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

Image courtesy of wentzwu

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

**2. Drawing Cards**

Images courtesy of thoughtsco

**Event A:**Drawing a heart. $P(A) = \frac{13}{52}$.**Event B:**Drawing a club after a heart. $P(B \text{ given } A) = \frac{13}{51}$.**Combined:**P(A and B) = $P(A \text{ and } B) = \frac{13}{52} \times \frac{13}{51}$.**Conclusion:**Not independent, as P(A and B) differs from P(A) × P(B).

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.