# 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 Transcript Lecture 3

Size: px
Start display at page:

Download "6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 Transcript Lecture 3"

Transcription

1 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 Transcript Lecture 3 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Let us start. So as always, we're to have a quick review of what we discussed last time. And then today we're going to introduce just one new concept, the notion of independence of two events. And we will play with that concept. So what did we talk about last time? The idea is that we have an experiment, and the experiment has a sample space omega. And then somebody comes and tells us you know the outcome of the experiments happens to lie inside this particular event B. Given this information, it kind of changes what we know about the situation. It tells us that the outcome is going to be somewhere inside here. So this is essentially our new sample space. And now we need to we reassign probabilities to the various possible outcomes, because, for example, these outcomes, even if they had positive probability beforehand, now that we're told that B occurred, those outcomes out there are going to have zero probability. So we need to revise our probabilities. The new probabilities are called conditional probabilities, and they're defined this way. The conditional probability that A occurs given that we're told that B occurred is calculated by this formula, which tells us the following-- out of the total probability that was initially assigned to the event B, what fraction of that probability is assigned to outcomes that also make A to happen? So out of the total probability assigned to B, we see what fraction of that total probability is assigned to those elements here that will also make A happen. Conditional probabilities are left undefined if the denominator here is zero. An easy consequence of the definition is if we bring that term to the other side, then we can find the probability of two things happening by taking the probability that the first thing happens, and then, given that the first thing happened, the conditional probability that the second one happens. Then we saw last time that we can divide and conquer in calculating probabilities of mildly complicated events by breaking it down into different scenarios. So event B can happen in two ways. It can happen either together with A, which is this probability, or it can happen together with A complement, which is this probability. So basically what we're saying that the total probability of B is the probability of this, which is A intersection B, plus the probability of that, which is A complement intersection B. So these two facts here, multiplication rule and the total probability theorem, are basic tools that one uses to break down probability calculations into a simpler parts. So we find probabilities of two things happening by looking at each one at a time. And this is what we do to break up a situation with two different possible scenarios. 1

5 So this definition would be limited to cases where event A has positive probability, whereas this definition is something that you can write down always. We will say that two events are independent if and only if their probability of happening simultaneously is equal to the product of their two individual probabilities. And in particular, we can have events of zero probability. There's nothing wrong with that. If A has 0 probability, then A intersection B will also have zero probability, because it's an even smaller event. And so we're going to get zero is equal to zero. A corollary of what I just said, if an event A has zero probability, it's actually independent of any other event in our model, because we're going to get zero is equal to zero. And the definition is going to be satisfied. This is a little bit harder to reconcile with the intuition we have about independence, but then again, it's part of the mathematical definition. So what I want you to retain is this notion that the independence is something that you can check formally using this definition, but also you can check intuitively by if, in some cases, you can reason that whatever happens and determines whether A is going to occur or not, has nothing absolutely to do with whatever happens and determines whether B is going to occur or not. So if I'm doing a science experiment in this room, and it gets hit by some noise that's causes randomness. And then five years later, somebody somewhere else does the same science experiment somewhere else, it gets hit by other noise, you would usually say that these experiments are independent. So what events happen in one experiment are not going to change your beliefs about what might be happening in the other, because the sources of noise in these two experiments are completely unrelated. They have nothing to do with each other. So if I flip a coin here today, and I flip a coin in my office tomorrow, one shouldn't affect the other. So the events that I get from these should be independent. So that's usually how independence arises. By having distinct physical phenomena that do not interact. Sometimes you also get independence even though there is a physical interaction, but you just happen to have a numerical accident. A and B might be physically related very tightly, but a numerical accident happens and you get equality here, that's another case where we do get independence. Now suppose that we have two events that are laid out like this. Are these two events independent or not? The picture kind of tells you that one is separate from the other. But separate has nothing to do with independent. In fact, these two events are as dependent as Siamese twins. Why is that? If I tell you that A occurred, then you are certain that B did not occur. So information about the occurrence of A definitely affects your beliefs about the possible occurrence or non-occurrence of B. When the picture is like that, knowing that A occurred will change drastically my beliefs about B, because now I suddenly become certain that B did not occur. So a picture like this is a case actually of extreme dependence. So don't confuse independence with disjointness. They're very different types of properties. 5

6 AUDIENCE: Question. PROFESSOR: Yes? AUDIENCE: So I understand the explanation, but the probability of A intersect B [INAUDIBLE] to zero, because they're disjoint. PROFESSOR: Yes. AUDIENCE: But then the product of probability A and probability B, one of them is going to be 1. [INAUDIBLE] PROFESSOR: No, suppose that the probabilities are 1/3, 1/4, and the rest is out there. You check the definition of independence. Probability of A intersection B is zero. Probability of A times the probability of B is 1/12. The two are not equal. Therefore we do not have independence. AUDIENCE: Right. So what's wrong with the intuition of the probability of A being 1, and the other one being 0? [INAUDIBLE]. PROFESSOR: No. The probability of A given B is equal to 0. Probability of A is equal to 1/3. So again, these two are different. So we had some initial beliefs about A, but as soon as we are told that B occurred, our beliefs about A changed. And so since our beliefs changed, that means that B conveys information about A. AUDIENCE: So can you not draw independent [INAUDIBLE] on a Venn diagram? PROFESSOR: I can't hear you. AUDIENCE: Can you draw independence on a Venn diagram? PROFESSOR: No, the Venn diagram is never enough to decide independence. So the typical picture in which you're going to have independence would be one event this way, and another event this way. You need to take the probability of this times the probability of that, and check that, numerically, it's equal to the probability of this intersection. So it's more than a Venn diagram. Numbers need to come out right. Now we did say some time ago that conditional probabilities are just like ordinary probabilities, and whatever we do in probability theory can also be done in conditional universes. Talking about conditional probabilities. So since we have a notion of independence, then there should be also a notion of conditional independence. So independence was defined by the probability that A intersection B is equal to the probability of A times the probability of B. What would be a reasonable definition of conditional independence? Conditional independence would mean that this same property could be true, but in a conditional universe where we are told that the certain event happens. So if we're told that the event C has happened, then were transported in a conditional universe where the only thing that matters are conditional 6

9 OK, so I've been playing a little bit too loose with the language here, because we defined the concept of independence of two events. But here I have been referring to independent coin tosses, where I'm thinking about many coin tosses, like 10 or 11 of them. So to be proper, I should have defined for you also the notion of independence of multiple events, not just two. We don't want to just say coin toss one is independent from coin toss two. We want to be able to say something like, these 10 then coin tosses are all independent of each other. Intuitively what that means should be the same thing-- that information about some of the coin tosses doesn't change your beliefs about the remaining coin tosses. How do we translate that into a mathematical definition? Well, an ugly attempt would be to impose requirements such as this. Think of A1 being the event that the first flip was heads. A2 is the event of that the second flip was heads. A3, the third flip, was heads, and so on. Here is an event whose occurrence is not determined by the first three coin flips. And here's an event whose occurrence or not is determined by the fifth and sixth coin flip. If we think physically that all those coin flips have nothing to do with each other, information about the fifth and sixth coin flip are not going to change what we expect from the first three. So the probability of this event, the conditional probability, should be the same as the unconditional probability. And we would like a relation of this kind to be true, no matter what kind of formula you write down, as long as the events that show up here are different from the events that show up there. OK. That's sort of an ugly definition. The mathematical definition that actually does the job, and leads to all the formulas of this kind, is the following. We're going to say that the collection of events are independent if we can find the probability of their joint occurrence by just multiplying probabilities. And that will be true even if you look at sub-collections of these events. Let's make that more precise. If we have three events, the definition tells us that the three events are independent if the following are true. Probability A1 and A2 and A3, you can calculate this probability by multiplying individual probabilities. But the same is true even if you take fewer events. Just a few indices out of the indices that we have available. So we also require P(A1 intersection A2) is P(A1) times P(A2). And similarly for the other possibilities of choosing the indices. OK, so independence, mathematical definition, requires that calculating probabilities of any intersection of the events we have in our hands, that calculation can be done by just multiplying individual probabilities. And this has to apply to the case where we consider all of the events in our hands or just sub-collections of those events. Now these relations just by themselves are called pairwise independence. So this relation, for example, tells us that A1 is independent from A2. This tells us that A2 is independent from A3. This will tell us that A1 is independent from A3. But independence of all the events together actually requires a little more. One more equality that has to do with all three events being considered at the same time. 9

13 MIT OpenCourseWare SC Probabilistic Systems Analysis and Applied Probability Fall 2013 For information about citing these materials or our Terms of Use, visit:

### 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 Transcript Lecture 21

6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 Transcript Lecture 21 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare

### MITOCW watch?v=ogo1gpxsuzu

MITOCW watch?v=ogo1gpxsuzu The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To

### The following content is provided under a Creative Commons license. Your support

MITOCW Lecture 15 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a

### MITOCW ocw f99-lec19_300k

MITOCW ocw-18.06-f99-lec19_300k OK, this is the second lecture on determinants. There are only three. With determinants it's a fascinating, small topic inside linear algebra. Used to be determinants were

### Lesson 10 Notes. Machine Learning. Intro. Joint Distribution

Machine Learning Lesson 10 Notes Intro M: Hey Charles. C: Hey Michael. M: So like I get to lecture near you today. C: Yes you do. I can even see you. M: This is, this is crazy. I sort of don't have my

### MITOCW ocw f99-lec18_300k

MITOCW ocw-18.06-f99-lec18_300k OK, this lecture is like the beginning of the second half of this is to prove. this course because up to now we paid a lot of attention to rectangular matrices. Now, concentrating

### MITOCW watch?v=6pxncdxixne

MITOCW watch?v=6pxncdxixne The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To

### The following content is provided under a Creative Commons license. Your support

MITOCW Lecture 13 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a

### MITOCW ocw f08-rec10_300k

MITOCW ocw-18-085-f08-rec10_300k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free.

### CSSS/SOC/STAT 321 Case-Based Statistics I. Introduction to Probability

CSSS/SOC/STAT 321 Case-Based Statistics I Introduction to Probability Christopher Adolph Department of Political Science and Center for Statistics and the Social Sciences University of Washington, Seattle

### Lesson 09 Notes. Machine Learning. Intro

Machine Learning Lesson 09 Notes Intro C: Hi Michael. M: Hey how's it going? C: So I want to talk about something today Michael. I want to talk about Bayesian Learning, and I've been inspired by our last

### MITOCW MITRES18_006F10_26_0703_300k-mp4

MITOCW MITRES18_006F10_26_0703_300k-mp4 ANNOUNCER: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational

### The following content is provided under a Creative Commons license. Your support

MITOCW Lecture 14 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a

### MITOCW watch?v=4hrhg4euimo

MITOCW watch?v=4hrhg4euimo The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To

### Probability Foundations for Electrical Engineers Prof. Krishna Jagannathan Department of Electrical Engineering Indian Institute of Technology, Madras

Probability Foundations for Electrical Engineers Prof. Krishna Jagannathan Department of Electrical Engineering Indian Institute of Technology, Madras Lecture - 1 Introduction Welcome, this is Probability

### Module 02 Lecture - 10 Inferential Statistics Single Sample Tests

Introduction to Data Analytics Prof. Nandan Sudarsanam and Prof. B. Ravindran Department of Management Studies and Department of Computer Science and Engineering Indian Institute of Technology, Madras

### MITOCW Lec 2 MIT 6.042J Mathematics for Computer Science, Fall 2010

MITOCW Lec 2 MIT 6.042J Mathematics for Computer Science, Fall 2010 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high

### Torah Code Cluster Probabilities

Torah Code Cluster Probabilities Robert M. Haralick Computer Science Graduate Center City University of New York 365 Fifth Avenue New York, NY 006 haralick@netscape.net Introduction In this note we analyze

### >> Marian Small: I was talking to a grade one teacher yesterday, and she was telling me

Marian Small transcripts Leadership Matters >> Marian Small: I've been asked by lots of leaders of boards, I've asked by teachers, you know, "What's the most effective thing to help us? Is it -- you know,

### Semantic Entailment and Natural Deduction

Semantic Entailment and Natural Deduction Alice Gao Lecture 6, September 26, 2017 Entailment 1/55 Learning goals Semantic entailment Define semantic entailment. Explain subtleties of semantic entailment.

### Philosophy Epistemology Topic 5 The Justification of Induction 1. Hume s Skeptical Challenge to Induction

Philosophy 5340 - Epistemology Topic 5 The Justification of Induction 1. Hume s Skeptical Challenge to Induction In the section entitled Sceptical Doubts Concerning the Operations of the Understanding

### MITOCW watch?v=a8fbmj4nixy

MITOCW watch?v=a8fbmj4nixy The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To

### MITOCW watch?v=ppqrukmvnas

MITOCW watch?v=ppqrukmvnas The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To

### MITOCW watch?v=k2sc-wpdt6k

MITOCW watch?v=k2sc-wpdt6k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To

### Lesson 07 Notes. Machine Learning. Quiz: Computational Learning Theory

Machine Learning Lesson 07 Notes Quiz: Computational Learning Theory M: Hey, Charles. C: Oh, hi Michael. M: It's funny running into to you here. C: It is. It's always funny running in to you over the interwebs.

### 6.00 Introduction to Computer Science and Programming, Fall 2008

MIT OpenCourseWare http://ocw.mit.edu 6.00 Introduction to Computer Science and Programming, Fall 2008 Please use the following citation format: Eric Grimson and John Guttag, 6.00 Introduction to Computer

The St. Petersburg paradox & the two envelope paradox Consider the following bet: The St. Petersburg I am going to flip a fair coin until it comes up heads. If the first time it comes up heads is on the

### Detachment, Probability, and Maximum Likelihood

Detachment, Probability, and Maximum Likelihood GILBERT HARMAN PRINCETON UNIVERSITY When can we detach probability qualifications from our inductive conclusions? The following rule may seem plausible:

### Introduction Symbolic Logic

An Introduction to Symbolic Logic Copyright 2006 by Terence Parsons all rights reserved CONTENTS Chapter One Sentential Logic with 'if' and 'not' 1 SYMBOLIC NOTATION 2 MEANINGS OF THE SYMBOLIC NOTATION

### We know that numbers are important in the natural world and

SPIRITUAL SIGNIFICANCE TO NUMBER PHI (ϕ)? IS THERE A SPIRITUAL SIGNIFICANCE TO THE NUMBER PHI (ϕ)? * George Gantz INTRODUCTION We know that numbers are important in the natural world and particularly in

### Friends and strangers

1997 2009, Millennium Mathematics Project, University of Cambridge. Permission is granted to print and copy this page on paper for non commercial use. For other uses, including electronic redistribution,

### = (value of LEAVE if rain x chance of rain) + (value of LEAVE if dry x chance of dry) = -20 x x.5 = -9

3. PASCAL S WAGER Suppose you are facing a decision under conditions of uncertainty : say, whether to take an umbrella or not, on a day when the chance of rain is one half. e value of taking as opposed

### Philosophy 148 Announcements & Such. Inverse Probability and Bayes s Theorem II. Inverse Probability and Bayes s Theorem III

Branden Fitelson Philosophy 148 Lecture 1 Branden Fitelson Philosophy 148 Lecture 2 Philosophy 148 Announcements & Such Administrative Stuff I ll be using a straight grading scale for this course. Here

### MITOCW 3. V: Recursive Structures and Processes

MITOCW 3. V: Recursive Structures and Processes The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational

### The end of the world & living in a computer simulation

The end of the world & living in a computer simulation In the reading for today, Leslie introduces a familiar sort of reasoning: The basic idea here is one which we employ all the time in our ordinary

### Step 2: Multiply both the numerator and the denominator. Remember that you can multiply numbers

Rationalizing Denominators Here are the steps required to rationalize the denominator containing one terms: Step 1: To rationalize the denominator, you need to multiply both the numerator and denominator

### Introduction to Statistical Hypothesis Testing Prof. Arun K Tangirala Department of Chemical Engineering Indian Institute of Technology, Madras

Introduction to Statistical Hypothesis Testing Prof. Arun K Tangirala Department of Chemical Engineering Indian Institute of Technology, Madras Lecture 09 Basics of Hypothesis Testing Hello friends, welcome

### Classroom Voting Questions: Statistics

Classroom Voting Questions: Statistics General Probability Rules 1. In a certain semester, 500 students enrolled in both Calculus I and Physics I. Of these students, 82 got an A in calculus, 73 got an

### Lecture Notes on Classical Logic

Lecture Notes on Classical Logic 15-317: Constructive Logic William Lovas Lecture 7 September 15, 2009 1 Introduction In this lecture, we design a judgmental formulation of classical logic To gain an intuition,

### MITOCW watch?v=z6n7j7dlmls

MITOCW watch?v=z6n7j7dlmls The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To

### Rationalizing Denominators

Solver9 Rationalizing Denominators Get that pesky radical OUT of my denominator John Reece 7//00 What does Rationalize the Denominator mean? It means to change a fraction, without changing its value, so

### Think by Simon Blackburn. Chapter 6b Reasoning

Think by Simon Blackburn Chapter 6b Reasoning According to Kant, a sentence like: Sisters are female is A. a synthetic truth B. an analytic truth C. an ethical truth D. a metaphysical truth If you reach

### 6.00 Introduction to Computer Science and Programming, Fall 2008

MIT OpenCourseWare http://ocw.mit.edu 6.00 Introduction to Computer Science and Programming, Fall 2008 Please use the following citation format: Eric Grimson and John Guttag, 6.00 Introduction to Computer

### 175 Chapter CHAPTER 23: Probability

75 Chapter 23 75 CHAPTER 23: Probability According to the doctrine of chance, you ought to put yourself to the trouble of searching for the truth; for if you die without worshipping the True Cause, you

### MITOCW L21

MITOCW 7.014-2005-L21 So, we have another kind of very interesting piece of the course right now. We're going to continue to talk about genetics, except now we're going to talk about the genetics of diploid

### Mathematics. The BIG game Behind the little tricks

Mathematics The BIG game Behind the little tricks Marta Maria Casetti @mmcasetti (She/Her) Hi there! :-) The goal of this talk is to show maths is nothing to fear, but it's a tool to embrace to empower

### MITOCW MIT24_908S17_Creole_Chapter_06_Authenticity_300k

MITOCW MIT24_908S17_Creole_Chapter_06_Authenticity_300k AUDIENCE: I wanted to give an answer to 2. MICHEL DEGRAFF: OK, yeah. AUDIENCE: So to both parts-- like, one of the parts was, like, how do the discourse

### Grade 6 Math Connects Suggested Course Outline for Schooling at Home

Grade 6 Math Connects Suggested Course Outline for Schooling at Home I. Introduction: (1 day) Look at p. 1 in the textbook with your child and learn how to use the math book effectively. DO: Scavenger

### 2.1 Review. 2.2 Inference and justifications

Applied Logic Lecture 2: Evidence Semantics for Intuitionistic Propositional Logic Formal logic and evidence CS 4860 Fall 2012 Tuesday, August 28, 2012 2.1 Review The purpose of logic is to make reasoning

### Free Acts and Chance: Why the Rollback Argument Fails Lara Buchak, UC Berkeley

1 Free Acts and Chance: Why the Rollback Argument Fails Lara Buchak, UC Berkeley ABSTRACT: The rollback argument, pioneered by Peter van Inwagen, purports to show that indeterminism in any form is incompatible

### Introduction to Inference

Introduction to Inference Confidence Intervals for Proportions 1 On the one hand, we can make a general claim with 100% confidence, but it usually isn t very useful; on the other hand, we can also make

### TwiceAround Podcast Episode 7: What Are Our Biases Costing Us? Transcript

TwiceAround Podcast Episode 7: What Are Our Biases Costing Us? Transcript Speaker 1: Speaker 2: Speaker 3: Speaker 4: [00:00:30] Speaker 5: Speaker 6: Speaker 7: Speaker 8: When I hear the word "bias,"

### CHEM 105 & 106 UNIT ONE, LECTURE THREE 1 YESTERDAY WHEN WE LEFT OFF WE WERE TALKING ABOUT CHANGE AND OF COURSE ONE OF THE

CHEM 105 & 106 UNIT ONE, LECTURE THREE 1 CHM 105/106 Program 3: Unit 1 Lecture 3 YESTERDAY WHEN WE LEFT OFF WE WERE TALKING ABOUT CHANGE AND OF COURSE ONE OF THE WAYS THAT WE DETERMINE THAT CHANGE, WHETHER

### Artificial Intelligence: Valid Arguments and Proof Systems. Prof. Deepak Khemani. Department of Computer Science and Engineering

Artificial Intelligence: Valid Arguments and Proof Systems Prof. Deepak Khemani Department of Computer Science and Engineering Indian Institute of Technology, Madras Module 02 Lecture - 03 So in the last

### 1 Introduction. Cambridge University Press Epistemic Game Theory: Reasoning and Choice Andrés Perea Excerpt More information

1 Introduction One thing I learned from Pop was to try to think as people around you think. And on that basis, anything s possible. Al Pacino alias Michael Corleone in The Godfather Part II What is this

### Discussion Notes for Bayesian Reasoning

Discussion Notes for Bayesian Reasoning Ivan Phillips - http://www.meetup.com/the-chicago-philosophy-meetup/events/163873962/ Bayes Theorem tells us how we ought to update our beliefs in a set of predefined

### MLLunsford, Spring Activity: Conditional Probability and The Law of Total Probability

MLLunsford, Spring 2003 1 Activity: Conditional Probability and The Law of Total Probability Concepts: Conditional Probability, Independent Events, the Multiplication Rule, the Law of Total Probability

### HANDBOOK (New or substantially modified material appears in boxes.)

1 HANDBOOK (New or substantially modified material appears in boxes.) I. ARGUMENT RECOGNITION Important Concepts An argument is a unit of reasoning that attempts to prove that a certain idea is true by

### Logic & Proofs. Chapter 3 Content. Sentential Logic Semantics. Contents: Studying this chapter will enable you to:

Sentential Logic Semantics Contents: Truth-Value Assignments and Truth-Functions Truth-Value Assignments Truth-Functions Introduction to the TruthLab Truth-Definition Logical Notions Truth-Trees Studying

### Death: Lecture 4 Transcript

Death: Lecture 4 Transcript Chapter 1. Introduction to Plato's Phaedo [00:00:00] Professor Shelly Kagan: We've been talking about the question, "What arguments might be offered for the existence of a soul?"

### Logic and Pragmatics: linear logic for inferential practice

Logic and Pragmatics: linear logic for inferential practice Daniele Porello danieleporello@gmail.com Institute for Logic, Language & Computation (ILLC) University of Amsterdam, Plantage Muidergracht 24

### MISSOURI S FRAMEWORK FOR CURRICULAR DEVELOPMENT IN MATH TOPIC I: PROBLEM SOLVING

Prentice Hall Mathematics:,, 2004 Missouri s Framework for Curricular Development in Mathematics (Grades 9-12) TOPIC I: PROBLEM SOLVING 1. Problem-solving strategies such as organizing data, drawing a

### FREE ACTS AND CHANCE: WHY THE ROLLBACK ARGUMENT FAILS

The Philosophical Quarterly Vol. 63, No. 250 January 2013 ISSN 0031-8094 doi: 10.1111/j.1467-9213.2012.00094.x FREE ACTS AND CHANCE: WHY THE ROLLBACK ARGUMENT FAILS BY LARA BUCHAK The rollback argument,

### 1.2. What is said: propositions

1.2. What is said: propositions 1.2.0. Overview In 1.1.5, we saw the close relation between two properties of a deductive inference: (i) it is a transition from premises to conclusion that is free of any

### Grade 7 Math Connects Suggested Course Outline for Schooling at Home 132 lessons

Grade 7 Math Connects Suggested Course Outline for Schooling at Home 132 lessons I. Introduction: (1 day) Look at p. 1 in the textbook with your child and learn how to use the math book effectively. DO:

### Outline. The argument from so many arguments. Framework. Royall s case. Ted Poston

Outline The argument from so many arguments Ted Poston poston@southalabama.edu University of South Alabama Plantinga Workshop Baylor University Nov 6-8, 2014 1 Measuring confirmation Framework Log likelihood

### Artificial Intelligence Prof. Deepak Khemani Department of Computer Science and Engineering Indian Institute of Technology, Madras

(Refer Slide Time: 00:14) Artificial Intelligence Prof. Deepak Khemani Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 35 Goal Stack Planning Sussman's Anomaly

### CS 2104 Intro Problem Solving in Computer Science Test 1 READ THIS NOW!

READ THIS NOW! Print your name in the space provided below. There are 5 problems, priced as marked. The maximum score is 100. The grading of each question will take into account whether you obtained a

### Artificial Intelligence Prof. Deepak Khemani Department of Computer Science and Engineering Indian Institute of Technology, Madras

(Refer Slide Time: 00:26) Artificial Intelligence Prof. Deepak Khemani Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 06 State Space Search Intro So, today

### ON THE TRUTH CONDITIONS OF INDICATIVE AND COUNTERFACTUAL CONDITIONALS Wylie Breckenridge

ON THE TRUTH CONDITIONS OF INDICATIVE AND COUNTERFACTUAL CONDITIONALS Wylie Breckenridge In this essay I will survey some theories about the truth conditions of indicative and counterfactual conditionals.

### Searle vs. Chalmers Debate, 8/2005 with Death Monkey (Kevin Dolan)

Searle vs. Chalmers Debate, 8/2005 with Death Monkey (Kevin Dolan) : Searle says of Chalmers book, The Conscious Mind, "it is one thing to bite the occasional bullet here and there, but this book consumes

### HANDBOOK (New or substantially modified material appears in boxes.)

1 HANDBOOK (New or substantially modified material appears in boxes.) I. ARGUMENT RECOGNITION Important Concepts An argument is a unit of reasoning that attempts to prove that a certain idea is true by

### THE ROLE OF COHERENCE OF EVIDENCE IN THE NON- DYNAMIC MODEL OF CONFIRMATION TOMOJI SHOGENJI

Page 1 To appear in Erkenntnis THE ROLE OF COHERENCE OF EVIDENCE IN THE NON- DYNAMIC MODEL OF CONFIRMATION TOMOJI SHOGENJI ABSTRACT This paper examines the role of coherence of evidence in what I call

### >> THE NEXT CASE IS STATE OF FLORIDA VERSUS FLOYD. >> TAKE YOUR TIME. TAKE YOUR TIME. >> THANK YOU, YOUR HONOR. >> WHENEVER YOU'RE READY.

>> THE NEXT CASE IS STATE OF FLORIDA VERSUS FLOYD. >> TAKE YOUR TIME. TAKE YOUR TIME. >> THANK YOU, YOUR HONOR. >> WHENEVER YOU'RE READY. >> GOOD MORNING. MAY IT PLEASE THE COURT, ASSISTANT ATTORNEY GENERAL

### The Gift of the Holy Spirit. 1 Thessalonians 5:23. Sermon Transcript by Rev. Ernest O'Neill

The Gift of the Holy Spirit 1 Thessalonians 5:23 Sermon Transcript by Rev. Ernest O'Neill We've been discussing, loved ones, the question the past few weeks: Why are we alive? The real problem, in trying

### A Posteriori Necessities by Saul Kripke (excerpted from Naming and Necessity, 1980)

A Posteriori Necessities by Saul Kripke (excerpted from Naming and Necessity, 1980) Let's suppose we refer to the same heavenly body twice, as 'Hesperus' and 'Phosphorus'. We say: Hesperus is that star

### Module - 02 Lecturer - 09 Inferential Statistics - Motivation

Introduction to Data Analytics Prof. Nandan Sudarsanam and Prof. B. Ravindran Department of Management Studies and Department of Computer Science and Engineering Indian Institute of Technology, Madras

### What Is the Thingy Illusion and How Does It Mess Up Philosophy?

What Is the Thingy Illusion and How Does It Mess Up Philosophy? Mark F. Sharlow The following is a transcript of an impromptu talk. The transcript has been edited and references have been added. There's

### I thought I should expand this population approach somewhat: P t = P0e is the equation which describes population growth.

I thought I should expand this population approach somewhat: P t = P0e is the equation which describes population growth. To head off the most common objections:! This does take into account the death

### Fr. Copleston vs. Bertrand Russell: The Famous 1948 BBC Radio Debate on the Existence of God

Fr. Copleston vs. Bertrand Russell: The Famous 1948 BBC Radio Debate on the Existence of God Father Frederick C. Copleston (Jesuit Catholic priest) versus Bertrand Russell (agnostic philosopher) Copleston:

### McDougal Littell High School Math Program. correlated to. Oregon Mathematics Grade-Level Standards

Math Program correlated to Grade-Level ( in regular (non-capitalized) font are eligible for inclusion on Oregon Statewide Assessment) CCG: NUMBERS - Understand numbers, ways of representing numbers, relationships

### MITOCW 1. Intro-III: Welcome, Tools for Thinking, Formal Systems

MITOCW 1. Intro-III: Welcome, Tools for Thinking, Formal Systems The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality

### TRANSCRIPT. Contact Repository Implementation Working Group Meeting Durban 14 July 2013

TRANSCRIPT Contact Repository Implementation Working Group Meeting Durban 14 July 2013 Attendees: Cristian Hesselman,.nl Luis Diego Esponiza, expert (Chair) Antonette Johnson,.vi (phone) Hitoshi Saito,.jp

### Student: In my opinion, I don't think the Haitian revolution was successful.

Facilitating a Socratic Seminar Video Transcript In my opinion, I don't think the Haitian revolution was successful. Even though they gained their independence, they still had to pay back the \$150 million

### Just Another Day in the Life of a Dole Bludger

Just Another Day in the Life of a Dole Bludger (November 2003): This was published in Lesbian Network some time in 1994 although I don't know which issue. (The notes were added in November 2003). 'It is

### September Frank W. Nelte SOME SPECULATIONS ABOUT THE PLAN OF GOD

September 2000 Frank W. Nelte SOME SPECULATIONS ABOUT THE PLAN OF GOD God wants us to understand His mind, His intentions and His purposes. As the Apostle Paul wrote in Romans: For the invisible things

### NPTEL NPTEL ONLINE COURSES REINFORCEMENT LEARNING. UCB1 Explanation (UCB1)

NPTEL NPTEL ONLINE COURSES REINFORCEMENT LEARNING UCB1 Explanation (UCB1) Prof. Balaraman Ravindran Department of Computer Science and Engineering Indian Institute of Technology Madras So we are looking

### Clergy Appraisal The goal of a good clergy appraisal process is to enable better ministry

Revised 12/30/16 Clergy Appraisal The goal of a good clergy appraisal process is to enable better ministry Can Non-Clergy Really Do a Meaningful Clergy Appraisal? Let's face it; the thought of lay people

### How to Generate a Thesis Statement if the Topic is Not Assigned.

What is a Thesis Statement? Almost all of us--even if we don't do it consciously--look early in an essay for a one- or two-sentence condensation of the argument or analysis that is to follow. We refer

### WORLD UTILITARIANISM AND ACTUALISM VS. POSSIBILISM

Professor Douglas W. Portmore WORLD UTILITARIANISM AND ACTUALISM VS. POSSIBILISM I. Hedonistic Act Utilitarianism: Some Deontic Puzzles Hedonistic Act Utilitarianism (HAU): S s performing x at t1 is morally

### Prof. Bryan Caplan Econ 812

Prof. Bryan Caplan bcaplan@gmu.edu http://www.bcaplan.com Econ 812 Week 1: Efficiency and Probability I. The Many Meanings of Efficiency A. The Merriam-Webster College Dictionary defines "efficiency" as

### The Fixed Hebrew Calendar

The Fixed Hebrew Calendar Moshe Lerman moshe.lerman@cremejvm.com June, 2017 קול גלגל המתגלגל ממטה למעלה 0. Introduction The present paper is an extension of a paper entitled Gauss Formula for the Julian

### Executive Power and the School Chaplains Case, Williams v Commonwealth Karena Viglianti

TRANSCRIPT Executive Power and the School Chaplains Case, Williams v Commonwealth Karena Viglianti Karena Viglianti is a Quentin Bryce Law Doctoral scholar and a teaching fellow here in the Faculty of

### Coordination Problems

Philosophy and Phenomenological Research Philosophy and Phenomenological Research Vol. LXXXI No. 2, September 2010 Ó 2010 Philosophy and Phenomenological Research, LLC Coordination Problems scott soames

### Surveying Prof. Bharat Lohani Department of Civil Engineering Indian Institute of Technology, Kanpur. Module - 7 Lecture - 3 Levelling and Contouring

Surveying Prof. Bharat Lohani Department of Civil Engineering Indian Institute of Technology, Kanpur Module - 7 Lecture - 3 Levelling and Contouring (Refer Slide Time: 00:21) Welcome to this lecture series

### Houghton Mifflin MATHEMATICS

2002 for Mathematics Assessment NUMBER/COMPUTATION Concepts Students will describe properties of, give examples of, and apply to real-world or mathematical situations: MA-E-1.1.1 Whole numbers (0 to 100,000,000),

### Artificial Intelligence I

Artificial Intelligence I Matthew Huntbach, Dept of Computer Science, Queen Mary and Westfield College, London, UK E 4NS. Email: mmh@dcs.qmw.ac.uk. Notes may be used with the permission of the author.

### a0rxh/ On Van Inwagen s Argument Against the Doctrine of Arbitrary Undetached Parts WESLEY H. BRONSON Princeton University

a0rxh/ On Van Inwagen s Argument Against the Doctrine of Arbitrary Undetached Parts WESLEY H. BRONSON Princeton University Imagine you are looking at a pen. It has a blue ink cartridge inside, along with