Lecture 7: Complex Events and Conditional Probabilities
|
|
- Clemence Carter
- 7 years ago
- Views:
Transcription
1 Lecture 7: Complex Events and Conditional Probabilities Chapter 5: Probability and Sampling Distributions 1/30/12 Lecture 7 1
2 5.1 Chance Experiments A chance experiment (or a random experiment) - An activity or situation whose outcomes depend on chance to some degree. - To determine whether a given activity qualifies as a chance experiment, ask: Will I get exactly the same result if the experiment is repeated more than once? If the answer is No, then it is a chance experiment. - Examples: - determine whether a metal part withstands a stress test, - record if it rains tomorrow, - measure the yield of a chemical reaction, - assess the potency of a pharmaceutical product, - measure the volume of water flow in a drainage system.
3 Events To compute probabilities, we need to work with the outcomes of chance experiments, which can be divided Into two types: Simple Events: individual outcomes Events: collections of simple events Sample Space: Obtained by taking all simple events together Venn Diagrams useful for depicting relationships between events
4 Venn Diagrams See from Pg 195, the textbook
5 Example Consider: An experiment of tossing a coin three times Simple Events: e.g. HHT, THT,. How many in total? 8 simple events: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT What is the sample space? {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} What is event A that we get at least two heads? A={HHT, HTH, THH, HHH}
6 Example 5.1 Conducting a series of stress tests on four metal parts What is the sample space? Event A: at least two parts pass the stress test Event B: at most two parts pass the stress test Event C: exactly three parts pass the test How to express all the above 4 events? (Sample space is also one kind of event) 1/30/12 Lecture 7 6
7 Sample Space: Continued S = {PPPP, PPPF,PPFP,PFPP, FPPP, PPFF,PFPF, PFFP, FPPF, FPFP, FFPP, PFFF,FPFF, FFPF, FFFP, FFFF}, by the tree diagram. Event A: at least two parts pass the stress test, so A={PPPP, PPPF,PPFP,PFPP, FPPP, PPFF,PFPF, PFFP, FPPF, FPFP, FFPP} Event B: at most two parts pass the stress test, thus B={PPFF,PFPF, PFFP, FPPF, FPFP, FFPP, PFFF,FPFF, FFPF, FFFP, FFFF} Event C: exactly three parts pass the test, therefore 1/30/12 C={PPPF, PPFP, Lecture 7 PFPP, FPPP} 7
8 A Tree Diagram See it from Fig. 5.2 on Pg 194 1/30/12 Lecture 7 8
9 Complex Events Let A and B be any two events of a chance experiment Event A or B: consists of all simple events that are contained in either A or B. i.e. At least one of A or B occur. Event A and B: consists of all simple events that A and B have in common. i.e. Both A and B occur. Event A (Complement of A): consists of all simple events that are NOT contained in A Disjoint Events (Mutually Exclusive): two events that have no simple events in common 1/30/12 Self-reading for generalized Lecture 7 definition (Pg 197) 9
10 Still, Example 5.1 Review: Conducting a series of stress tests on four metal parts 1. Event A: at least two parts pass the stress test A={PPPP, PPPF,PPFP,PFPP, FPPP, PPFF,PFPF, PFFP, FPPF, FPFP, FFPP} 2. Event B: at most two parts pass the stress test B={PPFF,PFPF, PFFP, FPPF, FPFP, FFPP, PFFF,FPFF, FFPF, FFFP, FFFF} What are: Event A and B, Event A or B, and Event A, respectively? A Case of Disjoint Events: Event A and Event A Based on definition of A complement 1/30/12 Lecture 7 10
11 Probability Concepts Probability Axioms: 0 P(A) 1 for any event A P(S) = 1, where S is the sample space To Determine Probabilities? Frequencies of occurrence From subjective estimates Assume simple events are equally likely Use density and mass functions Note: Depending on the circumstances, each method has its merits. 1/30/12 Lecture 7 11
12 Addition Rules Addition Rule - For any two disjoint events A and B, P(A or B) = P(A)+P(B) Complementary Events: P(A ) = 1 - P(A) General Addition Rule: (for any two events A and B) P(A or B) = P(A)+P(B)-P(A and B) 1/30/12 Lecture 7 12
13 Example 5.1 Conducting a series of stress tests on four metal parts Event A: at least two parts pass the stress test A={PPPP, PPPF, PPFP, PFPP, FPPP, PPFF, PFPF, PFFP, FPPF, FPFP, FFPP} Event B: at most two parts pass the stress test B={PPFF, PFPF, PFFP, FPPF, FPFP, FFPP, PFFF, FPFF, FFPF, FFFP, FFFF} P(A and B) =? P(A or B) =? P( A ) =? 1/30/12 Lecture 7 13
14 Conditional Probability Let A and B are two events with P(B)>0. The conditional probability of A occurring given that event B has already occurred is denoted by P(A B), where P(A B) = P (A and B) / P(B) 1/30/12 Lecture 7 14
15 Independent Events Two events, A and B, are independent events if the probability that either one occurs is not affected by the occurrence of the other. In this case, P (A and B) = P(A) P(B) Self-Reading: Air Filter on Pg 205, bottom; Independence for more than 2 events: Pg 206 If A and B are independent, so are A and B. Question: If two events, A and B, are independent, what is P(A B)? 1/30/12 Lecture 7 15
16 Example 5.6 Suppose that the switches A and B in a twocomponent series system are closed about 60% and 80% of the time, respectively. If we assume that the closing of switch A occurs independently of switch B, what is the probability that the entire circuit is closed? Suppose that A and B are independent events with P (A)=0.5 and P(B) = 0.4. Can A and B be mutually exclusive? 1/30/12 Lecture 7 16
17 After Class Review Sec 5.1 through 5.3 Start Hw#3 Lab#2 will be held this Wed Hw#2 is due today, 5pm. Office hour Dropbox beside my officedoor 1/30/12 Lecture 7 17
Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314
Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space
More informationProbability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to
More informationChapter 4 - Practice Problems 1
Chapter 4 - Practice Problems SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) Compare the relative frequency formula
More informationFor two disjoint subsets A and B of Ω, say that A and B are disjoint events. For disjoint events A and B we take an axiom P(A B) = P(A) + P(B)
Basic probability A probability space or event space is a set Ω together with a probability measure P on it. This means that to each subset A Ω we associate the probability P(A) = probability of A with
More informationPROBABILITY. The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE
PROBABILITY 53 Chapter 3 PROBABILITY The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE 3. Introduction In earlier Classes, we have studied the probability as
More informationSample Space and Probability
1 Sample Space and Probability Contents 1.1. Sets........................... p. 3 1.2. Probabilistic Models.................... p. 6 1.3. Conditional Probability................. p. 18 1.4. Total Probability
More informationIntroduction to Probability
3 Introduction to Probability Given a fair coin, what can we expect to be the frequency of tails in a sequence of 10 coin tosses? Tossing a coin is an example of a chance experiment, namely a process which
More informationChapter 4: Probability and Counting Rules
Chapter 4: Probability and Counting Rules Learning Objectives Upon successful completion of Chapter 4, you will be able to: Determine sample spaces and find the probability of an event using classical
More informationDecision Making Under Uncertainty. Professor Peter Cramton Economics 300
Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate
More informationV. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE
V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay $ to play. A penny and a nickel are flipped. You win $ if either
More informationBayesian Tutorial (Sheet Updated 20 March)
Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that
More informationChapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.
Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More informationStatistics in Geophysics: Introduction and Probability Theory
Statistics in Geophysics: Introduction and Steffen Unkel Department of Statistics Ludwig-Maximilians-University Munich, Germany Winter Term 2013/14 1/32 What is Statistics? Introduction Statistics is the
More informationChapter 4 - Practice Problems 2
Chapter - Practice Problems 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. 1) If you flip a coin three times, the
More informationPROBABILITY. Chapter. 0009T_c04_133-192.qxd 06/03/03 19:53 Page 133
0009T_c04_133-192.qxd 06/03/03 19:53 Page 133 Chapter 4 PROBABILITY Please stand up in front of the class and give your oral report on describing data using statistical methods. Does this request to speak
More informationProbability and statistical hypothesis testing. Holger Diessel holger.diessel@uni-jena.de
Probability and statistical hypothesis testing Holger Diessel holger.diessel@uni-jena.de Probability Two reasons why probability is important for the analysis of linguistic data: Joint and conditional
More informationHomework Assignment #2: Answer Key
Homework Assignment #2: Answer Key Chapter 4: #3 Assuming that the current interest rate is 3 percent, compute the value of a five-year, 5 percent coupon bond with a face value of $,000. What happens if
More informationProbability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1 www.math12.com
Probability --QUESTIONS-- Principles of Math - Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..
More informationChapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.
Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationPattern matching probabilities and paradoxes A new variation on Penney s coin game
Osaka Keidai Ronshu, Vol. 63 No. 4 November 2012 Pattern matching probabilities and paradoxes A new variation on Penney s coin game Yutaka Nishiyama Abstract This paper gives an outline of an interesting
More informationSTATISTICS HIGHER SECONDARY - SECOND YEAR. Untouchability is a sin Untouchability is a crime Untouchability is inhuman
STATISTICS HIGHER SECONDARY - SECOND YEAR Untouchability is a sin Untouchability is a crime Untouchability is inhuman TAMILNADU TEXTBOOK CORPORATION College Road, Chennai- 600 006 i Government of Tamilnadu
More information2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.
Math 142 September 27, 2011 1. How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are
More informationRandom variables, probability distributions, binomial random variable
Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that
More informationSTAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia
STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that
More informationChapter 7 Probability. Example of a random circumstance. Random Circumstance. What does probability mean?? Goals in this chapter
Homework (due Wed, Oct 27) Chapter 7: #17, 27, 28 Announcements: Midterm exams keys on web. (For a few hours the answer to MC#1 was incorrect on Version A.) No grade disputes now. Will have a chance to
More informationDefinition and Calculus of Probability
In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationChapter 3. Probability
Chapter 3 Probability Every Day, each us makes decisions based on uncertainty. Should you buy an extended warranty for your new DVD player? It depends on the likelihood that it will fail during the warranty.
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics Probability and Probability Distributions 1. Introduction 2. Probability 3. Basic rules of probability 4. Complementary events 5. Addition Law for
More informationIntroduction to Probability
LECTURE NOTES Course 6.041-6.431 M.I.T. FALL 2000 Introduction to Probability Dimitri P. Bertsekas and John N. Tsitsiklis Professors of Electrical Engineering and Computer Science Massachusetts Institute
More informationPractice Problems for Homework #8. Markov Chains. Read Sections 7.1-7.3. Solve the practice problems below.
Practice Problems for Homework #8. Markov Chains. Read Sections 7.1-7.3 Solve the practice problems below. Open Homework Assignment #8 and solve the problems. 1. (10 marks) A computer system can operate
More informationIAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION
IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION 1 WHAT IS STATISTICS? Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics
More informationECE302 Spring 2006 HW1 Solutions January 16, 2006 1
ECE302 Spring 2006 HW1 Solutions January 16, 2006 1 Solutions to HW1 Note: These solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics
More informationHomework 3 Solution, due July 16
Homework 3 Solution, due July 16 Problems from old actuarial exams are marked by a star. Problem 1*. Upon arrival at a hospital emergency room, patients are categorized according to their condition as
More informationMATH 140 Lab 4: Probability and the Standard Normal Distribution
MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes
More informationLecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University
Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments
More informationBasic Probability. Probability: The part of Mathematics devoted to quantify uncertainty
AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.
More informationBasic Probability Concepts
page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes
More informationSession 8 Probability
Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome
More informationNotes on Probability. Peter J. Cameron
Notes on Probability Peter J. Cameron ii Preface Here are the course lecture notes for the course MAS108, Probability I, at Queen Mary, University of London, taken by most Mathematics students and some
More informationX X X X X X X X X X X
AP Statistics Solutions to Packet 6 Probability The Study of Randomness The Idea of Probability Probability Models General Probability Rules HW #37 3, 4, 8, 11, 14, 15, 17, 18 6.3 SHAQ The basketball player
More informationProbabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I
Victor Adamchi Danny Sleator Great Theoretical Ideas In Computer Science Probability Theory I CS 5-25 Spring 200 Lecture Feb. 6, 200 Carnegie Mellon University We will consider chance experiments with
More informationName Please Print MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review Problems for Mid-Term 1, Fall 2012 (STA-120 Cal.Poly. Pomona) Name Please Print MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether
More informationDiscrete Structures for Computer Science
Discrete Structures for Computer Science Adam J. Lee adamlee@cs.pitt.edu 6111 Sennott Square Lecture #20: Bayes Theorem November 5, 2013 How can we incorporate prior knowledge? Sometimes we want to know
More informationWhat Do You Expect?: Homework Examples from ACE
What Do You Expect?: Homework Examples from ACE Investigation 1: A First Look at Chance, ACE #3, #4, #9, #31 Investigation 2: Experimental and Theoretical Probability, ACE #6, #12, #9, #37 Investigation
More informationLecture 2: Discrete Distributions, Normal Distributions. Chapter 1
Lecture 2: Discrete Distributions, Normal Distributions Chapter 1 Reminders Course website: www. stat.purdue.edu/~xuanyaoh/stat350 Office Hour: Mon 3:30-4:30, Wed 4-5 Bring a calculator, and copy Tables
More informationElements of probability theory
2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted
More informationProbability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes
Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all
More informationProbability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2
Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability
More informationSection 6.2 Definition of Probability
Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will
More informationLecture 10: Depicting Sampling Distributions of a Sample Proportion
Lecture 10: Depicting Sampling Distributions of a Sample Proportion Chapter 5: Probability and Sampling Distributions 2/10/12 Lecture 10 1 Sample Proportion 1 is assigned to population members having a
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationUnit 19: Probability Models
Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,
More informationCONTINGENCY (CROSS- TABULATION) TABLES
CONTINGENCY (CROSS- TABULATION) TABLES Presents counts of two or more variables A 1 A 2 Total B 1 a b a+b B 2 c d c+d Total a+c b+d n = a+b+c+d 1 Joint, Marginal, and Conditional Probability We study methods
More information3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved.
3.4 The Binomial Probability Distribution Copyright Cengage Learning. All rights reserved. The Binomial Probability Distribution There are many experiments that conform either exactly or approximately
More informationA Few Basics of Probability
A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study
More informationSTA 371G: Statistics and Modeling
STA 371G: Statistics and Modeling Decision Making Under Uncertainty: Probability, Betting Odds and Bayes Theorem Mingyuan Zhou McCombs School of Business The University of Texas at Austin http://mingyuanzhou.github.io/sta371g
More informationECE-316 Tutorial for the week of June 1-5
ECE-316 Tutorial for the week of June 1-5 Problem 35 Page 176: refer to lecture notes part 2, slides 8, 15 A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same
More informationConditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of conditional probability and independence
More informationReliability Applications (Independence and Bayes Rule)
Reliability Applications (Independence and Bayes Rule ECE 313 Probability with Engineering Applications Lecture 5 Professor Ravi K. Iyer University of Illinois Today s Topics Review of Physical vs. Stochastic
More informationThe Calculus of Probability
The Calculus of Probability Let A and B be events in a sample space S. Partition rule: P(A) = P(A B) + P(A B ) Example: Roll a pair of fair dice P(Total of 10) = P(Total of 10 and double) + P(Total of
More informationChapter 4 Probability: Overview; Basic Concepts of Probability; Addition Rule; Multiplication Rule: Basics; Multiplication Rule: Complements and
Chapter 4 Probability: Overview; Basic Concepts of Probability; Addition Rule; Multiplication Rule: Basics; Multiplication Rule: Complements and Conditional Probability; Probabilities Through Simulation;
More informationAP Stats - Probability Review
AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose
More informationChapter 5 A Survey of Probability Concepts
Chapter 5 A Survey of Probability Concepts True/False 1. Based on a classical approach, the probability of an event is defined as the number of favorable outcomes divided by the total number of possible
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,
More informationComplement. If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A.
Complement If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A. For example, if A is the event UNC wins at least 5 football games, then A c is the
More information36 Odds, Expected Value, and Conditional Probability
36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face
More informationProbability. a number between 0 and 1 that indicates how likely it is that a specific event or set of events will occur.
Probability Probability Simple experiment Sample space Sample point, or elementary event Event, or event class Mutually exclusive outcomes Independent events a number between 0 and 1 that indicates how
More informationData Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1
Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2011 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields
More informationSTAT 35A HW2 Solutions
STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },
More informationProbability & Probability Distributions
Probability & Probability Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Probability & Probability Distributions p. 1/61 Probability & Probability Distributions Elementary Probability Theory Definitions
More informationProbability. Section 9. Probability. Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space)
Probability Section 9 Probability Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space) In this section we summarise the key issues in the basic probability
More informationMath 141. Lecture 2: More Probability! Albyn Jones 1. jones@reed.edu www.people.reed.edu/ jones/courses/141. 1 Library 304. Albyn Jones Math 141
Math 141 Lecture 2: More Probability! Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Outline Law of total probability Bayes Theorem the Multiplication Rule, again Recall
More informationProbability definitions
Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a data-generating
More informationChapter 13 & 14 - Probability PART
Chapter 13 & 14 - Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14 - Probability 1 / 91 Why Should We Learn Probability Theory? Dr. Joseph
More informationMAS108 Probability I
1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper
More informationIf, under a given assumption, the of a particular observed is extremely. , we conclude that the is probably not
4.1 REVIEW AND PREVIEW RARE EVENT RULE FOR INFERENTIAL STATISTICS If, under a given assumption, the of a particular observed is extremely, we conclude that the is probably not. 4.2 BASIC CONCEPTS OF PROBABILITY
More informationECE302 Spring 2006 HW4 Solutions February 6, 2006 1
ECE302 Spring 2006 HW4 Solutions February 6, 2006 1 Solutions to HW4 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in
More informationIntroduction to CEE 3030 page 1
Uncertainty in Engineering Analysis An Introduction By Gilberto E. Urroz, August 2003/updated August 2005 This course on uncertainty in engineering analysis can also be referred to as probability and statistics
More informationProbability for Estimation (review)
Probability for Estimation (review) In general, we want to develop an estimator for systems of the form: x = f x, u + η(t); y = h x + ω(t); ggggg y, ffff x We will primarily focus on discrete time linear
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected
More informationSTA 256: Statistics and Probability I
Al Nosedal. University of Toronto. Fall 2014 1 2 3 4 5 My momma always said: Life was like a box of chocolates. You never know what you re gonna get. Forrest Gump. Experiment, outcome, sample space, and
More informationUNIT 7A 118 CHAPTER 7: PROBABILITY: LIVING WITH THE ODDS
11 CHAPTER 7: PROBABILITY: LIVING WITH THE ODDS UNIT 7A TIME OUT TO THINK Pg. 17. Birth orders of BBG, BGB, and GBB are the outcomes that produce the event of two boys in a family of. We can represent
More informationChapter10. Probability. Contents: A Experimental probability
Chapter10 Proaility Contents: A Experimental proaility B Proailities from data C Life tales D Sample spaces E Theoretical proaility F Using 2-dimensional grids G Compound events H Events and Venn diagrams
More informationSTATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS
STATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS 1. If two events (both with probability greater than 0) are mutually exclusive, then: A. They also must be independent. B. They also could
More informationInformation Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding
More informationChapter 7 Probability and Statistics
Chapter 7 Probability and Statistics In this chapter, students develop an understanding of data sampling and making inferences from representations of the sample data, with attention to both measures of
More informationWithout data, all you are is just another person with an opinion.
OCR Statistics Module Revision Sheet The S exam is hour 30 minutes long. You are allowed a graphics calculator. Before you go into the exam make sureyou are fully aware of the contents of theformula booklet
More informationTOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE. Topic P2: Sample Space and Assigning Probabilities
TOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE Roulette is one of the most popular casino games. The name roulette is derived from the French word meaning small
More informationSection 5-3 Binomial Probability Distributions
Section 5-3 Binomial Probability Distributions Key Concept This section presents a basic definition of a binomial distribution along with notation, and methods for finding probability values. Binomial
More information8.8. Probability 8-1. Random Variables EXAMPLE 1 EXAMPLE 2
8.8 Proaility The outcome of some events, such as a heavy rock falling from a great height, can e modeled so that we can predict with high accuracy what will happen. On the other hand, many events have
More informationDistributions. GOALS When you have completed this chapter, you will be able to: 1 Define the terms probability distribution and random variable.
6 Discrete GOALS When you have completed this chapter, you will be able to: Probability 1 Define the terms probability distribution and random variable. 2 Distinguish between discrete and continuous probability
More informationInformatics 2D Reasoning and Agents Semester 2, 2015-16
Informatics 2D Reasoning and Agents Semester 2, 2015-16 Alex Lascarides alex@inf.ed.ac.uk Lecture 29 Decision Making Under Uncertainty 24th March 2016 Informatics UoE Informatics 2D 1 Where are we? Last
More informationProblem sets for BUEC 333 Part 1: Probability and Statistics
Problem sets for BUEC 333 Part 1: Probability and Statistics I will indicate the relevant exercises for each week at the end of the Wednesday lecture. Numbered exercises are back-of-chapter exercises from
More informationDiscrete Probability Distributions
blu497_ch05.qxd /5/0 :0 PM Page 5 Confirming Pages C H A P T E R 5 Discrete Probability Distributions Objectives Outline After completing this chapter, you should be able to Introduction Construct a probability
More informationLecture 4: BK inequality 27th August and 6th September, 2007
CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound
More informationLecture Notes for Introductory Probability
Lecture Notes for Introductory Probability Janko Gravner Mathematics Department University of California Davis, CA 9566 gravner@math.ucdavis.edu June 9, These notes were started in January 9 with help
More informationSection 6-5 Sample Spaces and Probability
492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)
More informationUniversity of Chicago Graduate School of Business. Business 41000: Business Statistics Solution Key
Name: OUTLINE SOLUTIONS University of Chicago Graduate School of Business Business 41000: Business Statistics Solution Key Special Notes: 1. This is a closed-book exam. You may use an 8 11 piece of paper
More information