Lecture on Mutually Exclusive Events

Definition of Mutually Exclusive Events

• Mutually Exclusive Events: Events that cannot occur simultaneously.
  • If two events do not share any outcomes, they are mutually exclusive.
  • Probability of mutually exclusive events occurring together is always zero.

Example with a Six-Sided Die

• Sample Space: 1 to 6

• Events:

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

• Analysis:

  • A & B: Mutually exclusive (No common outcomes)
  • A & C: Not mutually exclusive (Common outcome: 3)
  • B & C: Not mutually exclusive (Common outcome: 5)

Determining Mutually Exclusive Events

• Method: Calculate the probability of both events happening.
  • Mutually Exclusive: Probability is zero.
  • Not Mutually Exclusive: Probability is not zero.

Venn Diagrams for Mutually Exclusive Events

• Illustration:
  • Non-overlapping circles represent mutually exclusive events.
  • Overlapping circles represent events that are not mutually exclusive.

Probability Calculations

• Probability of A or B (General Case):
  • (P(A \cup B) = P(A) + P(B) - P(A \cap B))
  • If events A and B are mutually exclusive, (P(A \cap B) = 0), simplifying to (P(A) + P(B)).

Example Problem

• Event A: Outcomes 1, 2, 3, 4

• Event B: Outcomes 3, 4, 5

• Are A and B Mutually Exclusive?

  • No, they share outcomes 3 and 4.

• Probability Calculation:

  • (P(A \cup B) = P(A) + P(B) - P(A \cap B))
  • (P(A) = \frac{4}{6}), (P(B) = \frac{3}{6}), (P(A \cap B) = \frac{2}{6})
  • Result: (\frac{5}{6})

Adding Event C

• Event C: Outcome 6
• Probability of B or C:
  • (P(B \cup C) = P(B) + P(C) - P(B \cap C))
  • B and C are mutually exclusive.
  • (P(B) = \frac{3}{6}), (P(C) = \frac{1}{6}), (P(B \cap C) = 0)
  • Result: (\frac{2}{3})

Key Takeaway

• Formula for Mutually Exclusive Events:
  • (P(A \cup B) = P(A) + P(B)) when A and B are mutually exclusive.
• Always check if events share any outcomes to determine mutual exclusivity.