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CIE A-Level Maths Study Notes

4.3.3 Exclusive and Independent Events

In the realm of probability, the concepts of mutually exclusive and independent events form the bedrock of understanding complex probabilistic scenarios.

Mutually Exclusive Events

  • Definition: Two events that can't happen at the same time.
  • Example: Rolling a '2' and rolling a '5' on a single die.
  • Probability: P(A or B) = P(A) + P(B).

Independent Events

  • Definition: Two events where one doesn't affect the other's probability.
  • Example: Flipping a coin and rolling a die.
  • Probability: P(A and B) = P(A) × P(B).

Checking Independence

  • Concept: See if P(A and B) equals P(A) × P(B).
  • Method: Compare P(A and B) with P(A) × P(B).

Examples

1. Coin and Die

Coin and die

Coin and Die

Image courtesy of wentzwu

  • Event A: Flipping a head. P(A)=12P(A) = \frac{1}{2}.
  • Event B: Rolling a 3. P(B)=16P(B) = \frac{1}{6}.
  • Combined: P(A and B)=12×16=112P(A \text{ and } B) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12} .
  • Conclusion: Independent, as P(A and B) equals P(A) × P(B).

2. Drawing Cards

Card

Images courtesy of thoughtsco

  • Event A: Drawing a heart. P(A)=1352 P(A) = \frac{13}{52} .
  • Event B: Drawing a club after a heart. P(B given A)=1351P(B \text{ given } A) = \frac{13}{51}.
  • Combined: P(A and B) = P(A and B)=1352×1351P(A \text{ and } B) = \frac{13}{52} \times \frac{13}{51}.
  • Conclusion: Not independent, as P(A and B) differs from P(A) × P(B).
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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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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