Alzebra of Events - Probability and Statistics

Alzebra of Events

The algebra of events, often discussed in the context of probability theory, is a mathematical framework used to describe and manipulate sets of outcomes (events) in a sample space. It provides the foundational operations and properties to handle events systematically.

1. Basic Set Operations

• Union (A ∪ B): The event that either A or B or both occur.
  $$A \cup B = \{ \omega \in \Omega \mid \omega \in A \text{ or } \omega \in B \}$$

  Example: In a survey, let $A$ be the event that a person likes coffee, and $B$ be the event that a person likes tea. Then, $A \cup B$ is the event that a person likes either coffee or tea or both.

• Intersection (A ∩ B): The event that both A and B occur.
  $$A \cap B = \{ \omega \in \Omega \mid \omega \in A \text{ and } \omega \in B \}$$

  Example: Continuing with the survey, $A \cap B$ is the event that a person likes both coffee and tea.

• Complement (Aᶜ): The event that A does not occur.
  $$A^c = \{ \omega \in \Omega \mid \omega \notin A \}$$

  Example: If $A$ is the event that a person likes coffee, then $A^c$ is the event that a person does not like coffee.

• Difference (A - B): The event that A occurs and B does not.
  $$A - B = \{ \omega \in \Omega \mid \omega \in A \text{ and } \omega \notin B \}$$

  Example: If $A$ is the event that a person likes coffee, and $B$ is the event that a person likes tea, then $A - B$ is the event that a person likes coffee but does not like tea.

2. Properties of Events

• Commutative Properties:
  $$A \cup B = B \cup A$$
  $$A \cap B = B \cap A$$
• Associative Properties:
  $$(A \cup B) \cup C = A \cup (B \cup C)$$
  $$(A \cap B) \cap C = A \cap (B \cap C)$$
• Distributive Properties:
  $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$
  $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$
• De Morgan's Laws:
  $$(A \cup B)^c = A^c \cap B^c$$
  $$(A \cap B)^c = A^c \cup B^c$$

3. Special Types of Events

• Mutually Exclusive Events: Events A and B are mutually exclusive if they cannot both occur at the same time.
  $$A \cap B = \emptyset$$

  Example: In a job application process, let $A$ be the event that a candidate is accepted for Position X, and $B$ be the event that the same candidate is accepted for Position Y. If the company only hires for one position per candidate, $A$ and $B$ are mutually exclusive.

• Exhaustive Events: A set of events is exhaustive if at least one of the events must occur.
  $$A_1 \cup A_2 \cup \ldots \cup A_n = \Omega$$

  Example: In a weather forecasting system, let $A_1$ be the event that it will rain, $A_2$ be the event that it will be sunny, and $A_3$ be the event that it will be cloudy. These events are exhaustive because one of these weather conditions must occur.

Example:

Consider a sample space $\Omega$ representing the outcomes of a day at a theme park: $\Omega = \{\text{Sunny, Rainy, Cloudy}\}$.

• Basic Set Operations:
  • Let $A$ be the event that it is Sunny ($A = \{\text{Sunny}\}$).
  • Let $B$ be the event that it is Rainy ($B = \{\text{Rainy}\}$).
  • Union: $A \cup B = \{\text{Sunny, Rainy}\}$ (either Sunny or Rainy).
  • Intersection: $A \cap B = \emptyset$ (it cannot be both Sunny and Rainy).
  • Complement: $A^c = \{\text{Rainy, Cloudy}\}$ (not Sunny).
  • Difference: $A - B = \{\text{Sunny}\}$ (Sunny and not Rainy).
• Properties of Events:
  • Commutative: $A \cup B = B \cup A$ and $A \cap B = B \cap A$.
  • Associative: $(A \cup B) \cup C = A \cup (B \cup C)$ for any event $C$.
  • Distributive: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ for any event $C$.
  • De Morgan's: $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$.
• Special Types of Events:
  • Mutually Exclusive: If $A = \{\text{Sunny}\}$ and $B = \{\text{Rainy}\}$, then $A \cap B = \emptyset$.
  • Exhaustive: Events $A = \{\text{Sunny}\}$, $B = \{\text{Rainy}\}$, and $C = \{\text{Cloudy}\}$ are exhaustive because $A \cup B \cup C = \{\text{Sunny, Rainy, Cloudy}\} = \Omega$.
• Sigma-Algebra:
  • Consider $\mathcal{F} = \{\emptyset, \{\text{Sunny}\}, \{\text{Rainy, Cloudy}\}, \Omega\}$. This collection is a σ-algebra because it includes $\Omega$, is closed under complement, and is closed under countable unions.