Lecture on Mutually Exclusive Events

Definition of Mutually Exclusive Events

• Two events, A and B, are mutually exclusive if it's impossible for both to happen at the same time.
• Mathematically, this means the probability of both A and B occurring (P(A and B)) is zero because they don't share any outcomes.
• In a Venn diagram, A and B would not overlap, indicating no shared outcomes.

Example of Mutually Exclusive Events

• Example Events:
  • Event E: Rolling an even number on a die.
  • Event D: Rolling an odd number on a die.
• Outcomes:
  • For event E (even), possible outcomes with a six-sided die are 2, 4, 6.
  • For event D (odd), possible outcomes are 1, 3, 5.
• These two events are mutually exclusive because:
  • Rolling an even number means you cannot roll an odd number and vice versa.
  • No common outcomes exist between E and D.

Application to Probability Formula

• General Addition Formula for Probability:
  • ( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) )
• Special Case for Mutually Exclusive Events:
  • If A and B are mutually exclusive, ( P(A \text{ and } B) = 0 ).
  • Formula simplifies to: ( P(A \text{ or } B) = P(A) + P(B) ).

Important Considerations

• The simplified formula only works if A and B are mutually exclusive.
• If events are not mutually exclusive or their status is unknown, use the general addition formula.
• Always confirm the condition of mutual exclusivity before using the simplified formula.