GCSE Maths Practice: conditional-probability

Question 10 of 10

This question tests conditional probability where the first outcome affects the second.

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\( \begin{array}{l}\text{A jar contains 5 red, 6 green, and 4 blue balls.} \\ \text{A ball is drawn at random and not replaced.} \\ \text{What is the probability of drawing a green ball on the second draw, given the first draw was red?}\end{array} \)

Choose one option:

Always update the total number of items after the first draw before calculating the next probability.

Conditional Probability with Sequential Events

This question tests conditional probability in a situation involving two events that happen one after the other. The phrase given the first draw was red tells us that information from the first event must be used to update the situation before calculating the probability of the second event.

Because the ball is drawn without replacement, the first event permanently changes the contents of the jar. This is a key distinction from situations where items are replaced and probabilities remain unchanged.

Understanding What Changes and What Does Not

After the first draw, two things must be reconsidered: the total number of balls and the number of balls of each colour. Only the colour that was drawn is affected. All other colours remain unchanged.

In this case, a red ball is removed. The total number of balls decreases by one, and the number of red balls decreases by one. The numbers of green and blue balls remain the same. Identifying this correctly is essential for solving the problem.

Step-by-Step Method

  1. Write down the original number of items and categories.
  2. Use the given condition to remove the known outcome.
  3. Update the total number of remaining items.
  4. Identify how many remaining items satisfy the required outcome.
  5. Form the probability as favourable ÷ remaining total.

Worked Example (Different Context)

A box contains 7 apples, 5 bananas, and 3 oranges. One apple is removed. What is the probability that the next fruit chosen is a banana?

After removing one apple, the total number of fruits decreases, but the number of bananas remains unchanged. The probability must be calculated using the updated total.

Another Example

A deck contains red and black cards. One red card is removed. The probability of drawing a black card next must be calculated using the reduced number of cards.

Common Mistakes

  • Forgetting to reduce the total number of items after the first draw.
  • Reducing the number of favourable outcomes incorrectly.
  • Using the original probability instead of updating the situation.
  • Assuming probabilities stay the same without replacement.

Why This Is Higher Tier

Higher-tier probability questions often require careful reasoning about how one event affects another. This problem tests logical thinking, attention to detail, and the ability to update a sample space based on given information.

Real-Life Applications

Sequential conditional probability is used in areas such as quality testing, games of chance, and data analysis. For example, probabilities change as items are removed from a batch during inspection.

Study Tip

After the first event in a probability question, pause and rewrite the situation using the updated numbers. Treat the second event as a new problem based on what remains.

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