4.2 - Compound Events (3)

Elementary Probability Theory

Overview

• Probability: A branch of mathematics concerning the likelihood of events occurring.

Section 4.1: Fundamentals of Probability Theory

• Introduction to key concepts and basic terminologies in probability.

Section 4.2: Key Topics in Probability

• Covers essential components such as events, sets, and various operations involving them.

Sets and Events

Definition of Sets

• Sets: Collections of unique elements sharing a common characteristic.

  • Examples:

    • A = {1, 2, 3, 4, 5}

    • B = {Blue, Green, Red}

    • C = {‘!’, ‘?’, ‘.’}

• Types of Sets:

  • Singleton: A set with one element (e.g., S = {8}).

  • Multiple Elements: A set with multiple elements (e.g., S = {2, 4, 6, 8}).

  • Empty Set: A set with no elements (e.g., S = {} or S = Ø).

Operations on Sets

Union of Sets

• Union: Combines elements of sets, denoted by ∪.

  • E.g., if A = {1, 2, 3, 4} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}.

Intersection of Sets

• Intersection: Includes only elements present in both sets, denoted by ∩.

  • E.g., for sets A and B as defined earlier, A ∩ B = {3, 4}.

Difference of Sets

• Difference: Elements present in one set but not the other, denoted by ''.

  • E.g., A \ B = {1, 2}.

Symmetric Difference of Sets

• Symmetric Difference: Elements that are in either set but not both, denoted by Δ.

  • E.g., A Δ B = {1, 2, 5}.

Venn Diagrams

• Venn Diagrams: Illustrations showing relationships between sets, where the rectangle represents the sample space, and circles represent events.

  • Overlap of circles indicates the intersection of events.

Complementary Events

Definition

• Complement: The set of all outcomes not in the specified event.

  • Notation: Aᶜ = {outcomes not included in event A}.

Calculation of Probability

• Probability of the complement: P(Aᶜ) = 1 - P(A).

At Least One Event

Probability Calculation

• To determine the probability of at least one event occurring: P(At Least One) = 1 - P(None).

Addition Rule

Concept

• The probability of either of the two events occurring, denoted by P(A ∪ B).

For Mutually Exclusive Events

• If events A and B are mutually exclusive: P(A ∪ B) = P(A) + P(B).

Generalized Addition Rule

• For non-mutually exclusive events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Multiplication Rule

Definition

• Used to calculate the probability of both events occurring, denoted by P(A ∩ B).

For Independent Events

• Formula: P(A ∩ B) = P(A) * P(B).

For Conditional Probability

• Formula: P(B | A) = P(A ∩ B) / P(A).

Conditional Probability with Tree Diagrams

• Utilize tree diagrams to determine probabilities of dependent events.

Example: Drug Abuse and Hospitalization

• Calculate probabilities based on given data using a structured approach.

Contingency Tables

Definition

• Contingency tables summarize two categorical variables in a matrix format, showing frequencies for combinations of categories.

Example Calculations

• Various examples on computing probabilities using contingency tables to analyze pet ownership data.

Test Sensitivity and Specificity

Definitions

• Sensitivity: Ability of a test to correctly identify a positive condition (disease).

• Specificity: Ability of a test to correctly dismiss a negative condition.

Bayes' Theorem

• Formula for determining conditional probabilities involving two events.

• P(B | A) = P(B | A)⋅P(A) / (P(B | A)⋅P(A) + P(B | Aᶜ)⋅P(Aᶜ)).

Example of Bayesian Probability

• Illustrate calculations using Bayes’ theorem in context of real-life scenarios.