Probability Formulas Lecture

Introduction

• Overview of important probability formulas for school.
• Key topics:
  • Symbols and notations
  • Types of probabilities: marginal, union, joint, conditional, negation
  • Independent vs. dependent events
  • Mutually exclusive events

Marginal Probability

• Definition: Probability of a single event occurring independent of other events.
• Formula: Probability of an event = (Number of successful outcomes) / (Total possible outcomes)
• Example:
  • Sample space: Numbers 1 to 9
  • Event A: Outcomes 1, 2, 3, 4
  • Event B: Outcomes 6, 7, 8
  • Event C: Outcomes 1, 7, 8, 9
  • Probability of A: 4/9, B: 1/3 (0.333), C: 4/9 (0.444)

Symbols

• Union (∪): Combines elements from both sets
• Intersection (∩): Common elements in both sets

Union Probability

• Definition: Probability that either A, B, or both occur.
• Formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
  • Mutually exclusive events: P(A and B) = 0
• Example:
  • A ∪ B: Elements 1, 2, 3, 4, 6, 7, 8
  • Probability: 7/9 (77.7%)

Mutually Exclusive Events

• A and B: No common elements, P(A and B) = 0
• B and C: Common elements (7, 8), not mutually exclusive

Joint Probability

• Definition: Probability that both A and B occur simultaneously.
• Formula: P(A ∩ B)
• Calculation:
  • P(A and B) = P(A|B) * P(B) = P(B|A) * P(A)

Independent vs. Dependent Events

• Independent: P(A|B) = P(A), P(B|A) = P(B)
• Dependent: Probability changes based on previous events
• Example:
  • Marbles with/without replacement
  • Coin toss outcomes

Conditional Probability

• Definition: Probability of A given B has occurred.
• Formula:
  • P(A|B) = P(A ∩ B) / P(B)
  • P(B|A) = P(A ∩ B) / P(A)

Bayes’ Theorem

• Formula: P(A|B) = [P(B|A) * P(A)] / P(B)
• Related to conditional probability*

Negation Probability

• Definition: Probability that the complement of an event occurs.
• Formula: P(A') = 1 - P(A)
• Example: Probability of not getting A when A has 4/9 chance
  • A': 5/9

Conclusion

• Summary of key probability formulas for academic purposes.
• Encouragement to check additional resources for practice problems.