GCSE Maths Practice: conditional-probability

Conditional probability is used when we want to find the probability of an event occurring given that another event has already happened. The key idea is that the condition changes what we are counting from. Instead of looking at the whole group, we focus only on the part of the group that satisfies the given condition.

In probability questions, words such as "given that", "knowing that", or "if we know" are strong indicators that conditional probability is involved. When you see these phrases, the first step should always be to identify the condition and use it to define a new sample space.

A common mistake is to continue using the total number of outcomes even after a condition has been applied. Once a condition is given, everything outside that condition is ignored. Only the outcomes that meet the condition are considered, and these form the new denominator in the probability calculation.

For example, imagine a group of 25 people where 10 play football, 15 play basketball, and 5 play both sports. If you are told that a randomly chosen person plays football, then your sample space becomes the 10 football players, not all 25 people. To find the probability that the person also plays basketball, you would count how many of the football players also play basketball and divide by 10.

Conditional probability can also be explained using subjects at school. Suppose some students study History, some study Geography, and some study both. If a student is chosen at random and you are told that the student studies Geography, then only Geography students matter. The probability that the student also studies History depends on how many Geography students are also in the History group.

At Foundation level, conditional probability questions usually involve simple numbers and are designed to test whether you can correctly change the denominator of the fraction. The numerator counts how many outcomes satisfy both conditions, while the denominator counts how many outcomes satisfy the given condition.

Visual tools such as Venn diagrams can be helpful for understanding conditional probability. By shading or circling the group that represents the condition, it becomes easier to see which outcomes should be included and which should be ignored.

To answer conditional probability questions confidently, follow three steps. First, identify the condition. Second, restrict your sample space to only those outcomes that meet the condition. Third, find the fraction of this restricted group that also meets the event you are interested in. Using this approach helps avoid common mistakes and makes conditional probability much clearer.