GCSE Maths Practice: two-way-tables

Probability questions based on surveys often involve two groups that overlap. This means that some people belong to both groups at the same time. When the question asks for the chance that someone is in one group or the other, you must count everyone who is in at least one group, but you must not count anyone twice.

A very common mistake is to add the two group totals and stop there. This does not work when there is overlap, because the people who are in both groups get counted twice: once in the first group and again in the second. If you do not correct this, your final total will be too large, which then makes your probability incorrect.

A good way to avoid confusion is to draw a simple Venn diagram. Draw two circles that overlap. Label one circle with the first drink and the other circle with the second drink. The middle overlapping part represents people who chose both. The non-overlapping parts represent people who chose only one option. Even if you do not fill in every number, the diagram helps you remember that the overlap needs special attention.

Another method is to think of counting in stages. Start by counting everyone in the first group. Then add everyone in the second group. At that point, ask yourself: “Who have I counted twice?” The answer will be the people in the overlap. To fix the double counting, subtract the overlap once. This leaves you with the correct number of people who are in at least one of the groups.

Once you have that corrected total, finding the probability is straightforward. Probability is the number of favourable outcomes divided by the number of possible outcomes. In survey questions, the number of possible outcomes is usually the total number of people surveyed. This is why the total surveyed becomes the denominator in your fraction.

These questions are popular at GCSE Foundation because they test careful reading and accurate counting. They also appear in real life. For example, a school might survey students about drink choices, club attendance, or transport methods. The results can be used to plan resources such as what to stock in a canteen or how many places are needed in an after-school club.

To improve accuracy in exams, underline words like “both”, “either”, and “or”. These words tell you whether you need to include an overlap and how to count students properly. If you build the habit of checking for overlap, you will avoid one of the most common errors in two-group probability questions.