GCSE Maths Practice: conditional-probability

Conditional Probability Using Elimination

This question is an example of conditional probability where the condition allows us to remove some outcomes before calculating the probability. Instead of dealing with two separate events, we are given extra information about the single event that has already occurred.

The phrase "given that it is not black" is the most important part of the question. It tells us that some outcomes are impossible and must be excluded from the sample space. Once these outcomes are removed, probability is calculated using only what remains.

How the Sample Space Changes

Originally, the box contains marbles of three different colours. However, once we know the marble is not black, every black marble becomes irrelevant. The probability must now be calculated using a smaller group consisting only of the marbles that could still have been selected.

This idea is central to conditional probability: the sample space is reduced by the condition.

Step-by-Step Strategy

1. Identify the condition in the question.
2. Remove all outcomes that contradict the condition.
3. Count how many outcomes remain in total.
4. Count how many of the remaining outcomes match the desired result.
5. Form the probability using the reduced total.

Worked Example 1

A bag contains 6 blue beads, 4 green beads, and 2 yellow beads. One bead is chosen at random. What is the probability the bead is green, given that it is not blue?

Answer: Removing blue beads leaves 4 green and 2 yellow beads, giving 6 beads. The probability of green is \(\frac{4}{6} = \frac{2}{3}\).

Worked Example 2

A box contains 3 pencils, 5 pens, and 2 markers. One item is selected. What is the probability the item is a marker, given that it is not a pen?

Answer: Removing pens leaves 3 pencils and 2 markers, giving 5 items. The probability of a marker is \(\frac{2}{5}\).

Common Mistakes

• Using the original total instead of the reduced total.
• Forgetting to remove all excluded outcomes.
• Thinking a second draw is involved when it is not.
• Mixing up this type of question with replacement problems.

Real-Life Context

This type of conditional probability is used whenever you receive information that rules something out. For example, if you know a product is not defective, your analysis focuses only on the remaining quality categories. Similarly, if a survey respondent is known not to belong to one group, probabilities are recalculated using only the remaining groups.

Frequently Asked Questions

Is this different from probability without conditions?
Yes. The condition reduces the number of possible outcomes.

Do I need to use formulas?
No. At Foundation level, careful counting is enough.

What words signal this type of question?
Phrases like “given that”, “knowing that”, and “it is not”.

Study Tip

Before calculating, rewrite the problem using only the outcomes that are still possible. This avoids almost all common errors.