GCSE Maths Practice: conditional-probability

Question 9 of 10

This question tests conditional probability by excluding a specific group before calculating probability.

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\( \begin{array}{l}\text{A group contains 12 Year 10 students, 8 Year 9 students, and 10 Year 11 students.} \\ \text{One student is selected at random.} \\ \text{What is the probability that the student is in Year 10, given that they are not in Year 11?}\end{array} \)

Choose one option:

Conditional probability often works by removing excluded groups before forming the probability.

Conditional Probability Using Group Exclusion

This question tests conditional probability by restricting the sample space based on given information. The phrase given that the student is not in Year 11 is a condition that removes certain outcomes before the probability is calculated.

At Higher GCSE level, students are expected to recognise that probabilities often need to be recalculated after applying a condition. Rather than working with the full group, the probability must be found using only the subset that satisfies the condition.

Understanding the Condition

Before the condition is applied, any student in the group could be selected. Once we are told that the student is not in Year 11, all Year 11 students must be excluded from the sample space. Probabilities should never include outcomes that have been ruled out by the condition.

This step is essential and is a common source of error when students continue to use the original total.

Step-by-Step Method

  1. Identify the total number of individuals in the group.
  2. Apply the condition to remove excluded individuals.
  3. Recalculate the total number of possible outcomes.
  4. Count how many remaining outcomes satisfy the event of interest.
  5. Form the probability using favourable ÷ remaining total.

Worked Example (Different Context)

A club has 18 members: 7 juniors, 6 seniors, and 5 adults. One member is chosen at random. Find the probability that the member is a junior, given that they are not an adult.

The condition removes all adult members from the sample space. Only juniors and seniors remain, and the probability must be calculated using this reduced group.

Another Example

A survey records participants as beginners, intermediate, or advanced. If it is known that a selected participant is not advanced, probabilities should be calculated using only the beginner and intermediate categories.

Common Mistakes

  • Using the original total instead of the reduced total.
  • Forgetting to remove all excluded outcomes.
  • Confusing conditional probability with a two-stage selection.
  • Forming the probability before applying the condition.

Why This Is Higher Tier

Although the numbers are simple, this question requires careful interpretation of language. Higher-tier GCSE questions often assess reasoning skills and the ability to translate written conditions into mathematical restrictions.

Real-Life Application

This type of conditional probability is common in data analysis, education statistics, and surveys, where probabilities are often calculated within specific subgroups.

Study Tip

Whenever you see given that, rewrite the sample space first. If the condition is applied correctly, the probability calculation becomes straightforward.

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