GCSE Maths Practice: conditional-probability

Question 3 of 13

This question focuses on conditional probability, where the result of the first event changes the possible outcomes of the second.

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\( \begin{array}{l}\text{A bag contains 10 balls: 4 red, 3 green, and 3 blue.} \\ \text{One ball is drawn at random and not replaced.} \\ \text{What is the probability the second ball is blue, given the first was green?}\end{array} \)

Choose one option:

Always update the total number of outcomes after the first event before calculating the new probability.

Understanding Conditional Probability

Conditional probability is used when the outcome of one event affects the probability of another event. In GCSE Maths, this often appears in problems involving drawing items from a bag, selecting cards from a deck, or choosing objects without replacement. The key idea is that once the first event happens, the original situation changes, and you must work with a new sample space.

When a question uses words like "given that", "knowing that", or "after", it is a strong signal that you are dealing with conditional probability. Instead of thinking about what could have happened originally, you focus only on what remains possible after the condition is applied.

Step-by-Step Method

  1. Identify the condition given in the question.
  2. Update the total number of possible outcomes based on that condition.
  3. Count how many favourable outcomes remain.
  4. Form the probability using the new total.

Worked Example 1

A bag contains 5 red and 5 yellow counters. One yellow counter is taken out and not replaced. What is the probability that the next counter is red?

Answer: After removing one yellow counter, there are 9 counters left. The number of red counters is still 5, so the probability is \(\frac{5}{9}\).

Worked Example 2

A deck contains 8 cards: 3 hearts and 5 spades. One heart is removed. What is the probability the next card is a spade?

Answer: There are now 7 cards left, with 5 spades remaining. The probability is \(\frac{5}{7}\).

Common Mistakes to Avoid

  • Using the original total instead of the updated total.
  • Reducing the wrong colour or category.
  • Forgetting that the first item is not replaced.
  • Assuming events are independent when they are not.

Real-Life Application

Conditional probability appears in everyday situations such as quality control in factories, card games, and even medical testing. For example, if a faulty product is removed from a batch, the probability that the next product is faulty changes. Understanding how the sample space updates helps make better decisions based on new information.

Frequently Asked Questions

Do I always need a formula?
No. At Foundation level, conditional probability is usually solved by carefully updating the totals rather than using a formal formula.

What if the item is replaced?
If the item is replaced, the total and probabilities stay the same, and the events become independent.

How can I spot conditional probability quickly?
Look for phrases like “given that”, “after”, or “knowing that”.

Study Tip

Always rewrite the situation after the first event happens. If you can clearly describe what remains, the probability calculation becomes much easier.

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