GCSE Maths Practice: two-way-tables

Higher-tier probability questions often use phrases such as “at least one”, “either”, or “or”. These phrases have a precise mathematical meaning and are closely linked to the idea of combining overlapping sets.

When two subjects are mentioned, some students may take only the first subject, some may take only the second, and some may take both. The phrase “takes either subject” includes all three of these groups. Understanding this language is essential for choosing the correct method.

The inclusion–exclusion principle provides a structured way to count all students who are in at least one group. Adding the totals for each subject includes every student, but it also includes students who take both subjects twice. Subtracting the overlap once corrects this double counting.

At Higher tier, it is useful to think about this process algebraically. If one group has size A, another has size B, and the overlap has size C, then the number in at least one group is A + B − C. This structure stays the same even when the context changes.

A Venn diagram can help confirm your reasoning. The two circles represent the two subjects, and the overlapping region represents students who take both. Counting all regions inside the circles gives the number of students who take at least one subject.

Consider a different scenario involving qualifications or training courses. Some people may complete one course, some another, and some both. If an organisation wants to know the probability that a randomly chosen person has completed at least one course, the same reasoning applies.

Once the number of favourable outcomes has been found, the probability is calculated by dividing by the total number of possible outcomes. In survey questions, this total is usually the total number of students or participants.

Common Higher-tier errors include misinterpreting “either” as meaning “exactly one”, or forgetting that students who take both subjects should still be included in the final count. Carefully reading the wording of the question is just as important as carrying out the calculation.

A strong exam strategy is to write a short statement such as “at least one = total of both − overlap” before inserting numbers. This helps keep your working organised and reduces the chance of mistakes.