GCSE Maths Practice: conditional-probability

Question 6 of 10

This question tests conditional probability by excluding a specific group from the sample space.

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\( \begin{array}{l}\text{A box contains 3 red, 2 green, and 5 blue balls.} \\ \text{A ball is drawn at random.} \\ \text{What is the probability of drawing a red ball, given that the ball drawn is not blue?}\end{array} \)

Choose one option:

Conditional probability often requires removing impossible outcomes before calculating probabilities.

Conditional Probability Through Sample Space Reduction

This question focuses on a key Higher GCSE skill: recognising when a probability must be calculated using a restricted sample space. The phrase given that the ball drawn is not blue provides information that immediately removes certain outcomes from consideration.

Rather than describing two separate events happening one after the other, this question uses a condition that limits which outcomes are possible. Understanding this distinction is essential for success at Higher tier.

How the Condition Changes the Problem

Before the condition is applied, all colours in the box are possible outcomes. However, once we are told that the ball is not blue, every blue ball becomes an impossible outcome. Probabilities must then be recalculated using only the remaining colours.

This step is often overlooked by students, leading to incorrect use of the original total instead of the reduced total.

Structured Method

  1. Identify all outcomes in the original situation.
  2. Apply the condition by removing excluded outcomes.
  3. Recount the total number of remaining outcomes.
  4. Identify which remaining outcomes satisfy the event of interest.
  5. Form the probability using the reduced sample space.

Worked Example (Different Scenario)

A drawer contains 7 socks: 3 black, 2 white, and 2 grey. One sock is chosen at random. Find the probability that the sock is black, given that it is not grey.

The condition removes all grey socks from the sample space. Only black and white socks are considered when calculating the probability.

Another Example

A multiple-choice test has questions labelled A, B, C, and D. If it is known that a randomly selected question is not labelled D, probabilities must be calculated using only A, B, and C as possible outcomes.

Common Mistakes

  • Using the original total instead of the reduced total.
  • Removing the wrong group when applying the condition.
  • Assuming the question involves two draws when it does not.
  • Forming the probability before applying the condition.

Why This Fits Higher Tier

Higher-tier probability questions often rely more on interpretation than calculation. This problem requires students to translate language into mathematical restrictions and reason carefully about what information is relevant.

Real-Life Applications

Conditional probability of this type is common in data analysis, where probabilities are calculated only within filtered data sets. For example, analysts may calculate outcomes only for people who meet certain criteria.

Study Tip

Always apply the condition first. If the phrase given that appears, rewrite the sample space before forming any probability.

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