GCSE Maths Practice: tree-diagrams

Tree diagrams at Higher GCSE are often used when an experiment does not always follow the same sequence of events. In this question, the key idea is that the second event (rolling a die) only happens if a specific outcome occurs in the first event (the coin landing on Tails). This makes the problem more demanding than simple independent-event questions.

When constructing a tree diagram, the first stage should always represent the first random event. Here, that is the coin flip. The probabilities for Heads and Tails are both 1/2. At this point, it is essential to read the question carefully and decide what happens next on each branch. The wording tells us that if the coin lands on Heads, the experiment stops. This means the Heads branch must end and should not be extended further.

The Tails branch continues to a second stage, where the die is rolled. A standard six-sided die has three even numbers (2, 4 and 6) and three odd numbers (1, 3 and 5). Therefore, the probability of rolling an even number is 3/6, which simplifies to 1/2. These probabilities are written only on the Tails branch, because the die is not rolled after Heads.

Higher GCSE questions often test whether students incorrectly extend every branch of a tree diagram. A very common mistake is to draw the die outcomes after both Heads and Tails, which would suggest that the die is always rolled. This would not match the experiment described and would lead to an incorrect probability. At Higher tier, students are expected to recognise when a branch should stop.

Once the tree diagram has been drawn correctly, finding the required probability is usually straightforward. The question asks for the probability that the experiment results in an even number. The only way this can happen is by following the path Tails → Even. To find the probability of this path, the probabilities along the branch are multiplied together.

In some Higher questions, there may be several valid paths and their probabilities must be added. In this case, however, there is only one valid path, so no addition is needed. Understanding whether to multiply one path or add several paths is another important Higher-tier skill.

Tree diagrams like this help organise conditional probability problems clearly. By carefully deciding which branches continue and which stop, students can avoid many common errors and handle more complex probability questions with confidence.