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: $\frac{26}{52}$ (half the deck).
  • Queen Probability: $\frac{4}{52}$ (one per suit).
• Add Probabilities: $\frac{1}{2} + \frac{4}{52}$.
• Result: Simplify to $\frac{15}{26} \approx 57.69\%$.

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: $\frac{4}{52}$.
  • King after Ace Probability: $\frac{4}{51}$ (one ace gone).
  • Multiply Probabilities: $\left(\frac{4}{52}\right) \times \left(\frac{4}{51}\right)$.
• Result: ≈ $\approx 0.60\%$.

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: $\frac{13}{52}$.
  • Ace Probability: $\frac{4}{52}$.
  • Subtract Ace of Hearts Overlap: $-\frac{1}{52}$.
• Result: ≈ $\approx 30.77\%$.