GCSE Maths Practice: conditional-probability

Question 4 of 10

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

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\( \begin{array}{l}\text{A bag contains 4 green balls, 3 yellow balls, and 5 blue balls.} \\ \text{A ball is drawn at random.} \\ \text{What is the probability the ball is blue, given that it is not yellow?}\end{array} \)

Choose one option:

Conditional probability often works by removing impossible outcomes before forming the probability.

Conditional Probability by Restricting the Sample Space

This question focuses on a subtle but important form of conditional probability. Instead of describing two separate events happening in sequence, it uses a condition that restricts which outcomes are possible. The phrase given that it is not yellow tells us that some outcomes must be completely excluded before any probability is calculated.

At Higher GCSE level, students are expected to recognise that conditional probability does not always involve multiple draws. Sometimes the condition simply narrows the sample space, and all probabilities must be recalculated using only the remaining outcomes.

Understanding the Key Idea

When a condition is applied, you should imagine removing all outcomes that do not satisfy it. In this question, the condition removes every yellow ball from consideration. Once this is done, the probability is no longer based on the original total but on a smaller, updated set of possible outcomes.

This approach prevents a very common error: using the original total when the question clearly restricts which outcomes are allowed.

Step-by-Step Approach

  1. Start by identifying the total number of outcomes.
  2. Use the condition to exclude outcomes that are not allowed.
  3. Count how many outcomes remain.
  4. Identify how many of those remaining outcomes satisfy the event you are interested in.
  5. Form the probability as favourable ÷ remaining total.

Worked Example (Different Context)

A box contains 6 apples, 2 oranges, and 4 pears. A fruit is chosen at random. Find the probability that the fruit is a pear, given that it is not an orange.

The condition removes all oranges from the sample space. Only apples and pears remain, and the probability must be calculated using this reduced set.

Another Example

A spinner has 10 equal sections: 3 red, 4 blue, and 3 green. Find the probability that the spinner lands on blue, given that it does not land on green.

The green sections are excluded before the probability is calculated, showing again how the condition reshapes the sample space.

Common Mistakes

  • Using the original total instead of the restricted total.
  • Subtracting the wrong outcomes when applying the condition.
  • Treating this as a two-step experiment rather than a single conditional situation.

Why This Is Higher Tier

Foundation-level questions often focus on direct probabilities. Higher-tier questions require interpreting language precisely and translating conditions into mathematical restrictions. This problem tests logical reasoning rather than routine calculation.

Real-Life Application

Conditional probability of this type appears in areas such as data filtering, survey analysis, and risk assessment. For example, analysts may calculate probabilities only within a specific subgroup that meets certain criteria.

Study Tip

Whenever you see given that, stop and redraw the sample space using only outcomes that satisfy the condition before doing any calculations.

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