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Introduction

Conditional probability, an essential topic in GCSE Mathematics, calculates the probability of an event happening given that another event has already occurred. It helps us make informed decisions based on real-world situations.

Key Concepts and Definitions

• Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other.
• Dependent Events: Two events are dependent if the occurrence of one affects the probability of the other.
• Conditional Probability: The conditional probability of event A happening given that event B has already occurred is written as P(A|B).

Step-by-Step Explanations

To calculate conditional probability:

1. Draw a tree diagram or Venn diagram to represent the events.

2. Count or calculate the number of outcomes that satisfy both events.

3. Divide the number from step 2 by the total number of outcomes.

Common Mistakes to Avoid

• Assuming events are independent when they are dependent.
• Using the multiplication rule when events are dependent.

Practice Problems with Solutions

• 1. A bag contains 5 red and 3 blue marbles. If a blue marble is drawn, what is the probability of drawing a red marble next?
• P(red|blue) = P(red) / P(blue)
• P(red) = 5/8 (5 red marbles out of 8 total marbles)
• P(blue) = 3/8 (3 blue marbles out of 8 total marbles)
• P(red|blue) = 5/8 / 3/8 = 5/3 = 1.67
• 2. A fair coin is tossed twice. What is the probability of getting a head on the second toss given that a tail was obtained on the first toss?
• Since the coin is fair, P(head) = 1/2 and P(tail) = 1/2.
• P(head|tail) = P(head and tail) / P(tail)
• P(head and tail) = 1/4 (only one way to get head and tail)
• P(head|tail) = 1/4 / 1/2 = 1/2

Conclusion

Conditional probability is a powerful tool for understanding and making decisions in real-world scenarios. By mastering the concepts presented here, you can confidently tackle problems and excel in your GCSE Mathematics exam.

Exam Tips

• Understand the difference between independent and dependent events.
• Use tree diagrams or Venn diagrams to organize information.
• Practice regularly to improve your confidence.

FAQ

• Q: When do I use the multiplication rule for probability?
• A: Use the multiplication rule for independent events.
• Q: How do I know if events are independent?
• A: Two events are independent if the occurrence of one does not change the probability of the other.

Resources

• [Conditional Probability Practice Problems](https://www.khanacademy.org/math/apstatistics/x2eef969c74e0d802:probability/x2eef969c74e0d802:conditionalprobabilityandindependence/v/conditionalprobabilityproblem1)
• [GCSE Mathematics Revision Notes: Conditional Probability](https://www.bbc.co.uk/bitesize/guides/z34j7ty/revision/3)