GCSE Maths Practice: conditional-probability

Question 7 of 10

This question tests conditional probability by restricting the sample space using a given condition.

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\( \begin{array}{l}\text{A box contains 8 white, 6 red, and 4 blue balls.} \\ \text{A ball is drawn at random.} \\ \text{What is the probability the ball is white, given that it is not blue?}\end{array} \)

Choose one option:

Conditional probability often requires removing impossible outcomes before calculating probabilities.

Conditional Probability with a Restricted Sample Space

This question tests a key Higher GCSE concept: calculating probability after the sample space has been restricted by a condition. The phrase given that it is not blue means that some outcomes are impossible and must be removed before any probability is calculated.

Conditional probability does not always involve multiple draws. In many cases, it involves rethinking the situation using only the outcomes that satisfy the condition. This question focuses on that exact skill.

Applying the Condition Correctly

Before the condition is applied, the box contains balls of three different colours. However, once we are told that the ball is not blue, all blue balls must be excluded from the sample space. Probabilities should never be calculated using outcomes that are no longer possible.

After removing the excluded outcomes, the probability must be formed using only the remaining balls. This step is essential and is where many mistakes occur.

Systematic Approach

  1. Identify the original total number of outcomes.
  2. Apply the condition by removing excluded outcomes.
  3. Recalculate the total number of remaining outcomes.
  4. Identify how many remaining outcomes satisfy the event of interest.
  5. Form the probability using the reduced sample space.

Worked Example (Different Context)

A drawer contains 10 batteries: 4 AA, 3 AAA, and 3 rechargeable. One battery is chosen at random. Find the probability that it is AA, given that it is not rechargeable.

The condition removes the rechargeable batteries from the sample space. Only AA and AAA batteries are considered when calculating the probability.

Another Example

A survey records students’ preferred subjects: Maths, Science, and English. If it is known that a randomly selected student does not prefer English, probabilities should be calculated using only Maths and Science preferences.

Common Errors

  • Using the original total instead of the reduced total.
  • Including excluded outcomes in the probability calculation.
  • Misreading the condition as a second event.
  • Assuming probabilities must add to 1 without checking the condition.

Why This Is Higher Tier

Higher-tier probability questions often test interpretation rather than computation. This problem requires careful reading, logical exclusion of outcomes, and correct construction of a new sample space.

Real-Life Application

This type of conditional reasoning appears in data analysis, quality control, and risk assessment, where probabilities are often calculated only within a filtered group.

Study Tip

Always apply the condition first. If you correctly restrict the sample space, the probability calculation becomes straightforward.

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