Special Case of Mutually Exclusive Events

Key Concepts

• Mutually Exclusive Events: Two events that cannot occur at the same time.
• Complementary Events: The complement of an event A (notated as A^C) includes all outcomes not in A.
• Sample Space: All possible outcomes of an experiment.

Example

• Rolling a Die:
  • Sample Space: {1, 2, 3, 4, 5, 6} for a standard six-sided die.
  • Event M: Rolling a number greater than 4, i.e., rolling a 5 or 6.
  • M Complement (M^C): Rolling a 1, 2, 3, or 4.

Properties of Complementary Events

1. Mutually Exclusive: A and A^C cannot happen simultaneously.
2. Fill the Sample Space: Every outcome is either in A or in A^C.

Addition Rule for Complementary Events

• Probability of A or A^C: Since they are mutually exclusive, use the addition rule:
  • Probability(A) + Probability(A^C) = 1
  • This indicates a 100% chance that one of these events occurs.

Applications

• Winning and Losing in Games:
  • If winning and losing are the only two outcomes, they are complementary.
  • Probability(Win) + Probability(Lose) = 1
  • Assumes no other outcomes (e.g., no draws in Tic-Tac-Toe or Rock Paper Scissors).

Summary

• Complementary events like winning/losing or rolling specific sets of numbers on a die illustrate the concept of mutually exclusive events that cover the entire sample space.