CHAPTER 5 Probability: What Are the Chances?
|
|
|
- Morgan Melton
- 5 years ago
- Views:
Transcription
1 CHAPTER 5 Probability: What Are the Chances? 5.3 Conditional Probability and Independence The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers
2 Conditional Probability and Independence Learning Objectives After this section, you should be able to: CALCULATE and INTERPRET conditional probabilities. USE the general multiplication rule to CALCULATE probabilities. USE tree diagrams to MODEL a chance process and CALCULATE probabilities involving two or more events. DETERMINE if two events are independent. When appropriate, USE the multiplication rule for independent events to COMPUTE probabilities. The Practice of Statistics, 5 th Edition 2
3 What is Conditional Probability? The probability we assign to an event can change if we know that some other event has occurred. This idea is the key to many applications of probability. When we are trying to find the probability that one event will happen under the condition that some other event is already known to have occurred, we are trying to determine a conditional probability. The probability that one event happens given that another event is already known to have happened is called a conditional probability. Suppose we know that event A has happened. Then the probability that event B happens given that event A has happened is denoted by P(B A). Read as given that or under the condition that The Practice of Statistics, 5 th Edition 3
4 Calculating Conditional Probabilities Calculating Conditional Probabilities To find the conditional probability P(A B), use the formula P(A B) = P(AÇ B) P(B) The conditional probability P(B A) is given by P(B A) = P(BÇ A) P(A) The Practice of Statistics, 5 th Edition 4
5 Calculating Conditional Probabilities Consider the two-way table on page 321. Define events E: the grade comes from an EPS course, and L: the grade is lower than a B. Total Total Find P(L) Find P(E L) Find P(L E) P(L) = 3656 / = P(E L) = 800 / 3656 = P(L E) = 800 / 1600 = The Practice of Statistics, 5 th Edition 5
6 The General Multiplication Rule General Multiplication Rule The probability that events A and B both occur can be found using the general multiplication rule P(A B) = P(A) P(B A) where P(B A) is the conditional probability that event B occurs given that event A has already occurred. In words, this rule says that for both of two events to occur, first one must occur, and then given that the first event has occurred, the second must occur. The Practice of Statistics, 5 th Edition 6
7 Tree Diagrams The general multiplication rule is especially useful when a chance process involves a sequence of outcomes. In such cases, we can use a tree diagram to display the sample space. Consider flipping a coin twice. What is the probability of getting two heads? Sample Space: HH HT TH TT So, P(two heads) = P(HH) = 1/4 The Practice of Statistics, 5 th Edition 7
8 Example: Tree Diagrams The Pew Internet and American Life Project finds that 93% of teenagers (ages 12 to 17) use the Internet, and that 55% of online teens have posted a profile on a social-networking site. What percent of teens are online and have posted a profile? P(online) = 0.93 P(profile online) = 0.55 P(online and have profile) = P(online) P(profile online) = (0.93)(0.55) = % of teens are online and have posted a profile. The Practice of Statistics, 5 th Edition 8
9 Conditional Probability and Independence When knowledge that one event has happened does not change the likelihood that another event will happen, we say that the two events are independent. Two events A and B are independent if the occurrence of one event does not change the probability that the other event will happen. In other words, events A and B are independent if P(A B) = P(A) and P(B A) = P(B). When events A and B are independent, we can simplify the general multiplication rule since P(B A) = P(B). Multiplication rule for independent events If A and B are independent events, then the probability that A and B both occur is P(A B) = P(A) P(B) The Practice of Statistics, 5 th Edition 9
10 Multiplication Rule for Independent Events Following the Space Shuttle Challenger disaster, it was determined that the failure of O-ring joints in the shuttle s booster rockets was to blame. Under cold conditions, it was estimated that the probability that an individual O-ring joint would function properly was Assuming O-ring joints succeed or fail independently, what is the probability all six would function properly? P( joint 1 OK and joint 2 OK and joint 3 OK and joint 4 OK and joint 5 OK and joint 6 OK) By the multiplication rule for independent events, this probability is: P(joint 1 OK) P(joint 2 OK) P (joint 3 OK) P (joint 6 OK) = (0.977)(0.977)(0.977)(0.977)(0.977)(0.977) = 0.87 There s an 87% chance that the shuttle would launch safely under similar conditions (and a 13% chance that it wouldn t). The Practice of Statistics, 5 th Edition 10
11 Conditional Probabilities and Independence Section Summary In this section, we learned how to CALCULATE and INTERPRET conditional probabilities. USE the general multiplication rule to CALCULATE probabilities. USE tree diagrams to MODEL a chance process and CALCULATE probabilities involving two or more events. DETERMINE if two events are independent. When appropriate, USE the multiplication rule for independent events to COMPUTE probabilities. The Practice of Statistics, 5 th Edition 11
CHAPTER 7 Sampling Distributions
CHAPTER 7 Sampling Distributions 7.3 Sample Means The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Sample Means Learning Objectives After this section,
CHAPTER 10 Comparing Two Populations or Groups
CHAPTER 10 Comparing Two Populations or Groups 10.1 Comparing Two Proportions The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Comparing Two Proportions
CHAPTER 10 Comparing Two Populations or Groups
CHAPTER 10 Comparing Two Populations or Groups 10.2 Comparing Two Means The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Comparing Two Means Learning
CHAPTER 10 Comparing Two Populations or Groups
CHAPTER 10 Comparing Two Populations or Groups 10. Comparing Two Means The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Comparing Two Means Learning
Lecture 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
Sociology 6Z03 Topic 10: Probability (Part I)
Sociology 6Z03 Topic 10: Probability (Part I) John Fox McMaster University Fall 2014 John Fox (McMaster University) Soc 6Z03: Probability I Fall 2014 1 / 29 Outline: Probability (Part I) Introduction Probability
CHAPTER 2 Modeling Distributions of Data
CHAPTER 2 Modeling Distributions of Data 2.1 Describing Location in a Distribution The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Describing Location
Chapter 2: Probability Part 1
Engineering Probability & Statistics (AGE 1150) Chapter 2: Probability Part 1 Dr. O. Phillips Agboola Sample Space (S) Experiment: is some procedure (or process) that we do and it results in an outcome.
I - 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)
CS4705. Probability Review and Naïve Bayes. Slides from Dragomir Radev
CS4705 Probability Review and Naïve Bayes Slides from Dragomir Radev Classification using a Generative Approach Previously on NLP discriminative models P C D here is a line with all the social media posts
CHAPTER 3 Describing Relationships
CHAPTER 3 Describing Relationships 3.1 Scatterplots and Correlation The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Scatterplots and Correlation Learning
2011 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
Probability 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
CS626 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
BASICS OF PROBABILITY CHAPTER-1 CS6015-LINEAR ALGEBRA AND RANDOM PROCESSES
BASICS OF PROBABILITY CHAPTER-1 CS6015-LINEAR ALGEBRA AND RANDOM PROCESSES COMMON TERMS RELATED TO PROBABILITY Probability is the measure of the likelihood that an event will occur Probability values are
Experiment -- the process by which an observation is made. Sample Space -- ( S) the collection of ALL possible outcomes of an experiment
A. 1 Elementary Probability Set Theory Experiment -- the process by which an observation is made Ex. Outcome The result of a chance experiment. Ex. Sample Space -- ( S) the collection of ALL possible outcomes
Chapter Learning Objectives. Random Experiments Dfiii Definition: Dfiii Definition:
Chapter 2: Probability 2-1 Sample Spaces & Events 2-1.1 Random Experiments 2-1.2 Sample Spaces 2-1.3 Events 2-1 1.4 Counting Techniques 2-2 Interpretations & Axioms of Probability 2-3 Addition Rules 2-4
A survey of Probability concepts. Chapter 5
A survey of Probability concepts Chapter 5 Learning Objectives Define probability. Explain the terms experiment, event, outcome. Define the terms conditional probability and joint probability. Calculate
The 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
Presentation 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
Fundamentals 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
SDS 321: Introduction to Probability and Statistics
SDS 321: Introduction to Probability and Statistics Lecture 2: Conditional probability Purnamrita Sarkar Department of Statistics and Data Science The University of Texas at Austin www.cs.cmu.edu/ psarkar/teaching
Probability Theory and Simulation Methods
Feb 28th, 2018 Lecture 10: Random variables Countdown to midterm (March 21st): 28 days Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters
Introduction to probability
Introduction to probability 4.1 The Basics of Probability Probability The chance that a particular event will occur The probability value will be in the range 0 to 1 Experiment A process that produces
CHAPTER 3 Describing Relationships
CHAPTER 3 Describing Relationships 3.1 Scatterplots and Correlation The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Scatterplots and Correlation Learning
Ch 14 Randomness and Probability
Ch 14 Randomness and Probability We ll begin a new part: randomness and probability. This part contain 4 chapters: 14-17. Why we need to learn this part? Probability is not a portion of statistics. Instead
Lecture 3 Probability Basics
Lecture 3 Probability Basics Thais Paiva STA 111 - Summer 2013 Term II July 3, 2013 Lecture Plan 1 Definitions of probability 2 Rules of probability 3 Conditional probability What is Probability? Probability
Section 13.3 Probability
288 Section 13.3 Probability Probability is a measure of how likely an event will occur. When the weather forecaster says that there will be a 50% chance of rain this afternoon, the probability that it
Introduction to Informatics
Introduction to Informatics Lecture 19: Probability Readings until now Lecture notes Posted online http://informatics.indiana.edu/rocha/i101 The Nature of Information Technology Modeling the World @ infoport
Independence. 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,
Elementary Discrete Probability
Elementary Discrete Probability MATH 472 Financial Mathematics J Robert Buchanan 2018 Objectives In this lesson we will learn: the terminology of elementary probability, elementary rules of probability,
Lecture 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
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
Business 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:
Outline. 1. Define likelihood 2. Interpretations of likelihoods 3. Likelihood plots 4. Maximum likelihood 5. Likelihood ratio benchmarks
Outline 1. Define likelihood 2. Interpretations of likelihoods 3. Likelihood plots 4. Maximum likelihood 5. Likelihood ratio benchmarks Likelihood A common and fruitful approach to statistics is to assume
STA 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
CHAPTER - 16 PROBABILITY Random Experiment : If an experiment has more than one possible out come and it is not possible to predict the outcome in advance then experiment is called random experiment. Sample
Statistics for Managers Using Microsoft Excel (3 rd Edition)
Statistics for Managers Using Microsoft Excel (3 rd Edition) Chapter 4 Basic Probability and Discrete Probability Distributions 2002 Prentice-Hall, Inc. Chap 4-1 Chapter Topics Basic probability concepts
Chapter 10: Comparing Two Populations or Groups
Chapter 10: Comparing Two Populations or Groups Sectio0.1 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 10 Comparing Two Populations or Groups 10.1 10.2 Comparing Two Means
Conditional Probability & Independence. Conditional Probabilities
Conditional Probability & Independence Conditional Probabilities Question: How should we modify P(E) if we learn that event F has occurred? Definition: the conditional probability of E given F is P(E F
CHAPTER 3 PROBABILITY: EVENTS AND PROBABILITIES
CHAPTER 3 PROBABILITY: EVENTS AND PROBABILITIES PROBABILITY: A probability is a number between 0 and 1, inclusive, that states the long-run relative frequency, likelihood, or chance that an outcome will
Chapter 10: Comparing Two Populations or Groups
Chapter 10: Comparing Two Populations or Groups Sectio0.1 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 10 Comparing Two Populations or Groups 10.1 10. Comparing Two Means
Lecture Slides. Elementary Statistics Eleventh Edition. by Mario F. Triola. and the Triola Statistics Series 4.1-1
Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola 4.1-1 4-1 Review and Preview Chapter 4 Probability 4-2 Basic Concepts of Probability 4-3 Addition
Announcements. 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
STAT 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):
When working with probabilities we often perform more than one event in a sequence - this is called a compound probability.
+ Independence + Compound Events When working with probabilities we often perform more than one event in a sequence - this is called a compound probability. Compound probabilities are more complex than
Outline 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
Econ 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
Review Basic Probability Concept
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.9008 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
P A = h n. Probability definitions, axioms and theorems. Classical approach to probability. Examples of classical probability definition
Probability definitions, axioms and theorems CEE 3030 A LECTURE PREPARED BY Classical approach to probability Sample space S is known Let n = number of total outcomes in sample space Let h = number of
COMPSCI 240: Reasoning Under Uncertainty
COMPSCI 240: Reasoning Under Uncertainty Nic Herndon and Andrew Lan University of Massachusetts at Amherst Spring 2019 Lecture 7: Counting (cont.) Summary of Counting Problems Structure Permutation k-permutation
Data Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber
Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2017 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields
Probability the chance that an uncertain event will occur (always between 0 and 1)
Quantitative Methods 2013 1 Probability as a Numerical Measure of the Likelihood of Occurrence Probability the chance that an uncertain event will occur (always between 0 and 1) Increasing Likelihood of
MAE 493G, CpE 493M, Mobile Robotics. 6. Basic Probability
MAE 493G, CpE 493M, Mobile Robotics 6. Basic Probability Instructor: Yu Gu, Fall 2013 Uncertainties in Robotics Robot environments are inherently unpredictable; Sensors and data acquisition systems are
TA Qinru Shi: Based on poll result, the first Programming Boot Camp will be: this Sunday 5 Feb, 7-8pm Gates 114.
TA Qinru Shi: Based on poll result, the first Programming Boot Camp will be: this Sunday 5 Feb, 7-8pm Gates 114. Prob Set 1: to be posted tomorrow. due in roughly a week Finite probability space S 1) a
Conditional Probability 2 Solutions COR1-GB.1305 Statistics and Data Analysis
Conditional Probability 2 Solutions COR-GB.305 Statistics and Data Analysis The Birthday Problem. A class has 50 students. What is the probability that at least two students have the same birthday? Assume
CHAPTER 1 Exploring Data
CHAPTER 1 Exploring Data 1.2 Displaying Quantitative Data with Graphs The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Displaying Quantitative Data
3 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
Preliminary 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
UNIT 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
Chapter 10: Comparing Two Populations or Groups
Chapter 10: Comparing Two Populations or Groups Sectio0.1 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 10 Comparing Two Populations or Groups 10.1 10.2 Comparing Two Means
Lecture 6: The Pigeonhole Principle and Probability Spaces
Lecture 6: The Pigeonhole Principle and Probability Spaces Anup Rao January 17, 2018 We discuss the pigeonhole principle and probability spaces. Pigeonhole Principle The pigeonhole principle is an extremely
1 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
Lecture 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.
Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions
Statistics for Managers Using Microsoft Excel/SPSS Chapter 4 Basic Probability And Discrete Probability Distributions 1999 Prentice-Hall, Inc. Chap. 4-1 Chapter Topics Basic Probability Concepts: Sample
Introduction to Probability
Introduction to Probability Gambling at its core 16th century Cardano: Books on Games of Chance First systematic treatment of probability 17th century Chevalier de Mere posed a problem to his friend Pascal.
Chapter 2.5 Random Variables and Probability The Modern View (cont.)
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
Chapter 7: Random Variables
Chapter 7: Random Variables Section 7.2 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Section 7.2 2 Learning Objectives After this section, you should be able to DESCRIBE the effect
Dept. 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
Unit 7 Probability M2 13.1,2,4, 5,6
+ Unit 7 Probability M2 13.1,2,4, 5,6 7.1 Probability n Obj.: I will be able to determine the experimental and theoretical probabilities of an event, or its complement, occurring. n Vocabulary o Outcome
Topic -2. Probability. Larson & Farber, Elementary Statistics: Picturing the World, 3e 1
Topic -2 Probability Larson & Farber, Elementary Statistics: Picturing the World, 3e 1 Probability Experiments Experiment : An experiment is an act that can be repeated under given condition. Rolling a
What are the Findings?
What are the Findings? James B. Rawlings Department of Chemical and Biological Engineering University of Wisconsin Madison Madison, Wisconsin April 2010 Rawlings (Wisconsin) Stating the findings 1 / 33
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
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
Independence Solutions STAT-UB.0103 Statistics for Business Control and Regression Models
Independence Solutions STAT-UB.003 Statistics for Business Control and Regression Models The Birthday Problem. A class has 70 students. What is the probability that at least two students have the same
Probability Review Lecturer: Ji Liu Thank Jerry Zhu for sharing his slides
Probability Review Lecturer: Ji Liu Thank Jerry Zhu for sharing his slides slide 1 Inference with Bayes rule: Example In a bag there are two envelopes one has a red ball (worth $100) and a black ball one
Probability 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
Conditional Probability & Independence. Conditional Probabilities
Conditional Probability & Independence Conditional Probabilities Question: How should we modify P(E) if we learn that event F has occurred? Definition: the conditional probability of E given F is P(E F
What 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
Bayesian Networks. Motivation
Bayesian Networks Computer Sciences 760 Spring 2014 http://pages.cs.wisc.edu/~dpage/cs760/ Motivation Assume we have five Boolean variables,,,, The joint probability is,,,, How many state configurations
Basic Statistics and Probability Chapter 3: Probability
Basic Statistics and Probability Chapter 3: Probability Events, Sample Spaces and Probability Unions and Intersections Complementary Events Additive Rule. Mutually Exclusive Events Conditional Probability
Discrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #24: Probability Theory Based on materials developed by Dr. Adam Lee Not all events are equally likely
Chap 4 Probability p227 The probability of any outcome in a random phenomenon is the proportion of times the outcome would occur in a long series of
Chap 4 Probability p227 The probability of any outcome in a random phenomenon is the proportion of times the outcome would occur in a long series of repetitions. (p229) That is, probability is a long-term
Chapter 7: Sampling Distributions
Chapter 7: Sampling Distributions Section 7.1 What is a Sampling Distribution? The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 7 Sampling Distributions 7.1 What is a Sampling
5.3 Conditional Probability and Independence
28 CHAPTER 5. PROBABILITY 5. Conditional Probability and Independence 5.. Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
Second Problem Assignment
EECS 401 Due on January 19, 2007 PROBLEM 1 (30 points) Bo and Ci are the only two people who will enter the Rover Dog Fod jingle contest. Only one entry is allowed per contestant, and the judge (Rover)
Statistics 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:
CIVL 7012/8012. Basic Laws and Axioms of Probability
CIVL 7012/8012 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
Discrete Probability Distribution Tables
Section 5 A : Discrete Probability Distributions Introduction Discrete Probability Distribution ables A probability distribution table is like the relative frequency tables that we constructed in chapter.
Probability 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
STA 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
UNIT Explain about the partition of a sampling space theorem?
UNIT -1 1. Explain about the partition of a sampling space theorem? PARTITIONS OF A SAMPLE SPACE The events B1, B2. B K represent a partition of the sample space 'S" if (a) So, when the experiment E is
4.2 Probability Models
4.2 Probability Models Ulrich Hoensch Tuesday, February 19, 2013 Sample Spaces Examples 1. When tossing a coin, the sample space is S = {H, T }, where H = heads, T = tails. 2. When randomly selecting a
STT When trying to evaluate the likelihood of random events we are using following wording.
Introduction to Chapter 2. Probability. When trying to evaluate the likelihood of random events we are using following wording. Provide your own corresponding examples. Subjective probability. An individual
Outline. 1. Define likelihood 2. Interpretations of likelihoods 3. Likelihood plots 4. Maximum likelihood 5. Likelihood ratio benchmarks
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
Probability: Why do we care? Lecture 2: Probability and Distributions. Classical Definition. What is Probability?
Probability: Why do we care? Lecture 2: Probability and Distributions Sandy Eckel seckel@jhsph.edu 22 April 2008 Probability helps us by: Allowing us to translate scientific questions into mathematical
CMPSCI 240: Reasoning Under Uncertainty
CMPSCI 240: Reasoning Under Uncertainty Lecture 5 Prof. Hanna Wallach wallach@cs.umass.edu February 7, 2012 Reminders Pick up a copy of B&T Check the course website: http://www.cs.umass.edu/ ~wallach/courses/s12/cmpsci240/
Econ 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
4. Conditional Probability P( ) CSE 312 Autumn 2012 W.L. Ruzzo
4. Conditional Probability P( ) CSE 312 Autumn 2012 W.L. Ruzzo 1 conditional probability Conditional probability of E given F: probability that E occurs given that F has occurred. Conditioning on F S Written
Chapter 2. Probability. Math 371. University of Hawai i at Mānoa. Summer 2011
Chapter 2 Probability Math 371 University of Hawai i at Mānoa Summer 2011 W. DeMeo (williamdemeo@gmail.com) Chapter 2: Probability math.hawaii.edu/ williamdemeo 1 / 8 Outline 1 Chapter 2 Examples Definition
Discrete probability distributions
Discrete probability s BSAD 30 Dave Novak Fall 08 Source: Anderson et al., 05 Quantitative Methods for Business th edition some slides are directly from J. Loucks 03 Cengage Learning Covered so far Chapter