GCSE Maths Practice: two-way-tables

Survey-based probability questions often involve people choosing between activities, hobbies, or interests. In many cases, some people enjoy more than one option. When this happens, the groups overlap, and this overlap must be handled carefully to avoid counting the same person more than once.

A common mistake is to add the number of people in each group and assume this gives the correct total. This only works when the groups do not overlap. If some people belong to both groups, they will be included twice in the total unless you correct for it.

To understand this better, imagine two lists. One list shows people who like the first activity, and another list shows people who like the second activity. Some names may appear on both lists. If you combine the lists without checking, those names appear twice. In probability questions, each person should only be counted once.

One helpful method is to draw a simple diagram with two overlapping circles. Each circle represents one activity. The overlapping section shows people who like both activities. This visual reminder makes it easier to see where double counting can happen.

Another approach is to count step by step. First, count everyone who likes the first activity. Next, include everyone who likes the second activity. Then, identify the people who were counted twice because they like both. Subtracting this overlap once gives the correct total number of people who like at least one activity.

Once you have the correct total, you can find the probability. Probability compares how many outcomes match the condition with how many outcomes are possible in total. In survey questions, the total number of people asked is usually the number of possible outcomes.

These types of questions are common in GCSE Foundation exams because they test careful reading and organisation rather than advanced mathematics. They also reflect real-life situations, such as choosing school clubs, leisure activities, or preferences in a survey.

To improve accuracy, always look for key words such as “both”, “either”, and “or”. These words give important clues about how the groups are related. Taking a moment to identify overlaps before calculating will help you avoid simple but costly errors.