Chapter 2.5 Random Variables and Probability The Modern View (cont.)
|
|
- Kathleen Brooks
- 5 years ago
- Views:
Transcription
1 Chapter 2.5 Random Variables and Probability The Modern View (cont.) I. Statistical Independence A crucially important idea in probability and statistics is the concept of statistical independence. Suppose that you have two random variables X1 and X2. These two random variables will be pairwise statistically independent if the realized value X1 takes on does not affect the probability of X2 taking on any possible value, and vice versa. Loosely speaking, movements in X1 cannot affect movements in X2. If the realized value of X1 somehow affected the probability that X2 would take on a particular value, then the two variables would be dependent. Independence greatly simplifies probability calculations for X1 since we need not worry about what is happening to X2. It is still somewhat murky, since we have not made clear how that we know two random variables are independent. But, a simple theorem called the factorization theorem (or factorization principle) makes things much simpler. Let A and B be random events. P(A and B) = P(A)P(B) if and only if A and B are independent. So, suppose that we are flipping a coin. This is a Bernoulli event. Assume the coin is fair and the probability of a heads is p, or P[H] = p. Now, flip it twice. What will be the probability of first getting a head and then getting another head? That's easy. Since the two flips are independent, P[H1 and H2] = p 2. But, this is just P[H1]P[H2] and so P[H1 and H2] = P[H1]P[H2] and factorization works. Now, try a tail on the first and a tail on the second. This is P[T1 and T2] = (1-p) 2 = P[T1]P[T2], factorization again occurs. What about a head on the first flip and a tail on the second. This will also show factorization, as will a tail on the first and a head on the second. Factorization is a very useful way of looking at statistical independence. Next, consider flipping a coin twice again, with P[Head] = 1. The first flip is independent of the second flip. Let A = 1 if the first flip is a head, and zero otherwise. Let B = 1 if the second flip is a head, and zero otherwise. Now, how do we compute the probability of A+B? Obviously, A+B can
2 be 0, 1, or 2. To get A+B = 0, both A and B must be zero. Therefore, by the factorization theorem we multiply to get P[A+B=0] = P[A = 0 and B = 0] = (1-p) 2. Similarly, for A+B to be 2, both A = 1 and B = 1 must occur. Again, employing the factorization idea, we get P[A+B=2] = P[A=1 and B=1]= p 2. However, to get A+B = 1, there are two separate ways this can happen. The first way is A = 1 and B = 0, while the second way is A = 0 and B = 1. Two mutually exclusive ways of getting A+B = 1. In this case, we ADD the probabilities of the two different, mutually exclusive ways together. Thus, P[A+B = 1] = P[A = 1 and B = 0] + P[A = 0 and B = 1] = P[A=1]P[B=0] + P[A=0]P[B=1] = p(1-p) + (1-p)p = 2p(1-p) The probabilities of mutually exclusive ways of an event occurring add together (called the addition principle), while independent events multiply together (called the factorization principle). II. Statistical Dependence Sometimes we must deal with two or more random variables that are not statistically independent. The probability of one is affect by what happens to the other. Let's try to understand this with a simple example. Think of the following highly contrived set of circumstances for random variables A and B. A : You flip a coin. A = 1 if a head, zero otherwise where P[A=1] = p. B : If A is a head, you flip the coin again. If A is a tail, you roll a die where P[X = i] = 1/6 for i = 1,...,6. This is the second event. Calculate the pdf for A+B.
3 Here is the way we can calculate the pdf of A+B and draw the graph, as shown below. The important point to note is that the probability of B = 3 or 4 or 1 depends on the value A assumes. For example, if A = 1, then P[B=3] = 0, whereas if A = 0, then P[B = 3] = 1/6. For sure, the world is complicated, and these dependent random variables show that randomness can be complicated also. Not all sets of random variables follow simple rules of combination. Some require us to think carefully how probability is determined. Sometimes, this involves us in counting arguments. In all cases we look at two important criteria to help us in calculating probability. First, what are the different, mutually exclusive ways of getting a particular outcome? Second, for each of these ways, are they composed of independent events? In the graph above, we look at an outcome (A+B) = 1 and find that there are two ways this can happen. Namely, A = 1 and B = 0, and also A = 0 and B = 1. Note that P[A = 1 and B = 0] = p(1-p) and P[A = 0 and B = 1] = (1-p)/6. These are two mutually exclusive ways of getting A+B = 1, so we add them together to get P[A+B = 1] = p(1-p) + (1-p)/6. It will be useful for you to work your way through the graph above. Questions:
4 #1. Consider the case where you draw two cards from a 52 card deck without replacement. Now consider the random event that you have drawn two red cards. Explain why that the first and second draws are not statistically independent. #2. Assume A and B are independent random variables. Therefore, we know that P[A and B] =? #3. Flip a coin twice. Let E1 be the outcome from the first flip and E2 be the outcome from the second flip. Use the factorization principle to prove E1 and E2 and statistically independent. #4. Let A and B be random variables. But, suppose B becomes certain and no longer random. Show that A and B must now be statistical independent. #5. Consider the following Venn diagram showing the probabilities of A and B. Is the equation as it is written correct? Draw the diagram again assuming A and B are independent. How does the equation below the diagram change? #6. Use the following Venn diagram to calculate P[A], P[B], P[A & B], P[A or B]
5 #7. Use the following Venn diagram to calculate P[A], P[B], P[C], P[A & B &C], P[A & B or C], and P[A&C] #8. In blackjack (or the card game 21), it is claimed that people can raise the odds of winning by counting cards. Does this mean that draws at a black jack table are not independent? Explain.
Mutually 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 informationProbability Year 10. Terminology
Probability Year 10 Terminology Probability measures the chance something happens. Formally, we say it measures how likely is the outcome of an event. We write P(result) as a shorthand. An event is some
More informationProbability and Independence Terri Bittner, Ph.D.
Probability and Independence Terri Bittner, Ph.D. The concept of independence is often confusing for students. This brief paper will cover the basics, and will explain the difference between independent
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 informationProbability Year 9. Terminology
Probability Year 9 Terminology Probability measures the chance something happens. Formally, we say it measures how likely is the outcome of an event. We write P(result) as a shorthand. An event is some
More informationChapter 7: Section 7-1 Probability Theory and Counting Principles
Chapter 7: Section 7-1 Probability Theory and Counting Principles D. S. Malik Creighton University, Omaha, NE D. S. Malik Creighton University, Omaha, NE Chapter () 7: Section 7-1 Probability Theory and
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 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 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 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 informationLecture 1. ABC of Probability
Math 408 - Mathematical Statistics Lecture 1. ABC of Probability January 16, 2013 Konstantin Zuev (USC) Math 408, Lecture 1 January 16, 2013 1 / 9 Agenda Sample Spaces Realizations, Events Axioms of Probability
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 informationn N CHAPTER 1 Atoms Thermodynamics Molecules Statistical Thermodynamics (S.T.)
CHAPTER 1 Atoms Thermodynamics Molecules Statistical Thermodynamics (S.T.) S.T. is the key to understanding driving forces. e.g., determines if a process proceeds spontaneously. Let s start with entropy
More informationWhat is Probability? Probability. Sample Spaces and Events. Simple Event
What is Probability? Probability Peter Lo Probability is the numerical measure of likelihood that the event will occur. Simple Event Joint Event Compound Event Lies between 0 & 1 Sum of events is 1 1.5
More informationProblems and results for the ninth week Mathematics A3 for Civil Engineering students
Problems and results for the ninth week Mathematics A3 for Civil Engineering students. Production line I of a factor works 0% of time, while production line II works 70% of time, independentl of each other.
More informationUNIT 5 ~ Probability: What Are the Chances? 1
UNIT 5 ~ Probability: What Are the Chances? 1 6.1: Simulation Simulation: The of chance behavior, based on a that accurately reflects the phenomenon under consideration. (ex 1) Suppose we are interested
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 informationIndependence. P(A) = P(B) = 3 6 = 1 2, and P(C) = 4 6 = 2 3.
Example: A fair die is tossed and we want to guess the outcome. The outcomes will be 1, 2, 3, 4, 5, 6 with equal probability 1 6 each. If we are interested in getting the following results: A = {1, 3,
More informationConditional Probability and Bayes
Conditional Probability and Bayes Chapter 2 Lecture 5 Yiren Ding Shanghai Qibao Dwight High School March 9, 2016 Yiren Ding Conditional Probability and Bayes 1 / 13 Outline 1 Independent Events Definition
More informationLecture Notes 1 Basic Probability. Elements of Probability. Conditional probability. Sequential Calculation of Probability
Lecture Notes 1 Basic Probability Set Theory Elements of Probability Conditional probability Sequential Calculation of Probability Total Probability and Bayes Rule Independence Counting EE 178/278A: Basic
More informationLecture 10: Probability distributions TUESDAY, FEBRUARY 19, 2019
Lecture 10: Probability distributions DANIEL WELLER TUESDAY, FEBRUARY 19, 2019 Agenda What is probability? (again) Describing probabilities (distributions) Understanding probabilities (expectation) Partial
More informationPresentation on Theo e ry r y o f P r P o r bab a il i i l t i y
Presentation on Theory of Probability Meaning of Probability: Chance of occurrence of any event In practical life we come across situation where the result are uncertain Theory of probability was originated
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 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 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 information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 3 Probability Contents 1. Events, Sample Spaces, and Probability 2. Unions and Intersections 3. Complementary Events 4. The Additive Rule and Mutually Exclusive
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 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 informationBasic Concepts of Probability
Probability Probability theory is the branch of math that deals with random events Probability is used to describe how likely a particular outcome is in a random event the probability of obtaining heads
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 informationIf S = {O 1, O 2,, O n }, where O i is the i th elementary outcome, and p i is the probability of the i th elementary outcome, then
1.1 Probabilities Def n: A random experiment is a process that, when performed, results in one and only one of many observations (or outcomes). The sample space S is the set of all elementary outcomes
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 informationProbability Theory Review
Cogsci 118A: Natural Computation I Lecture 2 (01/07/10) Lecturer: Angela Yu Probability Theory Review Scribe: Joseph Schilz Lecture Summary 1. Set theory: terms and operators In this section, we provide
More informationDiscrete Probability. Chemistry & Physics. Medicine
Discrete Probability The existence of gambling for many centuries is evidence of long-running interest in probability. But a good understanding of probability transcends mere gambling. The mathematics
More informationLecture notes for probability. Math 124
Lecture notes for probability Math 124 What is probability? Probabilities are ratios, expressed as fractions, decimals, or percents, determined by considering results or outcomes of experiments whose result
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 informationBasic Concepts of Probability
Probability Probability theory is the branch of math that deals with unpredictable or random events Probability is used to describe how likely a particular outcome is in a random event the probability
More informationMath 140 Introductory Statistics
Math 140 Introductory Statistics 5.1 Models of random behavior Outcome: Result or answer obtained from a chance process. Event: Collection of outcomes. Probability: Number between 0 and 1 (0% and 100%).
More informationDept. of Linguistics, Indiana University Fall 2015
L645 Dept. of Linguistics, Indiana University Fall 2015 1 / 34 To start out the course, we need to know something about statistics and This is only an introduction; for a fuller understanding, you would
More informationA Event has occurred
Statistics and probability: 1-1 1. Probability Event: a possible outcome or set of possible outcomes of an experiment or observation. Typically denoted by a capital letter: A, B etc. E.g. The result of
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 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 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 informationMAS108 Probability I
1 BSc Examination 2008 By Course Units 2:30 pm, Thursday 14 August, 2008 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators.
More informationOutline Conditional Probability The Law of Total Probability and Bayes Theorem Independent Events. Week 4 Classical Probability, Part II
Week 4 Classical Probability, Part II Week 4 Objectives This week we continue covering topics from classical probability. The notion of conditional probability is presented first. Important results/tools
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 informationConditional Probability and Bayes Theorem (2.4) Independence (2.5)
Conditional Probability and Bayes Theorem (2.4) Independence (2.5) Prof. Tesler Math 186 Winter 2019 Prof. Tesler Conditional Probability and Bayes Theorem Math 186 / Winter 2019 1 / 38 Scenario: Flip
More informationSTOR Lecture 4. Axioms of Probability - II
STOR 435.001 Lecture 4 Axioms of Probability - II Jan Hannig UNC Chapel Hill 1 / 23 How can we introduce and think of probabilities of events? Natural to think: repeat the experiment n times under same
More informationChapter 3 : Conditional Probability and Independence
STAT/MATH 394 A - PROBABILITY I UW Autumn Quarter 2016 Néhémy Lim Chapter 3 : Conditional Probability and Independence 1 Conditional Probabilities How should we modify the probability of an event when
More informationIntroduction to Probability 2017/18 Supplementary Problems
Introduction to Probability 2017/18 Supplementary Problems Problem 1: Let A and B denote two events with P(A B) 0. Show that P(A) 0 and P(B) 0. A A B implies P(A) P(A B) 0, hence P(A) 0. Similarly B A
More informationMATH STUDENT BOOK. 12th Grade Unit 9
MATH STUDENT BOOK 12th Grade Unit 9 Unit 9 COUNTING PRINCIPLES MATH 1209 COUNTING PRINCIPLES INTRODUCTION 1. PROBABILITY DEFINITIONS, SAMPLE SPACES, AND PROBABILITY ADDITION OF PROBABILITIES 11 MULTIPLICATION
More informationChapter 2 Class Notes
Chapter 2 Class Notes Probability can be thought of in many ways, for example as a relative frequency of a long series of trials (e.g. flips of a coin or die) Another approach is to let an expert (such
More informationSTAT:5100 (22S:193) Statistical Inference I
STAT:5100 (22S:193) Statistical Inference I Week 3 Luke Tierney University of Iowa Fall 2015 Luke Tierney (U Iowa) STAT:5100 (22S:193) Statistical Inference I Fall 2015 1 Recap Matching problem Generalized
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 informationthe time it takes until a radioactive substance undergoes a decay
1 Probabilities 1.1 Experiments with randomness Wewillusethetermexperimentinaverygeneralwaytorefertosomeprocess that produces a random outcome. Examples: (Ask class for some first) Here are some discrete
More informationIntroduction to Probability
Introduction to Probability Content Experiments, Counting Rules, and Assigning Probabilities Events and Their Probability Some Basic Relationships of Probability Conditional Probability Bayes Theorem 2
More informationConditional Probability
Conditional Probability Idea have performed a chance experiment but don t know the outcome (ω), but have some partial information (event A) about ω. Question: given this partial information what s the
More informationOrigins of Probability Theory
1 16.584: INTRODUCTION Theory and Tools of Probability required to analyze and design systems subject to uncertain outcomes/unpredictability/randomness. Such systems more generally referred to as Experiments.
More informationProbability & Random Variables
& Random Variables Probability Probability theory is the branch of math that deals with random events, processes, and variables What does randomness mean to you? How would you define probability in your
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 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 informationToday we ll discuss ways to learn how to think about events that are influenced by chance.
Overview Today we ll discuss ways to learn how to think about events that are influenced by chance. Basic probability: cards, coins and dice Definitions and rules: mutually exclusive events and independent
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 informationYear 10 Mathematics Probability Practice Test 1
Year 10 Mathematics Probability Practice Test 1 1 A letter is chosen randomly from the word TELEVISION. a How many letters are there in the word TELEVISION? b Find the probability that the letter is: i
More informationComputing Probability
Computing Probability James H. Steiger October 22, 2003 1 Goals for this Module In this module, we will 1. Develop a general rule for computing probability, and a special case rule applicable when elementary
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 informationRandom processes. Lecture 17: Probability, Part 1. Probability. Law of large numbers
Random processes Lecture 17: Probability, Part 1 Statistics 10 Colin Rundel March 26, 2012 A random process is a situation in which we know what outcomes could happen, but we don t know which particular
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 informationIntroduction and basic definitions
Chapter 1 Introduction and basic definitions 1.1 Sample space, events, elementary probability Exercise 1.1 Prove that P( ) = 0. Solution of Exercise 1.1 : Events S (where S is the sample space) and are
More informationChapter 8: An Introduction to Probability and Statistics
Course S3, 200 07 Chapter 8: An Introduction to Probability and Statistics This material is covered in the book: Erwin Kreyszig, Advanced Engineering Mathematics (9th edition) Chapter 24 (not including
More information(6, 1), (5, 2), (4, 3), (3, 4), (2, 5), (1, 6)
Section 7.3: 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
More informationStatistics for Engineers
Statistics for Engineers Antony Lewis http://cosmologist.info/teaching/stat/ Starter question Have you previously done any statistics? 1. Yes 2. No 54% 46% 1 2 BOOKS Chatfield C, 1989. Statistics for
More information1. When applied to an affected person, the test comes up positive in 90% of cases, and negative in 10% (these are called false negatives ).
CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 8 Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According to clinical trials,
More informationDiscrete Random Variables
Discrete Random Variables An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Introduction The markets can be thought of as a complex interaction of a large number of random
More informationAxiomatic Foundations of Probability. Definition: Probability Function
Chapter 1 sections We will SKIP a number of sections Set theory SKIP Real number uncountability Definition of probability Finite sample spaces Counting methods Combinatorial methods SKIP tennis tournament
More informationName: Exam 2 Solutions. March 13, 2017
Department of Mathematics University of Notre Dame Math 00 Finite Math Spring 07 Name: Instructors: Conant/Galvin Exam Solutions March, 07 This exam is in two parts on pages and contains problems worth
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 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 informationChapter 4 - Introduction to Probability
Chapter 4 - Introduction to Probability Probability is a numerical measure of the likelihood that an event will occur. Probability values are always assigned on a scale from 0 to 1. A probability near
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 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 informationGrades 7 & 8, Math Circles 24/25/26 October, Probability
Faculty of Mathematics Waterloo, Ontario NL 3G1 Centre for Education in Mathematics and Computing Grades 7 & 8, Math Circles 4/5/6 October, 017 Probability Introduction Probability is a measure of how
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More informationIntroduction to Probability. Ariel Yadin. Lecture 1. We begin with an example [this is known as Bertrand s paradox]. *** Nov.
Introduction to Probability Ariel Yadin Lecture 1 1. Example: Bertrand s Paradox We begin with an example [this is known as Bertrand s paradox]. *** Nov. 1 *** Question 1.1. Consider a circle of radius
More informationAQA Statistics 1. Probability. Section 1: Introducing Probability. Notation
Notes and Examples AQA Statistics 1 Probability Section 1: Introducing Probability These notes contain subsections on Notation The complement of an event Probability of either one event or another Notation
More information13-5 Probabilities of Independent and Dependent Events
CCSS REASONING Determine whether the events are independent or dependent. Then find the probability. 6. In a game, you roll an even number on a die and then spin a spinner numbered 1 through 5 and get
More informationSTAT 285 Fall Assignment 1 Solutions
STAT 285 Fall 2014 Assignment 1 Solutions 1. An environmental agency sets a standard of 200 ppb for the concentration of cadmium in a lake. The concentration of cadmium in one lake is measured 17 times.
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 informationAlgorithms for Uncertainty Quantification
Algorithms for Uncertainty Quantification Tobias Neckel, Ionuț-Gabriel Farcaș Lehrstuhl Informatik V Summer Semester 2017 Lecture 2: Repetition of probability theory and statistics Example: coin flip Example
More informationAnnouncements. Lecture 5: Probability. Dangling threads from last week: Mean vs. median. Dangling threads from last week: Sampling bias
Recap Announcements Lecture 5: Statistics 101 Mine Çetinkaya-Rundel September 13, 2011 HW1 due TA hours Thursday - Sunday 4pm - 9pm at Old Chem 211A If you added the class last week please make sure to
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 informationLecture 2: Repetition of probability theory and statistics
Algorithms for Uncertainty Quantification SS8, IN2345 Tobias Neckel Scientific Computing in Computer Science TUM Lecture 2: Repetition of probability theory and statistics Concept of Building Block: Prerequisites:
More informationEcon 113. Lecture Module 2
Econ 113 Lecture Module 2 Contents 1. Experiments and definitions 2. Events and probabilities 3. Assigning probabilities 4. Probability of complements 5. Conditional probability 6. Statistical independence
More informationExample: Suppose we toss a quarter and observe whether it falls heads or tails, recording the result as 1 for heads and 0 for tails.
Example: Suppose we toss a quarter and observe whether it falls heads or tails, recording the result as 1 for heads and 0 for tails. (In Mathematical language, the result of our toss is a random variable,
More informationMath , Fall 2012: HW 5 Solutions
Math 230.0, Fall 202: HW 5 Solutions Due Thursday, October 4th, 202. Problem (p.58 #2). Let X and Y be the numbers obtained in two draws at random from a box containing four tickets labeled, 2, 3, 4. Display
More informationLecture 11: Information theory THURSDAY, FEBRUARY 21, 2019
Lecture 11: Information theory DANIEL WELLER THURSDAY, FEBRUARY 21, 2019 Agenda Information and probability Entropy and coding Mutual information and capacity Both images contain the same fraction of black
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 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 informationOutline. Probability. Math 143. Department of Mathematics and Statistics Calvin College. Spring 2010
Outline Math 143 Department of Mathematics and Statistics Calvin College Spring 2010 Outline Outline 1 Review Basics Random Variables Mean, Variance and Standard Deviation of Random Variables 2 More Review
More informationM378K In-Class Assignment #1
The following problems are a review of M6K. M7K In-Class Assignment # Problem.. Complete the definition of mutual exclusivity of events below: Events A, B Ω are said to be mutually exclusive if A B =.
More information