GCSE Maths Practice: two-way-tables

This question brings together many of the key ideas tested in Higher-tier probability questions involving two-way tables and overlapping sets. The main challenge is deciding which students should be included in the final count before any calculation is carried out.

The phrase “studies either subject” means that a student is included if they study at least one of the subjects. This includes students who study only the first subject, only the second subject, and those who study both. Recognising this interpretation is essential at Higher tier.

A reliable way to approach this type of problem is to think in terms of sets. Each subject can be viewed as a set of students, and the group of students who study both subjects forms the intersection of these sets. The question asks for the size of the union of the two sets relative to the total number of students.

Adding the sizes of the two sets initially counts every student who studies either subject, but it also counts students in the intersection twice. Subtracting the overlap once corrects this and ensures that each student is counted exactly one time.

An alternative Higher-tier strategy is to think about complements. Instead of directly finding the probability that a student studies at least one subject, you could consider the students who study neither subject and subtract this from the total. While this approach is not always quicker, being aware of it demonstrates strong conceptual understanding.

Visual tools such as Venn diagrams can help organise thinking, especially in more complex problems. Even when a diagram is not drawn, mentally identifying the separate regions can prevent common mistakes.

Once the correct number of favourable outcomes has been found, the probability is calculated by dividing by the total number of possible outcomes. In survey questions, this is the total number of students surveyed. At Higher tier, probabilities are usually left as fractions unless another form is requested.

Common exam errors include interpreting “either” as meaning “exactly one”, forgetting to include students who study both subjects, or dividing by a subtotal instead of the full total. These mistakes are usually caused by misreading the question rather than poor calculation.

Questions like this are designed to test logical structure and interpretation as much as numerical skill. A strong exam habit is to write a short description such as “union = total of both − overlap” before substituting values.

Mastering these ideas prepares students for more advanced probability topics, including conditional probability and problems involving three or more overlapping sets.