GCSE Maths Practice: conditional-probability

Conditional Probability Using a Restricted Sample Space

This question focuses on conditional probability where additional information is provided about the outcome. The phrase given that the card is red is crucial, as it tells us that some outcomes are impossible and must be excluded before the probability is calculated.

At Higher GCSE level, students are expected to understand that conditional probability often involves redefining the sample space. Instead of working with all 52 cards, the condition restricts attention to a smaller group that satisfies the given requirement.

Understanding the Condition

In a standard deck of cards, red cards consist of two suits: hearts and diamonds. Once we are told that the card drawn is red, all black cards become impossible outcomes. Probabilities should never be calculated using outcomes that are ruled out by the condition.

This is a common source of error, where students incorrectly continue to use the original total of 52 cards rather than the reduced total defined by the condition.

Step-by-Step Strategy

1. Identify the original sample space.
2. Apply the condition to remove impossible outcomes.
3. Recalculate the total number of possible outcomes.
4. Identify how many of the remaining outcomes satisfy the event.
5. Form the probability using the reduced sample space.

Worked Example (Different Context)

A bag contains 10 sweets: 4 strawberry, 3 lemon, and 3 orange. A sweet is chosen at random. Find the probability that the sweet is strawberry, given that it is not lemon.

The condition removes all lemon sweets from the sample space. The probability must then be calculated using only the remaining sweets.

Another Example

A library shelf contains fiction, non-fiction, and reference books. If it is known that a randomly selected book is not a reference book, probabilities should be calculated using only fiction and non-fiction books.

Common Mistakes

• Using the original total instead of the restricted total.
• Ignoring the phrase given that.
• Confusing conditional probability with independent events.
• Including outcomes that have been ruled out by the condition.

Why This Is Higher Tier

Although the numbers involved are simple, the difficulty lies in interpreting the language of the question correctly. Higher-tier questions often assess reasoning and precision rather than arithmetic complexity.

Real-Life Relevance

Conditional probability is widely used in statistics, risk analysis, and data science. For example, probabilities are often calculated only within a specific group that meets certain criteria.

Study Tip

Whenever you see the words given that, pause and redefine the sample space before doing any calculations.