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.