psychology and economics
|
|
- Doris Short
- 8 years ago
- Views:
Transcription
1 psychology and economics lecture 9: biases in statistical reasoning tomasz strzalecki
2 failures of Bayesian updating how people fail to update in a Bayesian way how Bayes law fails to describe how people update
3 plan for today base rate fallacies: people under- or over- use their prior base rate neglect confirmation bias sample size fallacies: ppl are under-sensitive to sample size law of small numbers gambler s fallacy hot hand fallacy sampling variation fallacy non-belief in the law of large numbers
4 Bayes rule: tradeoff between signal and prior posterior = signal prior P(A B) P(notA B) = P(B A) P(B nota) P(A) P(notA) conditional odds = likelihood ratio unconditional odds base rate neglect: posterior not responsive to the prior confirmation bias: posterior is sticky
5 base rate neglect A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. You are given the following data: (a) 85% of the cabs in the city are Green and 15% are Blue (b) A witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time what is the probability that the cab involved in the accident was Blue rather than Green? % many people say 80%
6 base rate neglect P(B b) P(b B) = P(G b) P(b G) P(B) P(G) = < 1
7 confirmation bias the same information is interpreted in different ways, depending on the prior belief polarization Lord, Ross, and Lepper (1979) subjects either opposed (opponents) or favored capital punishment (proponents) read two articles that discussed whether the death penalty deters violent crimes one said yes, another no after reading: proponents were more in favor of capital punishment opponents less in favor of capital punishment proponents had greater belief in the deterrent effect of capital punishment opponents reported less belief in this deterrent effect
8 confirmation bias Plous (1991) replicated this study in the context of judgments about the safety of nuclear technology pro- and anti-nuclear subjects were given identical information and arguments regarding the Three Mile Island disaster belief change: 54 % of pronuclear subjects became more pronuclear only 7 % became less pronuclear 45 % of antinuclear subjects became more antinuclear only 7% became less antinuclear
9 sample size fallacies mathematical truth: noise goes down as sample size increases small samples are random in large samples more structure appears in the limit, frequecies converge to probabilities law of large numbers (LLN)
10 sample size fallacies noise true relationship 1 sample size general phenomenon: people misunderstand sample size
11 sample size fallacies noise true relationship LLN 1 sample size general phenomenon: people misunderstand sample size
12 sample size fallacies noise true relationship perception 1 sample size general phenomenon: people misunderstand sample size
13 sample size fallacies noise true relationship perception 1 sample size general phenomenon: people misunderstand sample size one extreme: law of small numbers (people think small samples are large and the LLN applies) in between: sampling variation fallacy (mispredict differences in randomness of samples of different size) another extreme: non-belief in the law of large numbers (underestimate how fast the LLN kicks in)
14 two manifestations of sample size fallacies noise informativeness true relationship true relationship 1 sample size 1 sample size people mis-predict randomness people mis-infer from samples
15 sample size and informativeness suppose that you have a coin with an unknown bias for simplicity, you assign equal probability to two scenarios: 3 5 Heads, 2 5 Tails (call this θ H) 2 5 Heads, 3 5 Tails (call this θ T ) toss 3 times and see HHT what are your posterior odds of θ H to θ L
16 sample size and informativeness so P(θ H HHHTT ) P(θ T HHHTT ) = P(HHHTT θ H) P(HHHTT θ T ) P(θ H) P(θ T ) = = 1.5 P(θ H HHHTT ) = 0.6
17 sample size and informativeness P(θ H (HHHTT ) n ) P(θ T (HHHTT ) n ) = P((HHHTT )n θ H ) P((HHHTT ) n θ T ) P(θ H) P(θ T ) ( 3 5 = ) 2 n = 1.5 n posterior belief in Θ H sample size with frequency 3 of H 5
18 two manifestations of sample size fallacies noise informativeness true relationship true relationship 1 sample size 1 sample size people mis-predict randomness people mis-infer from samples
19 two manifestations of sample size fallacies noise informativeness true relationship perception perception true relationship 1 sample size 1 sample size people mis-predict randomness people mis-infer from samples
20 the law of small numbers the false belief that in the LLN holds in small samples example: sex of six babies born in a sequence in a hospital which sequence is most likely? BBBGGG GGGGGG BGBBGB under independence and P(B) = B(G) = 1 2 : P(BBBGGG) = P(GGGGGG) = P(BGBBGB) =
21 the law of small numbers suppose you are tossing a fair coin and heads came up 7 times in a row what is more likely in the next toss: heads? tails? gambler s fallacy
22 gambler s fallacy gamblers fallacy: the false belief that in a sequence of independent draws from a distribution, an outcome that hasnt occurred for a while is more likely to come up on the next draw if red came up in roulette four times in a row, a black is due mean reversion; regression to the mean
23 Gold and Hester s (1987) experiment subjects were told that a coin with one black and one red side would be flipped 25 times the experimenter actually reported a pre-determined sequence of flips: 17 mixed, then 1 black, and then 4 red on the 23rd flip, participants were given a choice between 70 points for sure 100 points if the next flip was their color, 0 pts otherwise half of the subjects color was red and half s was black results: 24 of 29 red subjects took the sure thing 8 of 30 black subjects took the sure thing
24 hot hand fallacy Gilovich, Vallone and Tversky (1985) 91% of fans agreed that a player has a better chance of making a shot after having just made his last two or three shots 68% of fans agreed that a player has a better chance of making a shot after having just made his last two or three free throws 84% of fans agreed that it was important to pass the ball to someone who has just made several (two, three, or four) shots in a row. Gilovich et al. analyzed basketball statistics basically, P(H HHH) = P(H HMH) = P(H MMM)
25 hot hand fallacy
26 hot hand fallacy coach of the Boston Celtics on the Gilovich et al study: Who is this guy? So he makes a study. I couldn t care less.
27 hot hand fallacy likely explanation: over-inference people have the following model in mind: hot streaks P(H) is higher cold streaks P(L) is lower if we observe HHH this means that the chances are that the player is in a good shape today so the next shot will be a hit if we observe MMM this means that the chances are that the player is in a bad shape yesterday so the next shot will be a miss so also a law of small numbers phenomenon!
28 sampling variation fallacy A study of the incidence of kidney cancer in the 3,141 counties of the United States reveals a remarkable pattern. The counties in which the incidence of kidney cancer is lowest are mostly rural, sparsely populated, and located in traditionally Republican states in the Midwest, the South, and the West. What do you make of this?
29 typical replies sparsely populated located in traditionally Republican states rural typical responses focus on clean living of the rural lifestyle: no air pollution, no water pollution, healthy food
30 sampling variation fallacy A study of the incidence of kidney cancer in the 3,141 counties of the United States reveals a remarkable pattern. The counties in which the incidence of kidney cancer is highest are mostly rural, sparsely populated, and located in traditionally Republican states in the Midwest, the South, and the West. What do you make of this?
31 typical replies sparsely populated located in traditionally Republican states rural typical responses focus on poverty of the rural lifestyle: no access to good medical care, climate, they drink too much
32 sampling variation fallacy jar with 50% black and 50% white stones David draws 4 stones at the time David wins if all stones are of one color Tomasz draws 7 stones at the time Tomasz wins if all stones are of one color not a zero-sum game! Who is more likely to win? how much more likely?
33 sampling variation fallacy possible states for David: BBBB, BBBW, BBWB,..., total of 2 4 = 16 states we are interested in two: BBBB and WWWW probability that David wins: 2 16 possible states for Tomasz: BBBBBBB, BBBBBBW, BBBBBWB,..., total of 2 7 = 128 states we are interested in two: BBBBBBB and WWWWWWW 2 probability that Tomasz wins: 128 David wins = 8 times more frequently than Tomasz
34 sampling variation fallacy In a small town nearby, there are two hospitals. Hospital A has an average of 45 births per day; Hospital B is smaller, and has an average of 15 births per day. As we all know, overall the number of males born is 50%. Each hospital recorded the number of days in which, on that day, at least 60% of the babies born were male. which hospital recorded more such days: (a) hospital A (b) hospital B (c) both equal
35 sampling variation fallacy In a small town nearby, there are two hospitals. Hospital A has exactly 1000 births per day; Hospital B is smaller, and has exactly 1 birth per day. As we all know, overall the number of males born is 50%. Each hospital recorded the number of days in which, on that day, at least 90% of the babies born were male. Which hospital recorded more such days: (a) hospital A (b) hospital B (c) both equal
36 non-belief in the LLN suppose we toss a fair coin N times what is the probability that the fraction of heads is: between 45% and 55%? between 85% and 95%? etc...
37 Kahneman and Tversky (1971) reported probabilities
38 true probabilities Kahneman and Tversky (1971) reported probabilities
39 back to the coin example suppose that you have a coin with an unknown bias for simplicity, you assign equal probability to two scenarios: 3 5 Heads, 2 5 Tails (call this θ H) 2 5 Heads, 3 5 Tails (call this θ T ) you observed n heads and m tails what are your posterior odds of θ H to θ L
40 sample size P(θ H H n T m ) P(θ T H n T m ) = P(Hn T m θ H ) P(H n T m θ T ) P(θ H) P(θ T ) ) n ( 2 m 5) = = ( 3 5 ( 2 ) n ( 3 m 5 5) ( 3 n m ( 5) 2 ) n m = (1.5) n m 5 this is a case where the sample size should not matter this is pretty sophisticated!
41 Griffin and Tversky (1992)
42 perspective biases in statistical reasoning base rate fallacies sample size fallacies people don t do exactly what the Bayes law says how to model their learning? work of Matthew Rabin and coauthors work of Andrei Shleifer and coauthors
43 perspective Bayes theorem is 300 years old there are new theorems in statistics proven every day does this mean that economic agents must know them? at what point do these experiments become IQ tests? difference between experiments and problem sets? our statistical intuitions often point in the right direction bounded rationality which word to emphasize? Heuristics and Biases
44
45 going forward... next two classes will be about behavioral game theory one way to think about it using what we learned: in strategic situations lots of inferences to make people mis-infer from the behavior of the others
Law of Large Numbers. Alexandra Barbato and Craig O Connell. Honors 391A Mathematical Gems Jenia Tevelev
Law of Large Numbers Alexandra Barbato and Craig O Connell Honors 391A Mathematical Gems Jenia Tevelev Jacob Bernoulli Life of Jacob Bernoulli Born into a family of important citizens in Basel, Switzerland
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 informationIgnoring base rates. Ignoring base rates (cont.) Improving our judgments. Base Rate Neglect. When Base Rate Matters. Probabilities vs.
Ignoring base rates People were told that they would be reading descriptions of a group that had 30 engineers and 70 lawyers. People had to judge whether each description was of an engineer or a lawyer.
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 informationStatistical Fallacies: Lying to Ourselves and Others
Statistical Fallacies: Lying to Ourselves and Others "There are three kinds of lies: lies, damned lies, and statistics. Benjamin Disraeli +/- Benjamin Disraeli Introduction Statistics, assuming they ve
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 informationElementary Statistics and Inference. Elementary Statistics and Inference. 16 The Law of Averages (cont.) 22S:025 or 7P:025.
Elementary Statistics and Inference 22S:025 or 7P:025 Lecture 20 1 Elementary Statistics and Inference 22S:025 or 7P:025 Chapter 16 (cont.) 2 D. Making a Box Model Key Questions regarding box What numbers
More informationExpected Value. 24 February 2014. Expected Value 24 February 2014 1/19
Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery
More informationSection 7C: The Law of Large Numbers
Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half
More informationAmerican Economic Association
American Economic Association Does the Basketball Market Believe in the `Hot Hand,'? Author(s): Colin F. Camerer Source: The American Economic Review, Vol. 79, No. 5 (Dec., 1989), pp. 1257-1261 Published
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 informationREWARD System For Even Money Bet in Roulette By Izak Matatya
REWARD System For Even Money Bet in Roulette By Izak Matatya By even money betting we mean betting on Red or Black, High or Low, Even or Odd, because they pay 1 to 1. With the exception of the green zeros,
More informationBayesian Updating with Discrete Priors Class 11, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
1 Learning Goals Bayesian Updating with Discrete Priors Class 11, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1. Be able to apply Bayes theorem to compute probabilities. 2. Be able to identify
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 informationStatistics and Random Variables. Math 425 Introduction to Probability Lecture 14. Finite valued Random Variables. Expectation defined
Expectation Statistics and Random Variables Math 425 Introduction to Probability Lecture 4 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 9, 2009 When a large
More informationWeek 2: Conditional Probability and Bayes formula
Week 2: Conditional Probability and Bayes formula We ask the following question: suppose we know that a certain event B has occurred. How does this impact the probability of some other A. This question
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 informationWORKED EXAMPLES 1 TOTAL PROBABILITY AND BAYES THEOREM
WORKED EXAMPLES 1 TOTAL PROBABILITY AND BAYES THEOREM EXAMPLE 1. A biased coin (with probability of obtaining a Head equal to p > 0) is tossed repeatedly and independently until the first head is observed.
More informationChapter 6: Probability
Chapter 6: Probability In a more mathematically oriented statistics course, you would spend a lot of time talking about colored balls in urns. We will skip over such detailed examinations of probability,
More informationAMS 5 CHANCE VARIABILITY
AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and
More information(SEE IF YOU KNOW THE TRUTH ABOUT GAMBLING)
(SEE IF YOU KNOW THE TRUTH ABOUT GAMBLING) Casinos loosen the slot machines at the entrance to attract players. FACT: This is an urban myth. All modern slot machines are state-of-the-art and controlled
More informationBetting systems: how not to lose your money gambling
Betting systems: how not to lose your money gambling G. Berkolaiko Department of Mathematics Texas A&M University 28 April 2007 / Mini Fair, Math Awareness Month 2007 Gambling and Games of Chance Simple
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 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 informationFind an expected value involving two events. Find an expected value involving multiple events. Use expected value to make investment decisions.
374 Chapter 8 The Mathematics of Likelihood 8.3 Expected Value Find an expected value involving two events. Find an expected value involving multiple events. Use expected value to make investment decisions.
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 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 informationA probability experiment is a chance process that leads to well-defined outcomes. 3) What is the difference between an outcome and an event?
Ch 4.2 pg.191~(1-10 all), 12 (a, c, e, g), 13, 14, (a, b, c, d, e, h, i, j), 17, 21, 25, 31, 32. 1) What is a probability experiment? A probability experiment is a chance process that leads to well-defined
More informationMrMajik s Money Management Strategy Copyright MrMajik.com 2003 All rights reserved.
You are about to learn the very best method there is to beat an even-money bet ever devised. This works on almost any game that pays you an equal amount of your wager every time you win. Casino games are
More informationMathematical goals. Starting points. Materials required. Time needed
Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about
More informationProbability, statistics and football Franka Miriam Bru ckler Paris, 2015.
Probability, statistics and football Franka Miriam Bru ckler Paris, 2015 Please read this before starting! Although each activity can be performed by one person only, it is suggested that you work in groups
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 informationExpected Value and the Game of Craps
Expected Value and the Game of Craps Blake Thornton Craps is a gambling game found in most casinos based on rolling two six sided dice. Most players who walk into a casino and try to play craps for the
More informationLecture 8 The Subjective Theory of Betting on Theories
Lecture 8 The Subjective Theory of Betting on Theories Patrick Maher Philosophy 517 Spring 2007 Introduction The subjective theory of probability holds that the laws of probability are laws that rational
More informationPascal is here expressing a kind of skepticism about the ability of human reason to deliver an answer to this question.
Pascal s wager So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not Does God exist? but Should we believe in God? What is distinctive
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 informationJohn Kerrich s coin-tossing Experiment. Law of Averages - pg. 294 Moore s Text
Law of Averages - pg. 294 Moore s Text When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So, if the coin is tossed a large number of times, the number of heads and the
More informationContemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4?
Contemporary Mathematics- MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than
More information13.0 Central Limit Theorem
13.0 Central Limit Theorem Discuss Midterm/Answer Questions Box Models Expected Value and Standard Error Central Limit Theorem 1 13.1 Box Models A Box Model describes a process in terms of making repeated
More informationHow To Find Out If You Believe You Are More Likely To Be Right Or Wrong
Attitude Polarization: Theory and Evidence Jean-Pierre Benoît London Business School Juan Dubra Universidad de Montevideo July, 04 Abstract Numerous experiments have demonstrated the possibility of attitude
More informationLesson 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 informationRecursive Estimation
Recursive Estimation Raffaello D Andrea Spring 04 Problem Set : Bayes Theorem and Bayesian Tracking Last updated: March 8, 05 Notes: Notation: Unlessotherwisenoted,x, y,andz denoterandomvariables, f x
More informationElementary Statistics and Inference. Elementary Statistics and Inference. 17 Expected Value and Standard Error. 22S:025 or 7P:025.
Elementary Statistics and Inference S:05 or 7P:05 Lecture Elementary Statistics and Inference S:05 or 7P:05 Chapter 7 A. The Expected Value In a chance process (probability experiment) the outcomes of
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 informationTrading Binary Options Strategies and Tactics
Trading Binary Options Strategies and Tactics Binary options trading is not a gamble or a guessing game. By using both fundamental and technical market analysis, you are able to get a better understanding
More informationRemarks on the Concept of Probability
5. Probability A. Introduction B. Basic Concepts C. Permutations and Combinations D. Poisson Distribution E. Multinomial Distribution F. Hypergeometric Distribution G. Base Rates H. Exercises Probability
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 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 information2 What is Rational: Normative. 4 Lottery & Gambling
1 Decision Making 2 What is Rational: Normative Key Question: Are you Rational? Deviations from rationality: Value is subjective (utility, and more..). Value is multidimensional (multiple budgets, not
More informationAP Statistics 7!3! 6!
Lesson 6-4 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!
More informationProbability and Expected Value
Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are
More information3.2 Roulette and Markov Chains
238 CHAPTER 3. DISCRETE DYNAMICAL SYSTEMS WITH MANY VARIABLES 3.2 Roulette and Markov Chains In this section we will be discussing an application of systems of recursion equations called Markov Chains.
More informationQuestion 1 Formatted: Formatted: Formatted: Formatted:
In many situations in life, we are presented with opportunities to evaluate probabilities of events occurring and make judgments and decisions from this information. In this paper, we will explore four
More informationLecture 4: Psychology of probability: predictable irrationality.
Lecture 4: Psychology of probability: predictable irrationality. David Aldous September 15, 2014 Here are two extreme views of human rationality. (1) There is much evidence that people are not rational,
More informationIntroductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014
Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities
More informationBayesian Analysis for the Social Sciences
Bayesian Analysis for the Social Sciences Simon Jackman Stanford University http://jackman.stanford.edu/bass November 9, 2012 Simon Jackman (Stanford) Bayesian Analysis for the Social Sciences November
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 informationChapter 16. Law of averages. Chance. Example 1: rolling two dice Sum of draws. Setting up a. Example 2: American roulette. Summary.
Overview Box Part V Variability The Averages Box We will look at various chance : Tossing coins, rolling, playing Sampling voters We will use something called s to analyze these. Box s help to translate
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 informationDo Psychological Fallacies Influence Trading in Financial Markets? Evidence from the Foreign Exchange Market
Discussion Paper No. 2014-17 Michael Bleaney, Spiros Bougheas and Zhiyong Li November 2014 Do Psychological Fallacies Influence Trading in Financial Markets? Evidence from the Foreign Exchange Market CeDEx
More informationThe result of the bayesian analysis is the probability distribution of every possible hypothesis H, given one real data set D. This prestatistical approach to our problem was the standard approach of Laplace
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 informationFormula for Theoretical Probability
Notes Name: Date: Period: Probability I. Probability A. Vocabulary is the chance/ likelihood of some event occurring. Ex) The probability of rolling a for a six-faced die is 6. It is read as in 6 or out
More informationQuestion: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationThe Contrarian Let It Ride System For Trading Binary Options Profitably
The Contrarian Let It Ride System For Trading Binary Options Profitably by: William Hughes of Mass Money Machine This publication is subject to all relevant copyright laws. Reproduction or translation
More informationChapter 16: law of averages
Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................
More informationFree Human Memory Experiments For College Students
UNIVERSITY OF EDINBURGH COLLEGE OF SCIENCE AND ENGINEERING SCHOOL OF INFORMATICS COGNITIVE MODELLING (LEVEL 10) COGNITIVE MODELLING (LEVEL 11) Friday 1 April 2005 00:00 to 00:00 Year 4 Courses Convener:
More informationPrediction Markets, Fair Games and Martingales
Chapter 3 Prediction Markets, Fair Games and Martingales Prediction markets...... are speculative markets created for the purpose of making predictions. The current market prices can then be interpreted
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 informationIn the situations that we will encounter, we may generally calculate the probability of an event
What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead
More informationBanking on a Bad Bet: Probability Matching in Risky Choice Is Linked to Expectation Generation
Research Report Banking on a Bad Bet: Probability Matching in Risky Choice Is Linked to Expectation Generation Psychological Science 22(6) 77 711 The Author(s) 11 Reprints and permission: sagepub.com/journalspermissions.nav
More informationAlgebra 2 C Chapter 12 Probability and Statistics
Algebra 2 C Chapter 12 Probability and Statistics Section 3 Probability fraction Probability is the ratio that measures the chances of the event occurring For example a coin toss only has 2 equally likely
More informationStatistics 100A Homework 4 Solutions
Chapter 4 Statistics 00A Homework 4 Solutions Ryan Rosario 39. A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it is then replaced and another ball is drawn.
More informationCHAPTER 2 Estimating Probabilities
CHAPTER 2 Estimating Probabilities Machine Learning Copyright c 2016. Tom M. Mitchell. All rights reserved. *DRAFT OF January 24, 2016* *PLEASE DO NOT DISTRIBUTE WITHOUT AUTHOR S PERMISSION* This is a
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 informationLecture 13. Understanding Probability and Long-Term Expectations
Lecture 13 Understanding Probability and Long-Term Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).
More informationThe overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES
INTRODUCTION TO CHANCE VARIABILITY WHAT DOES THE LAW OF AVERAGES SAY? 4 coins were tossed 1600 times each, and the chance error number of heads half the number of tosses was plotted against the number
More informationStatistics and Data Analysis B01.1305
Statistics and Data Analysis B01.1305 Professor William Greene Phone: 212.998.0876 Office: KMC 7-78 Home page: www.stern.nyu.edu/~wgreene Email: wgreene@stern.nyu.edu Course web page: www.stern.nyu.edu/~wgreene/statistics/outline.htm
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 informationMath 58. Rumbos Fall 2008 1. Solutions to Review Problems for Exam 2
Math 58. Rumbos Fall 2008 1 Solutions to Review Problems for Exam 2 1. For each of the following scenarios, determine whether the binomial distribution is the appropriate distribution for the random variable
More informationChapter 5 Section 2 day 1 2014f.notebook. November 17, 2014. Honors Statistics
Chapter 5 Section 2 day 1 2014f.notebook November 17, 2014 Honors Statistics Monday November 17, 2014 1 1. Welcome to class Daily Agenda 2. Please find folder and take your seat. 3. Review Homework C5#3
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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability that the result
More informationLecture 9: Bayesian hypothesis testing
Lecture 9: Bayesian hypothesis testing 5 November 27 In this lecture we ll learn about Bayesian hypothesis testing. 1 Introduction to Bayesian hypothesis testing Before we go into the details of Bayesian
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 informationPractical Probability:
Practical Probability: Casino Odds and Sucker Bets Tom Davis tomrdavis@earthlink.net April 2, 2011 Abstract Gambling casinos are there to make money, so in almost every instance, the games you can bet
More information7.S.8 Interpret data to provide the basis for predictions and to establish
7 th Grade Probability Unit 7.S.8 Interpret data to provide the basis for predictions and to establish experimental probabilities. 7.S.10 Predict the outcome of experiment 7.S.11 Design and conduct an
More informationHomework 8 Solutions
CSE 21 - Winter 2014 Homework Homework 8 Solutions 1 Of 330 male and 270 female employees at the Flagstaff Mall, 210 of the men and 180 of the women are on flex-time (flexible working hours). Given that
More informationStat 20: Intro to Probability and Statistics
Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. Childers-Day UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum
More informationCausality and the Doomsday Argument
Causality and the Doomsday Argument Ivan Phillips The Futurists Guild January 24 th 2005 Abstract Using the Autodialer thought experiment, we show that the Self-Sampling Assumption (SSA) is too general,
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 informationStatistics and Probability
Statistics and Probability TABLE OF CONTENTS 1 Posing Questions and Gathering Data. 2 2 Representing Data. 7 3 Interpreting and Evaluating Data 13 4 Exploring Probability..17 5 Games of Chance 20 6 Ideas
More informationResponsible Gambling Education Unit: Mathematics A & B
The Queensland Responsible Gambling Strategy Responsible Gambling Education Unit: Mathematics A & B Outline of the Unit This document is a guide for teachers to the Responsible Gambling Education Unit:
More informationMONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010
MONT 07N Understanding Randomness Solutions For Final Examination May, 00 Short Answer (a) (0) How are the EV and SE for the sum of n draws with replacement from a box computed? Solution: The EV is n times
More informationStandard 12: The student will explain and evaluate the financial impact and consequences of gambling.
STUDENT MODULE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Simone, Paula, and Randy meet in the library every
More informationThis Method will show you exactly how you can profit from this specific online casino and beat them at their own game.
This Method will show you exactly how you can profit from this specific online casino and beat them at their own game. It s NOT complicated, and you DON T need a degree in mathematics or statistics to
More informationBeating Roulette? An analysis with probability and statistics.
The Mathematician s Wastebasket Volume 1, Issue 4 Stephen Devereaux April 28, 2013 Beating Roulette? An analysis with probability and statistics. Every time I watch the film 21, I feel like I ve made the
More informationACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Answers
ACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Answers Name DO NOT remove this answer page. DO turn in the entire exam. Make sure that you have all ten (10) pages
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Ch. 4 Discrete Probability Distributions 4.1 Probability Distributions 1 Decide if a Random Variable is Discrete or Continuous 1) State whether the variable is discrete or continuous. The number of cups
More informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6-sided dice. What s the probability of rolling at least one 6? There is a 1
More information