GCSE Maths Practice: two-way-tables

Probability questions that involve groups of people often require careful counting. When students are divided into groups based on subjects, hobbies, or choices, it is important to recognise that some people may belong to more than one group.

If you simply add the size of each group together, you may accidentally count the same person more than once. This happens when someone appears in both groups. Counting the same person twice will give a total that is too large, which then leads to an incorrect probability.

To avoid this mistake, you should always check whether the question mentions an overlap between the groups. An overlap means there are individuals who are included in both categories. These individuals should only be counted once when finding the total number of people involved.

A helpful way to visualise this situation is to draw a simple diagram. Two overlapping circles can represent the two groups. The overlapping section shows people who belong to both groups. The sections that do not overlap show people who belong to only one group. This makes it easier to see which people have been counted more than once.

Another useful approach is to think in steps. First, consider everyone who belongs to the first group. Next, include everyone from the second group. Finally, adjust the total so that no individual is counted twice. This step-by-step thinking helps reduce errors, especially under exam pressure.

Once you have the correct number of people who belong to at least one group, the probability can be found. Probability is always calculated by comparing a desired outcome with the total number of possible outcomes. In class-based problems, the total number of outcomes is usually the total number of students.

Students often lose marks in these questions by rushing or skipping the checking stage. A quick pause to ask, “Have I counted anyone twice?” can make a big difference. Examiners expect you to show awareness of overlaps, even at Foundation level.

These types of questions appear frequently in GCSE Maths because they reflect real-life situations, such as choosing subjects, attending clubs, or answering surveys. Learning to organise information clearly will help you not only in exams but also in everyday decision-making.

As a final tip, always read the question carefully and underline key phrases like “both”, “either”, or “or”. These words give important clues about how the groups relate to each other and how the probability should be calculated.