball cards. Two cards are taken one after another without replacement. Find the probability of picking a football card then a basketball card.

• P(football first) = 5/8
• After one football card is taken, total left = 7 and basketball cards still = 3
• P(basketball second | football first) = 3/7
• Multiply: 5/8 × 3/7

Worked Example 2 (Common Error)

A common mistake is to keep the denominator the same for the second probability. For example, writing (3/9) × (4/9) would be incorrect here because the total number of items becomes 8 after the first draw, not 9.

Tree Diagrams

Tree diagrams are particularly useful for these questions. Each branch shows an outcome and its probability. To find the probability of a specific sequence, you multiply the probabilities along that path. At Foundation level, it is often best to expand only the branch that leads to the required outcome.

Common Mistakes to Avoid

• Using the original total for the second draw
• Adding probabilities instead of multiplying them
• Including equivalent fractions as different answer choices
• Overcomplicating the tree diagram

Real-Life Use

This type of probability appears when sampling items for checks, selecting winners in competitions, or choosing items from a collection without putting them back. The key idea is always that the situation changes after the first choice.

Study tip: When you see “then” and “without replacement”, think: multiply, and reduce the total by one.