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).