GCSE Maths Practice: conditional-probability

Question 8 of 13

This question focuses on conditional probability by restricting attention to a specific subgroup.

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\( \begin{array}{l}\text{A class of 30 students includes 18 girls and 12 boys.} \\ \text{If a student is chosen at random and is a girl, what is the probability that she wears glasses,} \\ \text{given that 6 girls wear glasses?}\end{array} \)

Choose one option:

Once a condition is given, ignore everyone who does not meet it.

Conditional Probability Within a Subgroup

This question is a clear example of conditional probability where the condition restricts the group we are working with. Instead of selecting from the entire class, we are told that the student chosen is a girl. This information changes the sample space and is the key to solving the problem correctly.

In probability, whenever a question includes phrases such as "given that", "knowing that", or "if the student is", you should immediately think about limiting the group of possible outcomes. In this case, boys are no longer relevant because the condition tells us the student is a girl.

Understanding the Restricted Sample Space

Originally, the class contains 30 students. However, once we know the student is a girl, the sample space shrinks to include only the 18 girls. All probability calculations must now be based on this smaller group.

This is the defining feature of conditional probability: the total number of possible outcomes changes because of additional information.

Step-by-Step Approach

  1. Identify the condition in the question.
  2. Decide which group the condition refers to.
  3. Ignore all outcomes outside that group.
  4. Count how many individuals in the group meet the required condition.
  5. Form the probability using only the restricted group.

Worked Example 1

A school has 20 students in a club: 12 are Year 10 and 8 are Year 11. If a student is chosen and is known to be in Year 10, what is the probability that the student plays football, given that 5 Year 10 students play football?

Answer: Only Year 10 students are considered. The probability is calculated using 5 favourable outcomes out of 12 possible outcomes.

Worked Example 2

In a survey, 15 people prefer tea and 10 prefer coffee. If a person is selected and is known to prefer tea, what is the probability they take sugar, given that 6 tea drinkers take sugar?

Answer: The probability is found by comparing the number who take sugar with the total number of tea drinkers.

Common Mistakes

  • Using the total number of students instead of the subgroup.
  • Including outcomes that are excluded by the condition.
  • Confusing this type of question with replacement problems.
  • Dividing by the wrong total.

Real-Life Relevance

This type of conditional probability is used frequently in real life. For example, if a doctor knows a patient belongs to a particular age group, probabilities about symptoms or treatments are calculated only within that group. The same logic applies in education, surveys, and data analysis.

Frequently Asked Questions

Is this still conditional probability even though only one student is chosen?
Yes. The condition restricts the sample space, which makes the probability conditional.

Do I need to use a formula?
No. At Foundation level, careful counting and reasoning are sufficient.

How can I spot these questions quickly?
Look for information that tells you something specific about the chosen person or item.

Study Tip

Before calculating, ask yourself: Who am I allowed to choose from now? This helps you identify the correct total every time.

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