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CIE A-Level Maths Study Notes

4.3.2 Addition and Multiplication Rules

Probability, a core concept in mathematics, plays a vital role in understanding the likelihood of various events. The focus is on providing a thorough understanding of these rules, their applications, and problem-solving techniques.

Probability Basics

  • Probability of an Event: P(Event) = Favourable outcomes / Total outcomes.
  • Sum of Probabilities: All possible outcomes' probabilities add up to 1.

Mutually Exclusive Events

  • Definition: Can't happen at the same time. Example: Can't get heads and tails on one coin flip.
  • Addition Rule: P(A or B) = P(A) + P(B) for exclusive events A and B.

Addition Rule Example

  • Task: Probability of drawing a red card or a queen from a deck.
  • Steps:
    • Red Card Probability: 2652\frac{26}{52} (half the deck).
    • Queen Probability: 452\frac{4}{52} (one per suit).
  • Add Probabilities: 12+452\frac{1}{2} + \frac{4}{52}.
  • Result: Simplify to 152657.69%\frac{15}{26} \approx 57.69\%.

Independent Events

  • Definition: One event doesn't affect the other. Example: Rolling a die and flipping a coin.
  • Multiplication Rule: P(A and B) = P(A) × P(B) for independent events.

Multiplication Rule Example

  • Task: Probability of drawing an ace then a king, without replacement.
  • Steps:
    • Ace Probability: 452\frac{4}{52}.
    • King after Ace Probability: 451\frac{4}{51} (one ace gone).
    • Multiply Probabilities: (452)×(451)\left(\frac{4}{52}\right) \times \left(\frac{4}{51}\right).
  • Result: 0.60%\approx 0.60\%.

Choosing the Right Rule

  • Mutually Exclusive: Use Addition Rule.
  • Independent Events: Use Multiplication Rule.
  • Analyze Event Relationship: Decide based on whether events are exclusive or independent.

General Addition Formula Example

  • Task: Probability of drawing a heart or an ace.
  • Steps:
    • Heart Probability: 1352\frac{13}{52}.
    • Ace Probability: 452\frac{4}{52}.
    • Subtract Ace of Hearts Overlap: 152 -\frac{1}{52}.
  • Result:30.77%\approx 30.77\%.
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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