GCSE Maths Practice: two-way-tables

At Higher tier, probability questions often rely on precise interpretation of everyday language. Words such as “or”, “either”, and “at least one” have specific mathematical meanings that must be understood correctly in order to choose the right method.

When two preferences are described, the students involved can usually be divided into three distinct groups: those who prefer only the first option, those who prefer only the second option, and those who prefer both. The question asks for the probability that a student belongs to any of these groups.

The most common error is to assume that “or” means “one but not the other”. In mathematics, this is rarely the case unless the question explicitly says “exactly one”. At Higher tier, recognising this distinction is essential.

The inclusion–exclusion principle provides a reliable structure for dealing with overlapping groups. By adding the totals for each group and then subtracting the overlap once, each student is counted exactly one time. This approach works regardless of the context or the size of the numbers involved.

Thinking in terms of sets can help formalise this reasoning. Each preference can be thought of as a set of students, and the overlap represents the intersection of these sets. The question is asking for the size of the union of the two sets relative to the total group.

Consider a similar scenario involving membership of school clubs. Some students may belong to one club, some to another, and some to both. If the school wants to know the probability that a randomly chosen student belongs to at least one club, the same structure applies.

After finding the correct number of favourable outcomes, the probability is calculated by dividing by the total number of students surveyed. At Higher tier, probabilities are usually expressed as fractions unless another form is requested.

Examiners often design questions like this to test understanding rather than calculation. The arithmetic itself is straightforward, but only if the correct students have been included in the count.

A strong exam technique is to translate the wording of the question into a short mathematical statement before calculating. For example, identifying that “or” means “at least one” can immediately guide your method and prevent mistakes.