Class7

Class 7: Probability

• Introduction to Statistics for Social Sciences

  • Department of Statistics- UC3M

Chapter Overview

• Chapter 7: Probability

  1. Random experiments

  2. Definition and properties of probability

  3. Conditional probability and independence

  4. Supplementary notes:

  • The law of total probability

  • Bayes theorem

  • Recommended Watching: Video on introduction to probability

Objectives

• Connect descriptive statistics with inferential statistics.

• Quantify representativeness of a SAMPLE concerning the population.

• Approach probability as a measure of likelihood concerning events.

1. Random Experiments

• Definition: Conducting a random experiment to determine the probability of a specific event.

  • Example: Asking a Spanish adult who they voted for in the last general election

  • Sample space (S): Set of all possible outcomes:

    • {Didn’t vote, Cs, ERC, PNV, PP, PSOE, UP, VOX, ...}

  • Elementary event: Any specific outcome of the experiment.

  • Composite event: A grouping of elementary events.

    • Example: Voting for a left-wing party {ERC, PSOE, UP,...}

2. Probability: Definition and Properties

• Based on mathematical theory (Kolmogorov axioms) and different interpretations:

  1. Classical Probability

  2. Frequentist Probability

  3. Subjective Probability

  4. Philosophical Interpretations

Classical Probability (Laplace’s Rule)

• Consider experiments with equally likely elementary events.

• Formula: Probability of Event A = P(A) = (Number of events in A) x (1/K), where K = total elementary events

• Example: Probability of randomly selecting a PSOE president from a list.

Limitations of Classical Probability

• Situations where equal probability is unreasonable:

  • Answering a multiple-choice question (factors like preparation influence outcomes).

  • Estimating votes for a political party in elections.

  • Predicting future pandemics.

• Alternative interpretations must be utilized in these cases.

Frequentist Probability

• Repeated experiments yield relative frequencies that approximate true probabilities.

• The probability is the limit of relative frequencies.

Subjective Probability

• Individual probabilities based on personal knowledge, experience, and uncertainty.

3. Properties of Probability

Venn Diagram Application

• Areas in Venn diagrams adhere to probability rules:

  • P(S) = 1 (total probability of all outcomes is 1)

  • For any event A: 0 ≤ P(A) ≤ 1

• Complementary Events:

  • P(Ā) = 1 - P(A)

  • A is the event; Ā is its complement.

Union and Intersection of Events

• Combined Event Probability:

  • P(A∪B) = P(A) + P(B) - P(A∩B)

• Incompatible Events:

  • If incompatible, P(A∪B) = P(A) + P(B)

Application of Probability with Political Parties

• Example of Absolute vs Relative frequency:

  • PSOE: 20 votes; Relative frequency = 20/66 = 0.30, etc.

• Determine sample preferences as probability estimates.

4. Conditional Probability and Independence

Conditional Probability

• Formula: P(A|B) = P(A ∩ B) / P(B)

• Example Data Table: Steps taken by political party members in relation to preferences.

• Questions to analyze probabilities based on preferences and actions.

Multiplication Law

• P(A ∩ B) = P(A | B) * P(B)

• Example Question: Probability of two students both walking less than 8000 steps daily.

Independence of Events

• Events A and B are independent if: P(A ∩ B) = P(A)P(B)

• If B occurs, the probability of A remains unchanged.

Independence in Practice

• Rare independent events exist (e.g., coin tosses).

• Example Questions: Investigate political preference and walking steps.

Exercises

Exercise 1: YouGov Survey on Ukrainian Refugees

• Data based on regions in the UK concerning support for refugee acceptance.

• Questions regarding probabilities based on respondent location and support.

Exercise 2: Political Preferences and Refugees

• Survey concerning political preferences and opinions on refugee acceptance.

• Questions regarding probabilities based on voting history and support of refugees.

5. Supplementary Material: Total Probability and Bayes' Theorem

Law of Total Probability

• Definition: Partitioning event S into disjoint subsets.

• Formula: P(A) = sum of P(A ∩ Bi) = sum of P(A|Bi)P(Bi)

Bayes' Theorem

• 
  

  Given event A, calculate the probability of Bi:P(Bi| A) = P(A ∩ Bi) / P(A) = P(A|Bi)P(Bi) / P(A)

• Examples: Analyze probabilities based on random selections from urns.