GCSE Maths Practice: conditional-probability

Question 5 of 10

This question tests conditional probability by removing an excluded category before calculating probability.

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\( \begin{array}{l}\text{A bag contains 6 red, 4 blue, and 5 green marbles.} \\ \text{One marble is selected at random.} \\ \text{What is the probability it is red, given that it is not green?}\end{array} \)

Choose one option:

Conditional probability often works by removing impossible outcomes before calculating probabilities.

Conditional Probability Using Exclusion

This question demonstrates a common but subtle form of conditional probability where information is used to exclude certain outcomes before calculating a probability. The phrase given that it is not green is critical and must be applied before any numerical work begins.

At Higher GCSE level, students are expected to recognise that probabilities may need to be recalculated using a restricted sample space. This is different from simply finding the probability of an event from the original total.

Interpreting the Condition

When a question states that an outcome is not a particular category, all items in that category must be removed from consideration. In this case, green marbles are impossible outcomes and should not appear in the probability calculation.

Once those outcomes are removed, the probability becomes a comparison between the remaining favourable outcomes and the remaining total number of possible outcomes.

Systematic Method

  1. Identify the original total number of outcomes.
  2. Use the given condition to remove excluded outcomes.
  3. Recalculate the total number of possible outcomes.
  4. Identify how many remaining outcomes satisfy the event.
  5. Form the probability as favourable ÷ remaining total.

Worked Example (Different Context)

A box contains 8 pencils, 5 pens, and 7 markers. One item is chosen at random. Find the probability that the item is a pen, given that it is not a marker.

The condition removes all markers from the sample space. Only pencils and pens are considered when forming the probability.

Another Example

A survey records people’s favourite transport: car, bus, or bicycle. If it is known that a randomly selected person does not prefer the bus, probabilities should be calculated using only car and bicycle preferences.

Common Mistakes

  • Using the original total instead of the reduced total.
  • Forgetting to remove all excluded outcomes.
  • Confusing this type of question with two-stage experiments.
  • Simplifying fractions before forming the correct probability.

Why This Is Higher Tier

Higher-tier probability questions often test interpretation rather than calculation. This problem requires careful reading, logical exclusion of outcomes, and a clear understanding of how conditions affect probability.

Real-Life Applications

Conditional exclusion appears in many real situations, such as filtering data sets, analysing test results within a subgroup, or calculating risk after known factors are removed.

Study Tip

Whenever a question includes the phrase given that, rewrite the sample space first. If you do this correctly, the probability calculation becomes straightforward.

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