GCSE Maths Practice: conditional-probability

Question 2 of 10

This question tests conditional probability where the outcome of the first event changes the sample space for the second.

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\( \begin{array}{l}\text{A bag contains 10 red, 8 green, and 4 blue balls.} \\ \text{Two balls are drawn without replacement.} \\ \text{What is the probability that the second ball is red, given that the first ball drawn was green?}\end{array} \)

Choose one option:

Always update the sample space after the first event before calculating the next probability.

Understanding Conditional Probability Without Replacement

Conditional probability looks at the chance of an event happening given that another event has already occurred. In questions involving drawing objects from a bag or cards from a deck, the phrase without replacement is crucial because it means the first outcome changes the situation for the second.

In this question, the first ball drawn is known to be green. That information immediately affects the total number of balls remaining in the bag. However, it does not affect the number of red balls. Understanding which quantities change and which stay the same is the key skill being tested.

Step-by-Step Method

  1. Start by identifying the total number of items originally.
  2. Use the given condition to update the situation (remove the known outcome).
  3. Count how many total items remain.
  4. Count how many favourable outcomes remain.
  5. Form the probability as favourable ÷ total.

Worked Example (Different Situation)

A box contains 6 pens and 4 pencils. One pen is removed. What is the probability that the next item chosen is a pencil?

After removing one pen, there are 9 items left in total. The number of pencils is still 4. The probability is therefore \(\frac{4}{9}\).

Another Example

A deck contains 52 cards. One heart is removed. What is the probability that the next card drawn is a spade?

The deck now contains 51 cards, and the number of spades remains 13. The probability is \(\frac{13}{51}\).

Common Mistakes to Avoid

  • Forgetting to reduce the total number after the first draw.
  • Incorrectly reducing the number of favourable outcomes when they are unaffected.
  • Ignoring the phrase given that, which signals conditional probability.

Why This Is a Higher-Tier Skill

At Higher level GCSE, students are expected to interpret conditions mathematically, not just follow a formula. This question tests logical reasoning, careful reading, and understanding how prior information changes probabilities.

Real-Life Context

Conditional probability is widely used in areas such as medical testing, quality control in manufacturing, and data science. For example, the probability that a product is faulty may depend on which production line it came from.

Study Tip

When you see a conditional probability question, always redraw or rewrite the situation after the first event. Treat it as a new problem with updated numbers.

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