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Probability & Statistics (31 of 62) Basic Theorems 11 - 14 By Michel van Biezen

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Probability & Statistics (31 of 62) Basic Theorems 11 - 14 By Michel van Biezen


Summary:


  1. Introduction: The lecturer introduces four basic theorems in probability, emphasizing the importance of using examples to understand them better.

  2. Theorem 11 - Relationship between Event A and the Universe: If all outcomes in event A are also in the universe, then the union of A and the universe equals the universe itself. Similarly, the intersection of A and the universe equals A.

  3. Example: Using a die as a sample space, if event A represents outcomes 1, 2, and 3, then A union universe equals universe, and A intersection universe equals A.

  4. Theorem 12 - Complement of Union: The complement of the union of events A and B is equivalent to the intersection of their complements.

  5. Example: With the same die sample space, if A represents outcomes 1, 2, and 3, and B represents outcomes 2, 3, and 4, then the complement of A union B is outcomes 4, 5, and 6, which equals the intersection of the complements of A and B.

  6. Theorem 13 - Complement of Intersection: The complement of the intersection of events A and B is equivalent to the union of their complements.

  7. Example: Continuing with the previous example, if A intersection B represents outcome 2, then the complement of A intersection B is outcomes 1, 3, 4, 5, and 6, which equals the union of the complements of A and B.

  8. Theorem 14 - Set Equality: Event A can be expressed as the union of the intersection of A and B and the intersection of A and the complement of B.

  9. Example: Using the same die example, if A represents outcomes 1 and 2, and B represents outcomes 2 and 3, then A equals the union of (A intersection B) and (A intersection not B).

  10. Conclusion: The lecturer confirms the validity of the theorems through examples and concludes that they provide additional terms in probability calculations.

This summary outlines the lecturer's explanation of four basic theorems in probability and demonstrates their application using examples involving a sample space of rolling a die.