Chapter 5: Probability in Our Daily Lives
|
|
- Buddy Sutton
- 5 years ago
- Views:
Transcription
1 Chapter 5: in Our Daily Lives These notes reflect material from our text, Statistics: The Art and Science of Learning from Data, Third Edition, by Alan Agresti and Catherine Franklin, published by Pearson, quantifies randomness. It is a formal framework with a very specific vocabulary and notation. Imagine an experiment with a specific set of outcomes (say, flipping a fair coin twice). S is the sample space of all possible outcomes. Subsets of S are called events and are denoted with letters like A and B. The empty set, φ, is the event that contains no outcomes. Two events are disjoint if their intersection is empty. The Russian mathematician Kolmogorov helped to clarify the essential properties of a probability function, P P (S) = 1 for the entire sample space S 0 P (A) 1 for any event A S P ( n i=1 A i) = n i=1 P (A i) for disjoint events A i First examples : flip a coin, flip three coins, roll a die, roll two dice If you roll a die once the result is completely uncertain, because the individual outcomes are equally likely. But now begin to methodically roll the die and after each toss calculate the total number of 6 s observed so far divided by the total number of rolls at this point. Call this a cumulative proportion and graph these cumulative proportions for a large number of rolls of the die, say 100,000 rolls. A computer did this and displayed the following graph. In this particular simulation, the first ten rolls of the die produced the sequence , where 1 means a 6 was rolled and 0 means something else appeared. Calculate the first ten cumulative sums for this short sequence and compare your results to the following chart. What is the height of the dotted red line? p^n ,000 10, ,000 n (number of rolls) Fig. Cumulative proportions of a 6 in 100,000 rolls of a fair die, from OpenIntro Statistics, chapter 2 Display discrete probabilities in a table Flip a fair coin outcome h t probability Spring 2015 Page 1 of 7
2 Venn diagram A B Rules of Mutually exclusive events. A B = φ Unions. P (A B) = P (A) + P (B) P (A B). Complements. P (A c ) = 1 P (A). Independent events. P (A B) = P (A)P (B) when A and B are independent. Conditional probability. P (A B) = P (A B)/P (B) when P (B) 0 Intersections. P (A B) = P (A B)P (B) Spring 2015 Page 2 of 7
3 Contingency tables and conditional probabilities Vocabulary for diagnostic testing, S medical state present, P OS test positive : sensitivity P (P OS S), specificity P (NEG S c ), incidence P (S) Consider the Triple Blood Test for Down Syndrome (Agresti and Franklin, chapter 5, pp ) Blood Test Status P OS NEG T otal D (Down) D c (unaffected) T otal Calculate the following probabilities based on the figures in this study: sensitivity P (P OS D), specificity P (NEG D c ), incidence P (D) false positives P (P OS D c ), false negatives P (NEG D) An individual being tested would be most concerned about P (D P OS). What is this probability? Why is it so small? Hint: Calculate P (D c P OS). Again, an individual being tested would want to know P (D NEG). How would that probability compare to the a priori P (D)? Triple Blood Test POS NEG status unaffected Down blood test Spring 2015 Page 3 of 7
4 Using R to Compute Conditional Probabilities Construct a data frame named down to represent the Down Syndrome contingency table, and then use addmargins(down) to compute its row and column totals. down <- c(48, 1307, 6, 3921) dim(down) <- c(2, 2) dimnames(down) <- list(status=c("down", "unaffected"), "blood test"=c("pos", "neg")) down # status pos neg # down 48 6 # unaffected addmargins(down) # status pos neg Sum # down # unaffected # Sum Then prop.table(down, 1) will divide each row by its row sum. The numbers in each row are conditional probabilities. And prop.table(down, 2) will divide each column by its column sum. The numbers in each column are conditional probabilities. Therefore, each of the eight numbers shown below is a conditional probability of the form P (A B) for some A and B. Identify the correct A and B for each number. prop.table(down, 1) # status pos neg # down # unaffected prop.table(down, 2) # status pos neg # down # unaffected What values do these tables indicate for P (pos down) and P (down pos)? Spring 2015 Page 4 of 7
5 Boston Smallpox Epidemic of 1721 The following contingency table (OpenIntro Statistics, pp.83 87) refers to the Boston smallpox epidemic of A total of 6224 residents of Boston contracted smallpox in this epidemic and 850 of them died. The epidemic was marked by vigorous public debate of the value (or lack thereof) of a type of inoculation known as variolation (which was dangerous). The Reverend Cotton Mather advocated inoculation but the physician William Douglass was firmly against it. See the article in Harvard s Contagion for more details. An effective smallpox vaccination procedure was eventually demonstrated by Edward Jenner in England in 1796, and succeeding efforts to eradicate smallpox from the world were finally declared to be successful in 1980 by the World Health Organization. Cotton Mather, on the other hand, lives on in infamy for his role in the Salem witch trials. Inoculated Result yes no T otal lived died T otal Smallpox Epidemic, Boston, 1721 yes no died result lived innoculated Spring 2015 Page 5 of 7
6 Tree Diagrams The following tree diagram, generated by OpenIntro software, summarizes the relevant statistics for the Boston smallpox epidemic of Here Inoculated is a categorical explanatory variable with levels yes and no. In the Inoculated column of the tree diagram are the probabilities P (yes) and P (no). The categorical response variable Result has levels lived and died. The conditional probabilities in the Result column are P (lived yes), P (died yes), P (lived no), P (died no). The probabilities calculated by the software in the third column are P (lived and yes), P (died and yes), P (lived and no), P (died and no), because P (A) P (B A) = P (A B). Innoculated yes, Result lived, died, * = * = no, lived, died, * = * = Fig. Smallpox in Boston, 1721, from OpenIntro Statistics, chapter 2, pp Spring 2015 Page 6 of 7
7 Exercises We will attempt to solve some of the following exercises as a community project in class today. Finish these solutions as homework exercises, write them up carefully and clearly, and hand them in at the beginning of class next Friday. Exercises for Chapter 5: 5.10 (coin), 5.12 (stock market), 5.21 (risky behavior), 5.23 (seat belts), 5.34 (free throws), 5.37 (Down), 5.38 (job), 5.40 (serves), 5.50 (Masters), 5.57 (mammogram) Class work 5a probability Exercises from Chapter 5: 5.10 (coin), 5.12 (stock market), 5.21 (risky behavior), 5.23 (seat belts), 5.34 (free throws) Class work 5b probability Exercises from Chapter 5: 5.37 (Down), 5.38 (job), 5.40 (serves), 5.50 (Masters), 5.57 (mammogram) Spring 2015 Page 7 of 7
Chapter 7: Sampling Distributions
Chapter 7: Sampling Distributions These notes reflect material from our text, Statistics: The Art and Science of Learning from Data, Third Edition, by Alan Agresti and Catherine Franklin, published by
More informationWeek 2. Section Texas A& M University. Department of Mathematics Texas A& M University, College Station 22 January-24 January 2019
Week 2 Section 1.2-1.4 Texas A& M University Department of Mathematics Texas A& M University, College Station 22 January-24 January 2019 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week2 1
More informationChapter 6: Probability The Study of Randomness
Chapter 6: Probability The Study of Randomness 6.1 The Idea of Probability 6.2 Probability Models 6.3 General Probability Rules 1 Simple Question: If tossing a coin, what is the probability of the coin
More information4. Probability of an event A for equally likely outcomes:
University of California, Los Angeles Department of Statistics Statistics 110A Instructor: Nicolas Christou Probability Probability: A measure of the chance that something will occur. 1. Random experiment:
More informationProbability- describes the pattern of chance outcomes
Chapter 6 Probability the study of randomness Probability- describes the pattern of chance outcomes Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long
More informationEcon 325: Introduction to Empirical Economics
Econ 325: Introduction to Empirical Economics Lecture 2 Probability Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 3-1 3.1 Definition Random Experiment a process leading to an uncertain
More informationChapter 1 (Basic Probability)
Chapter 1 (Basic Probability) What is probability? Consider the following experiments: 1. Count the number of arrival requests to a web server in a day. 2. Determine the execution time of a program. 3.
More informationStatistics 251: Statistical Methods
Statistics 251: Statistical Methods Probability Module 3 2018 file:///volumes/users/r/renaes/documents/classes/lectures/251301/renae/markdown/master%20versions/module3.html#1 1/33 Terminology probability:
More informationProbability Theory and Applications
Probability Theory and Applications Videos of the topics covered in this manual are available at the following links: Lesson 4 Probability I http://faculty.citadel.edu/silver/ba205/online course/lesson
More informationRecap. The study of randomness and uncertainty Chances, odds, likelihood, expected, probably, on average,... PROBABILITY INFERENTIAL STATISTICS
Recap. Probability (section 1.1) The study of randomness and uncertainty Chances, odds, likelihood, expected, probably, on average,... PROBABILITY Population Sample INFERENTIAL STATISTICS Today. Formulation
More information1 Probability Theory. 1.1 Introduction
1 Probability Theory Probability theory is used as a tool in statistics. It helps to evaluate the reliability of our conclusions about the population when we have only information about a sample. Probability
More informationThe probability of an event is viewed as a numerical measure of the chance that the event will occur.
Chapter 5 This chapter introduces probability to quantify randomness. Section 5.1: How Can Probability Quantify Randomness? The probability of an event is viewed as a numerical measure of the chance that
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 14 Introduction One of the key properties of coin flips is independence: if you flip a fair coin ten times and get ten
More informationMath 1313 Experiments, Events and Sample Spaces
Math 1313 Experiments, Events and Sample Spaces At the end of this recording, you should be able to define and use the basic terminology used in defining experiments. Terminology The next main topic in
More informationChapter 5 : Probability. Exercise Sheet. SHilal. 1 P a g e
1 P a g e experiment ( observing / measuring ) outcomes = results sample space = set of all outcomes events = subset of outcomes If we collect all outcomes we are forming a sample space If we collect some
More informationORF 245 Fundamentals of Statistics Chapter 5 Probability
ORF 245 Fundamentals of Statistics Chapter 5 Probability Robert Vanderbei Oct 2015 Slides last edited on October 14, 2015 http://www.princeton.edu/ rvdb Sample Spaces (aka Populations) and Events When
More informationMA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
More informationORF 245 Fundamentals of Statistics Chapter 1 Probability
ORF 245 Fundamentals of Statistics Chapter 1 Probability Robert Vanderbei Sept 2014 Slides last edited on September 19, 2014 http://www.princeton.edu/ rvdb Course Info Prereqs: Textbook: Three semesters
More informationChapter 14. From Randomness to Probability. Copyright 2012, 2008, 2005 Pearson Education, Inc.
Chapter 14 From Randomness to Probability Copyright 2012, 2008, 2005 Pearson Education, Inc. Dealing with Random Phenomena A random phenomenon is a situation in which we know what outcomes could happen,
More information4 Lecture 4 Notes: Introduction to Probability. Probability Rules. Independence and Conditional Probability. Bayes Theorem. Risk and Odds Ratio
4 Lecture 4 Notes: Introduction to Probability. Probability Rules. Independence and Conditional Probability. Bayes Theorem. Risk and Odds Ratio Wrong is right. Thelonious Monk 4.1 Three Definitions of
More informationChapter 8: Sampling Variability and Sampling Distributions
Chapter 8: Sampling Variability and Sampling Distributions These notes reflect material from our text, Statistics, Learning from Data, First Edition, by Roxy Peck, published by CENGAGE Learning, 2015.
More informationVenn Diagrams; Probability Laws. Notes. Set Operations and Relations. Venn Diagram 2.1. Venn Diagrams; Probability Laws. Notes
Lecture 2 s; Text: A Course in Probability by Weiss 2.4 STAT 225 Introduction to Probability Models January 8, 2014 s; Whitney Huang Purdue University 2.1 Agenda s; 1 2 2.2 Intersection: the intersection
More informationProbability: Terminology and Examples Class 2, Jeremy Orloff and Jonathan Bloom
1 Learning Goals Probability: Terminology and Examples Class 2, 18.05 Jeremy Orloff and Jonathan Bloom 1. Know the definitions of sample space, event and probability function. 2. Be able to organize a
More informationProbability: Part 1 Naima Hammoud
Probability: Part 1 Naima ammoud Feb 7, 2017 Motivation ossing a coin Rolling a die Outcomes: eads or ails Outcomes: 1, 2, 3, 4, 5 or 6 Defining Probability If I toss a coin, there is a 50% chance I will
More informationProperties of Probability
Econ 325 Notes on Probability 1 By Hiro Kasahara Properties of Probability In statistics, we consider random experiments, experiments for which the outcome is random, i.e., cannot be predicted with certainty.
More informationLast time. Numerical summaries for continuous variables. Center: mean and median. Spread: Standard deviation and inter-quartile range
Lecture 4 Last time Numerical summaries for continuous variables Center: mean and median Spread: Standard deviation and inter-quartile range Exploratory graphics Histogram (revisit modes ) Histograms Histogram
More informationSTAT 111 Recitation 1
STAT 111 Recitation 1 Linjun Zhang January 20, 2017 What s in the recitation This class, and the exam of this class, is a mix of statistical concepts and calculations. We are going to do a little bit of
More information3.2 Probability Rules
3.2 Probability Rules The idea of probability rests on the fact that chance behavior is predictable in the long run. In the last section, we used simulation to imitate chance behavior. Do we always need
More informationDiscrete Probability. Mark Huiskes, LIACS Probability and Statistics, Mark Huiskes, LIACS, Lecture 2
Discrete Probability Mark Huiskes, LIACS mark.huiskes@liacs.nl Probability: Basic Definitions In probability theory we consider experiments whose outcome depends on chance or are uncertain. How do we model
More informationProbability (special topic)
Chapter 2 Probability (special topic) Probability forms a foundation for statistics. You may already be familiar with many aspects of probability, however, formalization of the concepts is new for most.
More informationPreliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com
1 School of Oriental and African Studies September 2015 Department of Economics Preliminary Statistics Lecture 2: Probability Theory (Outline) prelimsoas.webs.com Gujarati D. Basic Econometrics, Appendix
More informationFundamentals of Probability CE 311S
Fundamentals of Probability CE 311S OUTLINE Review Elementary set theory Probability fundamentals: outcomes, sample spaces, events Outline ELEMENTARY SET THEORY Basic probability concepts can be cast in
More informationProbability. Chapter 1 Probability. A Simple Example. Sample Space and Probability. Sample Space and Event. Sample Space (Two Dice) Probability
Probability Chapter 1 Probability 1.1 asic Concepts researcher claims that 10% of a large population have disease H. random sample of 100 people is taken from this population and examined. If 20 people
More informationConditional Probability
Conditional Probability When we obtain additional information about a probability experiment, we want to use the additional information to reassess the probabilities of events given the new information.
More informationProbability (Devore Chapter Two)
Probability (Devore Chapter Two) 1016-345-01: Probability and Statistics for Engineers Fall 2012 Contents 0 Administrata 2 0.1 Outline....................................... 3 1 Axiomatic Probability 3
More informationSTAT 201 Chapter 5. Probability
STAT 201 Chapter 5 Probability 1 2 Introduction to Probability Probability The way we quantify uncertainty. Subjective Probability A probability derived from an individual's personal judgment about whether
More informationThe enumeration of all possible outcomes of an experiment is called the sample space, denoted S. E.g.: S={head, tail}
Random Experiment In random experiments, the result is unpredictable, unknown prior to its conduct, and can be one of several choices. Examples: The Experiment of tossing a coin (head, tail) The Experiment
More information1 Preliminaries Sample Space and Events Interpretation of Probability... 13
Summer 2017 UAkron Dept. of Stats [3470 : 461/561] Applied Statistics Ch 2: Probability Contents 1 Preliminaries 3 1.1 Sample Space and Events...........................................................
More informationAnnouncements. Topics: To Do:
Announcements Topics: In the Probability and Statistics module: - Sections 1 + 2: Introduction to Stochastic Models - Section 3: Basics of Probability Theory - Section 4: Conditional Probability; Law of
More informationThe Practice of Statistics Third Edition
The Practice of Statistics Third Edition Chapter 6: Probability and Simulation: The Study of Randomness Copyright 2008 by W. H. Freeman & Company Probability Rules True probability can only be found by
More information3.1 Events, Sample Spaces, and Probability
Chapter 3 Probability Probability is the tool that allows the statistician to use sample information to make inferences about or to describe the population from which the sample was drawn. 3.1 Events,
More informationProbabilistic models
Kolmogorov (Andrei Nikolaevich, 1903 1987) put forward an axiomatic system for probability theory. Foundations of the Calculus of Probabilities, published in 1933, immediately became the definitive formulation
More informationELEG 3143 Probability & Stochastic Process Ch. 1 Probability
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 1 Probability Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Applications Elementary Set Theory Random
More informationPERMUTATIONS, COMBINATIONS AND DISCRETE PROBABILITY
Friends, we continue the discussion with fundamentals of discrete probability in the second session of third chapter of our course in Discrete Mathematics. The conditional probability and Baye s theorem
More informationI - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability
What is Probability? the chance of an event occuring eg 1classical probability 2empirical probability 3subjective probability Section 2 - Probability (1) Probability - Terminology random (probability)
More informationSingle Maths B: Introduction to Probability
Single Maths B: Introduction to Probability Overview Lecturer Email Office Homework Webpage Dr Jonathan Cumming j.a.cumming@durham.ac.uk CM233 None! http://maths.dur.ac.uk/stats/people/jac/singleb/ 1 Introduction
More informationMATH 3C: MIDTERM 1 REVIEW. 1. Counting
MATH 3C: MIDTERM REVIEW JOE HUGHES. Counting. Imagine that a sports betting pool is run in the following way: there are 20 teams, 2 weeks, and each week you pick a team to win. However, you can t pick
More informationConditional Probability and Independence
Conditional Probability and Independence September 3, 2009 1 Restricting the Sample Space - Conditional Probability How do we modify the probability of an event in light of the fact that something is known?
More informationAxioms of Probability
Sample Space (denoted by S) The set of all possible outcomes of a random experiment is called the Sample Space of the experiment, and is denoted by S. Example 1.10 If the experiment consists of tossing
More informationAMS7: WEEK 2. CLASS 2
AMS7: WEEK 2. CLASS 2 Introduction to Probability. Probability Rules. Independence and Conditional Probability. Bayes Theorem. Risk and Odds Ratio Friday April 10, 2015 Probability: Introduction Probability:
More informationIntermediate Math Circles November 8, 2017 Probability II
Intersection of Events and Independence Consider two groups of pairs of events Intermediate Math Circles November 8, 017 Probability II Group 1 (Dependent Events) A = {a sales associate has training} B
More informationCS626 Data Analysis and Simulation
CS626 Data Analysis and Simulation Instructor: Peter Kemper R 104A, phone 221-3462, email:kemper@cs.wm.edu Today: Probability Primer Quick Reference: Sheldon Ross: Introduction to Probability Models 9th
More informationLecture 1. Chapter 1. (Part I) Material Covered in This Lecture: Chapter 1, Chapter 2 ( ). 1. What is Statistics?
Lecture 1 (Part I) Material Covered in This Lecture: Chapter 1, Chapter 2 (2.1 --- 2.6). Chapter 1 1. What is Statistics? 2. Two definitions. (1). Population (2). Sample 3. The objective of statistics.
More informationStatistics for Financial Engineering Session 2: Basic Set Theory March 19 th, 2006
Statistics for Financial Engineering Session 2: Basic Set Theory March 19 th, 2006 Topics What is a set? Notations for sets Empty set Inclusion/containment and subsets Sample spaces and events Operations
More informationElements of probability theory
The role of probability theory in statistics We collect data so as to provide evidentiary support for answers we give to our many questions about the world (and in our particular case, about the business
More informationCMPSCI 240: Reasoning about Uncertainty
CMPSCI 240: Reasoning about Uncertainty Lecture 2: Sets and Events Andrew McGregor University of Massachusetts Last Compiled: January 27, 2017 Outline 1 Recap 2 Experiments and Events 3 Probabilistic Models
More informationCIVL Why are we studying probability and statistics? Learning Objectives. Basic Laws and Axioms of Probability
CIVL 3103 Basic Laws and Axioms of Probability Why are we studying probability and statistics? How can we quantify risks of decisions based on samples from a population? How should samples be selected
More informationProbability Rules. MATH 130, Elements of Statistics I. J. Robert Buchanan. Fall Department of Mathematics
Probability Rules MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Introduction Probability is a measure of the likelihood of the occurrence of a certain behavior
More information4/17/2012. NE ( ) # of ways an event can happen NS ( ) # of events in the sample space
I. Vocabulary: A. Outcomes: the things that can happen in a probability experiment B. Sample Space (S): all possible outcomes C. Event (E): one outcome D. Probability of an Event (P(E)): the likelihood
More informationProbability deals with modeling of random phenomena (phenomena or experiments whose outcomes may vary)
Chapter 14 From Randomness to Probability How to measure a likelihood of an event? How likely is it to answer correctly one out of two true-false questions on a quiz? Is it more, less, or equally likely
More informationSTA 247 Solutions to Assignment #1
STA 247 Solutions to Assignment #1 Question 1: Suppose you throw three six-sided dice (coloured red, green, and blue) repeatedly, until the three dice all show different numbers. Assuming that these dice
More informationLECTURE 1. 1 Introduction. 1.1 Sample spaces and events
LECTURE 1 1 Introduction The first part of our adventure is a highly selective review of probability theory, focusing especially on things that are most useful in statistics. 1.1 Sample spaces and events
More informationLecture 2: Probability. Readings: Sections Statistical Inference: drawing conclusions about the population based on a sample
Lecture 2: Probability Readings: Sections 5.1-5.3 1 Introduction Statistical Inference: drawing conclusions about the population based on a sample Parameter: a number that describes the population a fixed
More informationSTATISTICAL INDEPENDENCE AND AN INVITATION TO THE Art OF CONDITIONING
STATISTICAL INDEPENDENCE AND AN INVITATION TO THE Art OF CONDITIONING Tutorial 2 STAT1301 Fall 2010 28SEP2010, MB103@HKU By Joseph Dong Look, imagine a remote village where there has been a long drought.
More informationLecture 1: Probability Fundamentals
Lecture 1: Probability Fundamentals IB Paper 7: Probability and Statistics Carl Edward Rasmussen Department of Engineering, University of Cambridge January 22nd, 2008 Rasmussen (CUED) Lecture 1: Probability
More information2.6 Tools for Counting sample points
2.6 Tools for Counting sample points When the number of simple events in S is too large, manual enumeration of every sample point in S is tedious or even impossible. (Example) If S contains N equiprobable
More informationRecitation 2: Probability
Recitation 2: Probability Colin White, Kenny Marino January 23, 2018 Outline Facts about sets Definitions and facts about probability Random Variables and Joint Distributions Characteristics of distributions
More informationDeep Learning for Computer Vision
Deep Learning for Computer Vision Lecture 3: Probability, Bayes Theorem, and Bayes Classification Peter Belhumeur Computer Science Columbia University Probability Should you play this game? Game: A fair
More informationMAT2377. Ali Karimnezhad. Version September 9, Ali Karimnezhad
MAT2377 Ali Karimnezhad Version September 9, 2015 Ali Karimnezhad Comments These slides cover material from Chapter 1. In class, I may use a blackboard. I recommend reading these slides before you come
More informationLecture 1: Basics of Probability
Lecture 1: Basics of Probability (Luise-Vitetta, Chapter 8) Why probability in data science? Data acquisition is noisy Sampling/quantization external factors: If you record your voice saying machine learning
More informationProbability: Sets, Sample Spaces, Events
Probability: Sets, Sample Spaces, Events Engineering Statistics Section 2.1 Josh Engwer TTU 01 February 2016 Josh Engwer (TTU) Probability: Sets, Sample Spaces, Events 01 February 2016 1 / 29 The Need
More informationMathematical Foundations of Computer Science Lecture Outline October 18, 2018
Mathematical Foundations of Computer Science Lecture Outline October 18, 2018 The Total Probability Theorem. Consider events E and F. Consider a sample point ω E. Observe that ω belongs to either F or
More informationProbability Experiments, Trials, Outcomes, Sample Spaces Example 1 Example 2
Probability Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application. However, probability models underlie
More informationNotes 1 Autumn Sample space, events. S is the number of elements in the set S.)
MAS 108 Probability I Notes 1 Autumn 2005 Sample space, events The general setting is: We perform an experiment which can have a number of different outcomes. The sample space is the set of all possible
More information3 PROBABILITY TOPICS
Chapter 3 Probability Topics 135 3 PROBABILITY TOPICS Figure 3.1 Meteor showers are rare, but the probability of them occurring can be calculated. (credit: Navicore/flickr) Introduction It is often necessary
More informationSTA 291 Lecture 8. Probability. Probability Rules. Joint and Marginal Probability. STA Lecture 8 1
STA 291 Lecture 8 Probability Probability Rules Joint and Marginal Probability STA 291 - Lecture 8 1 Union and Intersection Let A and B denote two events. The union of two events: A B The intersection
More informationChapter 7 Wednesday, May 26th
Chapter 7 Wednesday, May 26 th Random event A random event is an event that the outcome is unpredictable. Example: There are 45 students in this class. What is the probability that if I select one student,
More informationECE 340 Probabilistic Methods in Engineering M/W 3-4:15. Lecture 2: Random Experiments. Prof. Vince Calhoun
ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Lecture 2: Random Experiments Prof. Vince Calhoun Reading This class: Section 2.1-2.2 Next class: Section 2.3-2.4 Homework: Assignment 1: From the
More informationSTA Module 4 Probability Concepts. Rev.F08 1
STA 2023 Module 4 Probability Concepts Rev.F08 1 Learning Objectives Upon completing this module, you should be able to: 1. Compute probabilities for experiments having equally likely outcomes. 2. Interpret
More informationSTA111 - Lecture 1 Welcome to STA111! 1 What is the difference between Probability and Statistics?
STA111 - Lecture 1 Welcome to STA111! Some basic information: Instructor: Víctor Peña (email: vp58@duke.edu) Course Website: http://stat.duke.edu/~vp58/sta111. 1 What is the difference between Probability
More informationELEG 3143 Probability & Stochastic Process Ch. 1 Experiments, Models, and Probabilities
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 1 Experiments, Models, and Probabilities Dr. Jing Yang jingyang@uark.edu OUTLINE 2 Applications
More information2.4. Conditional Probability
2.4. Conditional Probability Objectives. Definition of conditional probability and multiplication rule Total probability Bayes Theorem Example 2.4.1. (#46 p.80 textbook) Suppose an individual is randomly
More informationCompound Events. The event E = E c (the complement of E) is the event consisting of those outcomes which are not in E.
Compound Events Because we are using the framework of set theory to analyze probability, we can use unions, intersections and complements to break complex events into compositions of events for which it
More informationCHAPTER 3 PROBABILITY TOPICS
CHAPTER 3 PROBABILITY TOPICS 1. Terminology In this chapter, we are interested in the probability of a particular event occurring when we conduct an experiment. The sample space of an experiment is the
More informationBusiness Statistics. Lecture 3: Random Variables and the Normal Distribution
Business Statistics Lecture 3: Random Variables and the Normal Distribution 1 Goals for this Lecture A little bit of probability Random variables The normal distribution 2 Probability vs. Statistics Probability:
More informationLecture 6 Probability
Lecture 6 Probability Example: When you toss a coin, there are only two possible outcomes, heads and tails. What if we toss a coin 4 times? Figure below shows the results of tossing a coin 5000 times twice.
More informationENGI 4421 Introduction to Probability; Sets & Venn Diagrams Page α 2 θ 1 u 3. wear coat. θ 2 = warm u 2 = sweaty! θ 1 = cold u 3 = brrr!
ENGI 4421 Introduction to Probability; Sets & Venn Diagrams Page 2-01 Probability Decision trees u 1 u 2 α 2 θ 1 u 3 θ 2 u 4 Example 2.01 θ 1 = cold u 1 = snug! α 1 wear coat θ 2 = warm u 2 = sweaty! θ
More informationProbabilistic models
Probabilistic models Kolmogorov (Andrei Nikolaevich, 1903 1987) put forward an axiomatic system for probability theory. Foundations of the Calculus of Probabilities, published in 1933, immediately became
More informationP [(E and F )] P [F ]
CONDITIONAL PROBABILITY AND INDEPENDENCE WORKSHEET MTH 1210 This worksheet supplements our textbook material on the concepts of conditional probability and independence. The exercises at the end of each
More informationTerm Definition Example Random Phenomena
UNIT VI STUDY GUIDE Probabilities Course Learning Outcomes for Unit VI Upon completion of this unit, students should be able to: 1. Apply mathematical principles used in real-world situations. 1.1 Demonstrate
More informationSTAT Chapter 3: Probability
Basic Definitions STAT 515 --- Chapter 3: Probability Experiment: A process which leads to a single outcome (called a sample point) that cannot be predicted with certainty. Sample Space (of an experiment):
More informationIntro to Stats Lecture 11
Outliers and influential points Intro to Stats Lecture 11 Collect data this week! Midterm is coming! Terms X outliers: observations outlying the overall pattern of the X- variable Y outliers: observations
More informationLecture Lecture 5
Lecture 4 --- Lecture 5 A. Basic Concepts (4.1-4.2) 1. Experiment: A process of observing a phenomenon that has variation in its outcome. Examples: (E1). Rolling a die, (E2). Drawing a card form a shuffled
More informationProbability and Sample space
Probability and Sample space We call a phenomenon random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions. The probability of any outcome
More informationProbability Notes. Definitions: The probability of an event is the likelihood of choosing an outcome from that event.
ability Notes Definitions: Sample Space: sample space is a set or collection of possible outcomes. Flipping a Coin: {Head, Tail} Rolling Two Die: {,,,, 6, 7, 8, 9, 0,, } Outcome: n outcome is an element
More informationMutually Exclusive Events
172 CHAPTER 3 PROBABILITY TOPICS c. QS, 7D, 6D, KS Mutually Exclusive Events A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes
More informationWhat is the probability of getting a heads when flipping a coin
Chapter 2 Probability Probability theory is a branch of mathematics dealing with chance phenomena. The origins of the subject date back to the Italian mathematician Cardano about 1550, and French mathematicians
More informationMATH MW Elementary Probability Course Notes Part I: Models and Counting
MATH 2030 3.00MW Elementary Probability Course Notes Part I: Models and Counting Tom Salisbury salt@yorku.ca York University Winter 2010 Introduction [Jan 5] Probability: the mathematics used for Statistics
More information1 The Basic Counting Principles
1 The Basic Counting Principles The Multiplication Rule If an operation consists of k steps and the first step can be performed in n 1 ways, the second step can be performed in n ways [regardless of how
More informationChapter 2 PROBABILITY SAMPLE SPACE
Chapter 2 PROBABILITY Key words: Sample space, sample point, tree diagram, events, complement, union and intersection of an event, mutually exclusive events; Counting techniques: multiplication rule, permutation,
More information