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3: Probability Topics

Last updated

Mar 17, 2025
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- 2.15: Solutions
- 3.0: Introduction to Probability

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You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study for an exam, you are using probability. In this chapter, you will learn how to solve probability problems using a systematic approach.

3.0: Introduction to Probability
It is often necessary to "guess" about the outcome of an event in order to make a decision. Politicians study polls to guess their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good.
3.1: Probability Terminology
This page explains probability, detailing the sample space of outcomes, events, and the range of probabilities (0 to 1). It highlights the Law of Large Numbers, representing outcomes through unions, intersections, and complements, and discusses conditional probabilities and odds, particularly in gambling.
3.2: Independent and Mutually Exclusive Events *
This page discusses probability concepts such as independent and dependent events, including definitions and examples of sampling with and without replacement. It emphasizes the importance of mutual exclusivity, illustrated by coin flips. Key topics include calculating joint and conditional probabilities, checking independence, and the relationships between events using practical examples.
3.3: Two Basic Rules of Probability
This page explains two key probability rules: the Multiplication Rule, which calculates the probability of two events happening together, and the Addition Rule, which assesses the chances of either event happening while considering overlaps. It notes that the multiplication rule simplifies for independent events, while the addition rule applies differently to mutually exclusive events. An example involving a student's library activities illustrates these concepts in practice.
3.4: Contingency Tables and Probability Trees
A contingency table provides a way of portraying data that can facilitate calculating probabilities. The table helps in determining conditional probabilities quite easily. The table displays sample values in relation to two different variables that may be dependent or contingent on one another.
3.5: Venn Diagrams
A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. Venn diagrams also help us to convert common English words into mathematical terms that help add precision.
3.6: Conditional Probability and Bayes' Rule
This page discusses conditional probability, detailing its definition and mathematical formulation, along with relevant examples like card drawing. It covers the Multiplication Law and Law of Total Probability for calculating probabilities across multiple events, and introduces Bayes' Rule for reverse conditional probabilities.
3.7: Independent Events
This page explains independence in conditional probability for pairs and larger collections of events. Two events, A and B, are independent if knowing one does not change the other's probability, meaning P(A|B) = P(A) and P(B|A) = P(B). This leads to P(A∩B) = P(A)P(B). For three or more events, independence can be pairwise (each pair is independent) or mutually independent (all combinations satisfy independence). While mutual independence guarantees pairwise independence, the reverse is not true.
3.8: Discrete Distribution Experiment (Worksheet) **
The student will use theoretical and empirical methods to estimate probabilities. The student will appraise the differences between the two estimates. The student will demonstrate an understanding of long-term relative frequencies.
3.9: Key Terms
This page provides a foundational overview of key terms in probability theory, covering concepts such as conditional probability, dependent and independent events, and mutually exclusive events. It discusses events and outcomes within a sample space, sampling methods, and essential operations like complement, union, and intersection. Additionally, it introduces visual tools like tree diagrams and Venn diagrams for graphical representation of these concepts.
3.10: Chapter Review
This page provides an overview of fundamental probability concepts, including terminology, independent and mutually exclusive events, and essential probability rules. It discusses sampling methods and introduces visual tools such as contingency tables, probability trees, and Venn diagrams. These tools assist in organizing data and clarifying relationships between events, enhancing the understanding of intersections, unions, complements, and conditional probabilities.
3.11: Formula Review
This page defines key probability terminology, including events $A$ and $B$ , sample space $S$ , and associated probabilities. It explains independent events (probabilities that multiply) and mutually exclusive events (no overlap). The page also outlines two fundamental rules: the multiplication rule for joint probabilities and the addition rule for the union of events.
3.12: Practice
3.13: Bringing It Together- Practice
This page summarizes a study on smoking habits among Californians and Hawaiians, highlighting self-reported ethnicity and cigarette consumption levels. It includes data on smokers categorized by daily consumption (1-10, 11-20, 21-30, and 31+ cigarettes). Additionally, it covers exercises on calculating probabilities, analyzing gender and age relationships among U.S. licensed drivers, commuting patterns, and a coin toss experiment.
3.14: Homework
This page covers a range of probability concepts including event definitions, calculation methods, independent and mutually exclusive events, and statistical interpretations. It features exercises on various scenarios, such as survey data, dice rolls, coin tosses, and U.S. political affiliations, requiring calculations and justifications.
3.15: Bringing It Together- Homework
This page provides statistical insights on San Francisco 49ers and Dallas Cowboys players' weights and shirt numbers, alongside probabilities linked to male cancer risks, including false positives. It includes exercises for calculating probabilities, such as unions and intersections of events, though some questions remain unanswered due to lack of information. The page highlights the significance of probability and conditional probability in interpreting statistical data.
3.16: Reference
This page offers a comprehensive list of references and sources on statistical and data analysis topics, including U.S. teachers' well-being, probability, contingency tables, and health statistics. It details authors, titles, publication years, and access URLs, featuring organizations like Gallup, the U.S. Census Bureau, and the American Red Cross. The sources cover public opinion, health, and crime statistics, while also recommending various online platforms for locating datasets.
3.17: Solutions
This page covers a range of probability concepts, including conditional and joint probabilities, independence of events, and the calculation of probabilities across various contexts. It explores scenarios like demographic characteristics, gambling, and specific examples such as card and coin selections, as well as blood type distributions.

Curated and edited by Kristin Kuter | Saint Mary's College, Notre Dame, IN

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