Lecture on Mutually Exclusive Events

Definition of Mutually Exclusive Events

• Mutually Exclusive Events: Events that cannot occur at the same time.
• Example: If we have two events, Event A and Event B, they are mutually exclusive if they do not share any outcomes.

Example with a Six-Sided Die

• Sample space: 1 to 6
• Event A: Outcomes 1, 2, 3
• Event B: Outcomes 5, 6
• Event C: Outcomes 3, 4, 5

Analysis of Events

• A and B:
  • No shared outcomes
  • Intersection is empty
  • Probability of A and B occurring together is 0
  • Thus, A and B are mutually exclusive
• A and C:
  • Share the outcome 3
  • Probability of A and C is 1/6
  • Not mutually exclusive
• B and C:
  • Share the outcome 5
  • Probability of B and C is 1/6
  • Not mutually exclusive

Understanding with Venn Diagrams

• Mutually Exclusive Events: Do not share any outcomes, represented by non-overlapping circles in Venn diagrams.
• Non-Mutually Exclusive Events: Share outcomes, represented by overlapping circles.

Calculating Probabilities

• For Non-Mutually Exclusive Events

  • Formula: P(A or B) = P(A) + P(B) - P(A and B)
  • Accounts for overlapping outcomes in A and B

• Example Problem with a Die

  • Event A: 1, 2, 3, 4
  • Event B: 3, 4, 5
  • Probabilities:
    • P(A) = 4/6
    • P(B) = 3/6
    • P(A and B) = 2/6
  • P(A or B) = P(A) + P(B) - P(A and B) = 5/6

• Adding Event C:

  • Event C: Outcome 6
  • Calculate P(B or C):
    • B and C are mutually exclusive
    • P(B) = 3/6
    • P(C) = 1/6
    • P(B and C) = 0
    • P(B or C) = 2/3

Key Points

• Mutually Exclusive Events: Probability of occurring together is always 0.
• Formula Simplification: For mutually exclusive events, P(A or B) = P(A) + P(B) since P(A and B) = 0.
• Use Venn diagrams for visualizing and confirming mutually exclusive or non-mutually exclusive relationships.