Lecture Notes: Combining Events in Probability

Introduction to Combining Events

• Sample Space & Events: Learnings about basic probability, sample space, events, and outcomes.
• Objective: Expand knowledge by combining two different events to create a new event.

Combining Events Using the "And" Operator

• Definition:
  • Given two events, A and B, in the same sample space, "A and B" is the event containing outcomes present in both A and B.
  • This is an operator, not just a word, hence the emphasis on uppercase.
• Example: Events A and B
  • Event A outcomes: 1, 2, 3, 4
  • Event B outcomes: 2, 4, 6, 8
  • Event A and B (Combined): Outcomes 2 and 4 (common in both A and B)
• Visualization:
  • Represented by a Venn diagram, where the intersection of circles A and B depicts "A and B."

Combining Events Using the "Or" Operator

• Definition:
  • Given two events, A and B, in the same sample space, "A or B" consists of outcomes present in either A or B, or in both.
  • The "or" is a mathematical operator.
• Example: Events A and B
  • Event A outcomes: 1, 2, 3, 4
  • Event B outcomes: 2, 4, 6, 8
  • Event A or B (Combined): Includes outcomes 1, 2, 3, 4, 6, 8
  • Note: Outcomes 2 and 4 are counted only once in the "or" event.
• Visualization:
  • In a Venn diagram, "A or B" encompasses all areas covered by circles A and B, including their union.

Summary

• Operators in Probability:
  • "And" involves only common outcomes; depicted by intersection in Venn diagrams.
  • "Or" includes all outcomes from both events; depicted by the union of circles in Venn diagrams.
• Practical Application: These concepts are fundamental in probability theory to understand how different events interact within a sample space.