Lecture Notes: Probability and Relationship between Political Affiliation and Vaccination Status

Key Topics Covered

• Political affiliation and vaccination status are used to understand probability rules.
• Emphasis on conceptual understanding of probability over memorizing formulas.

Basic Probability Concepts

• Definition: Probability is the number of successes over the total number.
• Simple Probability:
  • Number of specific outcomes divided by the total number of outcomes.
  • Probability of an event, P(A), is calculated as ( \frac{\text{Number of A outcomes}}{\text{Total number of outcomes}} ).

Types of Probability Statements

• And Statements (Intersection):
  • Represented by overlapping circles in Venn Diagrams.
  • P(A and B) represents the intersection of two events, requiring both conditions to be true.
• Or Statements (Union):
  • Represented by the union of areas in Venn Diagrams.
  • P(A or B) includes all outcomes in either event A or event B.
• Not Statements (Complement):
  • Represents everything outside a specific event.
  • P(Not A) is calculated by subtracting P(A) from 1.

Examples with Venn Diagrams

• Visual representation helps understand complex probabilities.
• Example: Women and left-handed people illustrated with Venn Diagrams.

Real Data Example

• Survey of 750 California adults on coronavirus vaccine safety perceptions and political affiliations.
• Use of a two-way table to organize data.
• Subtotals summarized: Definitely safe, Probably safe, Probably unsafe, Definitely unsafe, and Don't know categories.
• Political affiliations: Republicans, Democrats, Independents, and Others.

Exercises and Calculations

1. Probability Calculation: Single Category

   • Example: Probability of believing vaccine is definitely safe.
   • Calculated by dividing the number of people who believe it's definitely safe by the total number surveyed.

2. Probability Calculation: Intersection

   • Example: Probability of being both Republican and believing the vaccine is definitely safe.
   • Use the intersection of rows and columns in two-way tables.

3. Probability Calculation: Union

   • Example: Probability of being definitely safe or probably safe.
   • Add subtotals of both categories to find the union.

4. Mutually Exclusive Events

   • Definition: Events with no overlap, cannot occur simultaneously.
   • Example: Pregnancy status as mutually exclusive.

5. Probability Calculation: Not Statements

   • Example: Probability of not believing the vaccine is definitely safe.
   • Use complement rule to find probability.

6. Example Problem: Combining Political Affiliation and Vaccination Status

   • Calculate probabilities of intersecting categories like vaccinated and Republican or Democrat.
   • Use of union and intersection concepts to solve complex probability questions.

Important Formulas

• Union Formula: ( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) )
• Complement Formula: ( P(\text{Not } A) = 1 - P(A) )

Conceptual Takeaways

• Keep probability concepts simple for ease of understanding and application.
• Venn Diagrams are a useful tool for visualizing probability problems.
• Real-world data can help solidify understanding of probability concepts.