ECE 316 Probability Theory and Random Processes


 Shawn Campbell
 2 years ago
 Views:
Transcription
1 ECE 316 Probability Theory and Random Processes Chapter 4 Solutions (Part 2) Xinxin Fan Problems 20. A gambling book recommends the following winning strategy for the game of roulette. It recommends that a gambler bet $1 on red. If red appears (which has probability 18 ), then the gambler should take her $1 profit and quit. If the gambler loses this bet (which has probability 20 of occurring), she should make additional $1 bets on red on each of the next two spins of the roulette wheel and then quit. Let X denote the gambler s winnings when she quits. (a) Find P {X > 0}. (b) Are you convinced that the strategy is indeed a winning strategy? Explain your answer! (c) Find E[X]. Solution. (a) The event that X > 0 denotes that the gambler wins the first bet or he loses the first bet and wins the next two bets. Therefore, we get P {X > 0} = P {win first bet} + P {lose, win, win} = ( ) (b) The strategy described above is not a winning strategy because if the gambler wins then he or she wins $1. However, a loss would either be $1 or $3. (c) We first note that the random variable X can take on 1, 1 and 3, where denotes that the gambler loses money. Then we obtain P {X = 1} = P {win first bet} + P {lose, win, win} = P {X = 1} = P {lose, lose, win} + P {lose, win, lose} = 2 18 ( ) 20 3 P {X = 3} = P {lose, lose, lose} =. Therefore, we can compute the expectation as follows: ( ) 18 2, ( ) 20 2, E[X] = 1 P {X = 1} + ( 1) P {X = 1} + ( 3) P {X = 3} A typical slot machine has 3 dials, each with 20 symbols (cherries, lemons, plums, oranges, bells, and bars). A typical set of dials is shown in Table 1. According to this table, of the 20 slots on dial 1, are cherries, 3 are oranges, and so on. A typical payoff on a 1unit 1
2 Table 1: Slot machine dial setup Dial 1 Dial 2 Dial 3 Cherries 0 Oranges 3 6 Lemons Plums Bells Bars Table 2: Typical payoff on a 1unit bet Dial 1 Dial 2 Dial 3 Payoff Bar Bar Bar 60 Bell Bell Bell 20 Bell Bell Bar 18 Plum Plum Plum 14 Orange Orange Orange 10 Orange Orange Bar 8 Cherry Cherry Anything 2 Cherry No cherry Anything 0 Anything else 1 bet is shown in Table 2. Compute the player s expected winnings on a single play of the slot machine. Assume that each dial acts independently. Solution. Let X denote the payoff on a 1unit bet. Then we get P {X = 60} = P {Bar, Bar, Bar} = = 3 P {X = 20} = P {Bell, Bell, Bell} = = 12 P {X = 18} = P {Bell, Bell, Bar} = = 4 P {X = 14} = P {Plum, Plum, Plum} = = 24 P {X = 10} = P {Orange, Orange, Orange} = = 126 P {X = 8} = P {Orange, Orange, Bar} = 3 20 P {X = 2} = P {Cherry, Cherry, Anything} = = = , 2
3 P {X = 0} = P {Cherry, No cherry, Anything} = = , P {X = 1} = P {Anything else} = = Therefore, the player s expected winnings on a single play of the slot machine can be computed as follows: E[X] = 60 P {X = 60} + 20 P {X = 20} + 18 P {X = 18} + = 14 P {X = 14} + 10 P {X = 10} + 8 P {X = 8} + 2 P {X = 2} + 0 P {X = 0} + ( 1) P {X = 1} = To determine whether or not they have a certain disease, 100 people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to group the people in groups of 10. The blood samples of the 10 people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 10 people; whereas, if the test is positive each of the 10 people will also be individually tested and, in all, 11 tests will be made on this group. Assume the probability that a person has the disease is 0.1 for all people, independently of each other, and compute the expected number of tests necessary for each group. (Note that we are assuming that the pooled test will be positive if at least one person in the pool has the disease.) Solution. Let T be the number of tests for a group of 10 people. Then we know that T = 1 if the test is negative and T = 11 if the test is positive. Therefore, we get E[T ] = (0.9) [1 (0.9) 10 ] = (0.9) A newsboy purchases papers at 10 cents and sells them at 15 cents. However, he is not allowed to return unsold papers. If his daily demand is a binomial random variable with n = 10, p = 1 3, approximately how many papers should he purchase so as to maximize his expected profit? Solution. Let a be the number of papers the newsboy purchases, X be the daily demand, and Y be the profit the newsboy gets. Then we obtain the relation between random variables X and Y as follows: { 5a X a Y = 15X 10a X < a. Therefore, the expected profit that the newsboy obtains is E[Y ] = 5a P (X a) + (15X 10a) P (X < a) = 10 ( ) ( ) 10 1 i ( ) 2 10 i a 1 5a + (15i 10a) i 3 3 i=a i=0 ( 10 i ) ( 1 3 ) i ( ) 2 10 i 3 To find the approximate value for a, we can use Matlab to draw the following figure to show the relation between E[Y ] and a. From Fig. 1, we note that the newsboy should purchase 3 papers so as to maximize his expected profit. 3
4 E[Y] a Figure 1: The Expected Profit E[Y ]. If E[X] = 1 and Var(X) = 5, find (a) E[(2 + X) 2 ]; (b) Var(4 + 3X). Solution. (a) E[(2 + X) 2 ] = Var(2 + X) + (E[2 + X]) 2 = Var(X) + 9 = 14. (b) Var(4 + 3X) = 9 Var(X) = A satellite system consists of n components and functions on any given day if at least k of the n components function on that day. On a rainy day each of the components independently functions with probability p 1, whereas on a dry day they each independently function with probability p 2. If the probability of rain tomorrow is α, what is the probability that the satellite system will function? Solution. The probability that the satellite system will function can be computed as follows: P (system functions) = P (rain)p (system functions rain) + P (dry)p (system functions dry) n ( ) n n ( ) n = α p i i 1(1 p 1 ) n i + (1 α) p i i 2(1 p 2 ) n i. i=k 48. It is known that diskettes produced by a certain company will be defective with probability 0.01, independently of each other. The company sells the diskettes in packages of size 10 and offers a moneyback guarantee that at most 1 of the 10 diskettes in the package will be defective. If someone buys 3 packages, what is the probability that he or she will return exactly 1 of them? Solution. Let p be the probability that a package will be returned. The we obtain p = 1 (0.99) (0.99) 9 (0.01). Therefore, if someone buys 3 packages then the probability they will return exactly 1 package is 3p(1 p) When coin 1 is flipped, it lands heads with probability 0.4; when coin 2 is flipped, it lands heads with probability 0.. One of these coins is randomly chosen and flipped 10 times. i=k 4
5 (a) What is the probability that exactly of the 10 flips land on heads? (b) Given that the first of these ten flips lands heads, what is the conditional probability that exactly of the 10 flips land on heads?. Solution. (a) Note that one of two coins will be randomly chosen, we get P ( heads) = P (coin 1)P ( heads coin 1) + P (coin 2)P ( heads coin 2) = 1 ( ) 10 2 (0.4) (0.6) ( ) 10 2 (0.) (0.3) 3. (b) We can compute the conditional probability as follows: P ( heads 1st heads) = = 2 i=1 P (coin i)p ( heads, 1st heads coin i) P (1st heads) 1 2 (9 6 ) (0.4) (0.6) (9) 6 (0.) (0.3) Theoretical Exercises 4. If X has distribution function F, what is the distribution function of e X. Solution. Let G(x) be the distribution function of e X. Then we need to consider the following two cases: When x 0, we get G(x) = P {e X x} = 0. When x > 0, we obtain G(x) = P {e X x} = P {X ln x} = F (ln x). 5. If X has distribution function F, what is the distribution function of the random variable αx + β, where α and β are constants, α 0? Solution. Let G(x) be the distribution function of e X. Then we need to consider the following two cases: When α > 0, we get When α < 0, we obtain G(x) = P {αx + β x} = P G(x) = P {αx + β x} = P { X x β } = F α ( x β α ). { X x β } ( ) x β = 1 lim α F h 0 + α h. 5
6 8. Let X be such that Find c 1 such that E[c X ] = 1. P {X = 1} = p = 1 P {X = 1}. Solution. We have known the probability mass function of X: P {X = 1} = p and P {X = 1} = 1 p. Thus, E[c X ] = c 1 +c 1 (1 p). Let E[c X ] = cp + 1 p c = 1, it follows that c = p 1 p that E[c X ] = 1. or 1. Therefore, except 1, c = p 1 p satisfies 9. Let X be a random variable having expected value µ and variance σ 2. Find the expected value and variance of Y = X µ σ. Solution. Using the properties of expected value and variance, we get [ ] X µ E[Y ] = E (E[X] µ) = 0, = 1 σ σ E[X µ] = 1 σ ( ) X µ Var(Y ) = Var = σ ( ) 1 2 Var(X) = σ2 σ σ 2 = Let X be a binomial random variable with parameters (n, p). What value of p maximizes P {X = k}, k = 0, 1,..., n? This is an example of a statistical method used to estimate p when a binomial (n, p) random variable is observed to equal k. If we assume that n is known, then we estimate p by choosing that value of p that maximizes P {X = k}. This is known as the the method of maximum likelihood estimation. Solution. Note that when P {X = k} achieves the maximum value, log P {X = k} also gets the maximum value. Therefore, we can first take logarithm and then determine the p that maximizes log P {X = k}. More specifically, we first obtain ( ) n log P {X = k} = log + k log p + (n k) log(1 p). k Then we can find p by computing the derivative as follows: p log P {X = k} = k p n k 1 p = 0. Therefore, p = k n maximizes P {X = k} for k = 0, 1,..., n. 6
Statistics 100A Homework 3 Solutions
Chapter Statistics 00A Homework Solutions Ryan Rosario. Two balls are chosen randomly from an urn containing 8 white, black, and orange balls. Suppose that we win $ for each black ball selected and we
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a realvalued function defined on the sample space of some experiment. For instance,
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 informationMA 1125 Lecture 14  Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.
MA 5 Lecture 4  Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More informationECE316 Tutorial for the week of June 15
ECE316 Tutorial for the week of June 15 Problem 35 Page 176: refer to lecture notes part 2, slides 8, 15 A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same
More informationMath 425 (Fall 08) Solutions Midterm 2 November 6, 2008
Math 425 (Fall 8) Solutions Midterm 2 November 6, 28 (5 pts) Compute E[X] and Var[X] for i) X a random variable that takes the values, 2, 3 with probabilities.2,.5,.3; ii) X a random variable with the
More informationwww.problemgambling.sa.gov.au THE POKIES: BEFORE YOU PRESS THE BUTTON, KNOW THE FACTS.
www.problemgambling.sa.gov.au THE POKIES: BEFORE YOU PRESS THE BUTTON, KNOW THE FACTS. IMPORTANT INFORMATION FOR ANYONE WHO PLAYS THE POKIES The pokies are simply a form of entertainment. However, sometimes
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 informationMidterm Exam #1 Instructions:
Public Affairs 818 Professor: Geoffrey L. Wallace October 9 th, 008 Midterm Exam #1 Instructions: You have 10 minutes to complete the examination and there are 6 questions worth a total of 10 points. The
More informationExample. A casino offers the following bets (the fairest bets in the casino!) 1 You get $0 (i.e., you can walk away)
: Three bets Math 45 Introduction to Probability Lecture 5 Kenneth Harris aharri@umich.edu Department of Mathematics University of Michigan February, 009. A casino offers the following bets (the fairest
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 informationMidterm Exam #1 Instructions:
Public Affairs 818 Professor: Geoffrey L. Wallace October 9 th, 008 Midterm Exam #1 Instructions: You have 10 minutes to complete the examination and there are 6 questions worth a total of 10 points. The
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 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 informationHONORS STATISTICS. Mrs. Garrett Block 2 & 3
HONORS STATISTICS Mrs. Garrett Block 2 & 3 Tuesday December 4, 2012 1 Daily Agenda 1. Welcome to class 2. Please find folder and take your seat. 3. Review OTL C7#1 4. Notes and practice 7.2 day 1 5. Folders
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 informationBetting with the Kelly Criterion
Betting with the Kelly Criterion Jane June 2, 2010 Contents 1 Introduction 2 2 Kelly Criterion 2 3 The Stock Market 3 4 Simulations 5 5 Conclusion 8 1 Page 2 of 9 1 Introduction Gambling in all forms,
More informationUniversity of California, Los Angeles Department of Statistics. Random variables
University of California, Los Angeles Department of Statistics Statistics Instructor: Nicolas Christou Random variables Discrete random variables. Continuous random variables. Discrete random variables.
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 informationSolutions: Problems for Chapter 3. Solutions: Problems for Chapter 3
Problem A: You are dealt five cards from a standard deck. Are you more likely to be dealt two pairs or three of a kind? experiment: choose 5 cards at random from a standard deck Ω = {5combinations of
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 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 stateoftheart and controlled
More informationPLAYING GAMES OF CHANCE
SLOTS PLAYING GAMES OF CHANCE The one thing that all games of chance have in common is that winning or losing is based on randomness. While the dream of winning is exciting, it s important to know your
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 information.4 120 +.1 80 +.5 100 = 48 + 8 + 50 = 106.
Chapter 16. Risk and Uncertainty Part A 2009, Kwan Choi Expected Value X i = outcome i, p i = probability of X i EV = pix For instance, suppose a person has an idle fund, $100, for one month, and is considering
More informationYou can place bets on the Roulette table until the dealer announces, No more bets.
Roulette Roulette is one of the oldest and most famous casino games. Every Roulette table has its own set of distinctive chips that can only be used at that particular table. These chips are purchased
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 informationT: Here is a fruit machine with 3 DIALS and with 20 SYMBOLS on each dial. Each of the 20 symbols is equally likely to occur on each dial.
Activity Fruit Machines 1 Introduction T: Describe how fruit machines work T: Can you win? T: Here is a fruit machine with 3 DIALS and with 20 SYMBOLS on each dial. Each of the 20 symbols is equally likely
More informationWe { can see that if U = 2, 3, 7, 11, or 12 then the round is decided on the first cast, U = V, and W if U = 7, 11 X = L if U = 2, 3, 12.
How to Play Craps: Craps is a dice game that is played at most casinos. We will describe here the most common rules of the game with the intention of understanding the game well enough to analyze the probability
More informationFeb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172179)
Feb 7 Homework Solutions Math 151, Winter 2012 Chapter Problems (pages 172179) Problem 3 Three dice are rolled. By assuming that each of the 6 3 216 possible outcomes is equally likely, find the probabilities
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 informationECE302 Spring 2006 HW3 Solutions February 2, 2006 1
ECE302 Spring 2006 HW3 Solutions February 2, 2006 1 Solutions to HW3 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
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 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 informationRandom variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.
Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()
More informationפרויקט מסכם לתואר בוגר במדעים )B.Sc( במתמטיקה שימושית
המחלקה למתמטיקה Department of Mathematics פרויקט מסכם לתואר בוגר במדעים )B.Sc( במתמטיקה שימושית הימורים אופטימליים ע"י שימוש בקריטריון קלי אלון תושיה Optimal betting using the Kelly Criterion Alon Tushia
More informationExpectation Discrete RV  weighted average Continuous RV  use integral to take the weighted average
PHP 2510 Expectation, variance, covariance, correlation Expectation Discrete RV  weighted average Continuous RV  use integral to take the weighted average Variance Variance is the average of (X µ) 2
More informationSome special discrete probability distributions
University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Some special discrete probability distributions Bernoulli random variable: It is a variable that
More informationSOME ASPECTS OF GAMBLING WITH THE KELLY CRITERION. School of Mathematical Sciences. Monash University, Clayton, Victoria, Australia 3168
SOME ASPECTS OF GAMBLING WITH THE KELLY CRITERION Ravi PHATARFOD School of Mathematical Sciences Monash University, Clayton, Victoria, Australia 3168 In this paper we consider the problem of gambling with
More informationMinimax Strategies. Minimax Strategies. Zero Sum Games. Why Zero Sum Games? An Example. An Example
Everyone who has studied a game like poker knows the importance of mixing strategies With a bad hand, you often fold But you must bluff sometimes Lectures in MicroeconomicsCharles W Upton Zero Sum Games
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 evenmoney 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 informationBayesian logistic betting strategy against probability forecasting. Akimichi Takemura, Univ. Tokyo. November 12, 2012
Bayesian logistic betting strategy against probability forecasting Akimichi Takemura, Univ. Tokyo (joint with Masayuki Kumon, Jing Li and Kei Takeuchi) November 12, 2012 arxiv:1204.3496. To appear in Stochastic
More informationThe Math. P (x) = 5! = 1 2 3 4 5 = 120.
The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct
More informationLaw 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 informationSlide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value.
Slide 1 Math 1520, Lecture 23 This lecture covers mean, median, mode, odds, and expected value. Slide 2 Mean, Median and Mode Mean, Median and mode are 3 concepts used to get a sense of the central tendencies
More informationInside the pokies  player guide
Inside the pokies  player guide 3nd Edition  May 2009 References 1, 2, 3 Productivity Commission 1999, Australia s Gambling Industries, Report No. 10, AusInfo, Canberra. 4 Victorian Department of Justice,
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 informationFourth Problem Assignment
EECS 401 Due on Feb 2, 2007 PROBLEM 1 (25 points) Joe and Helen each know that the a priori probability that her mother will be home on any given night is 0.6. However, Helen can determine her mother s
More informationQuestions and Answers
MA3245 Financial Mathematics I Suggested Solutions of Tutorial 1 (Semester 2/0304) Questions and Answers 1. What is the difference between entering into a long forward contract when the forward price
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 informationDecision Making Under Uncertainty. Professor Peter Cramton Economics 300
Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate
More informationThe Kelly criterion for spread bets
IMA Journal of Applied Mathematics 2007 72,43 51 doi:10.1093/imamat/hxl027 Advance Access publication on December 5, 2006 The Kelly criterion for spread bets S. J. CHAPMAN Oxford Centre for Industrial
More informationRoulette. Math 5 Crew. Department of Mathematics Dartmouth College. Roulette p.1/14
Roulette p.1/14 Roulette Math 5 Crew Department of Mathematics Dartmouth College Roulette p.2/14 Roulette: A Game of Chance To analyze Roulette, we make two hypotheses about Roulette s behavior. When we
More informationTHE WINNING ROULETTE SYSTEM by http://www.webgoldminer.com/
THE WINNING ROULETTE SYSTEM by http://www.webgoldminer.com/ Is it possible to earn money from online gambling? Are there any 100% sure winning roulette systems? Are there actually people who make a living
More informationAP Statistics 7!3! 6!
Lesson 64 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 informationHomework 4  KEY. Jeff Brenion. June 16, 2004. Note: Many problems can be solved in more than one way; we present only a single solution here.
Homework 4  KEY Jeff Brenion June 16, 2004 Note: Many problems can be solved in more than one way; we present only a single solution here. 1 Problem 21 Since there can be anywhere from 0 to 4 aces, the
More information1 Interest rates, and riskfree investments
Interest rates, and riskfree investments Copyright c 2005 by Karl Sigman. Interest and compounded interest Suppose that you place x 0 ($) in an account that offers a fixed (never to change over time)
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 11: Choice Under Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 11: Choice Under Uncertainty Tuesday, October 21, 2008 Last class we wrapped up consumption over time. Today we
More informationAustralian Reels Slot Machine By Michael Shackleford, A.S.A. January 13, 2009
Australian Reels Slot Machine By Michael Shackleford, A.S.A. January 13, 2009 Game Description This is a threereel, singleline, stepper slot. URL: wizardofodds.com/play/slotausreels/ For academic use
More informationMathematical Expectation
Mathematical Expectation Properties of Mathematical Expectation I The concept of mathematical expectation arose in connection with games of chance. In its simplest form, mathematical expectation is the
More informationEthical Gambling: policies of gambling in modern societies
Ethical Gambling: policies of gambling in modern societies Mariano Chóliz, PhD Psychology School University of Valencia Spain Ethical Gambling: a necessary concept Can be ethical an economic activity which
More information510  2040 80160  320640  12802560  512010,24020,48040,96081,920
Progression Betting With horse betting, or any kind of betting, anything other than flat betting is in fact a kind of progression... but the subject in this article is (as it should be) a bit controversial:
More informationRANDOM VARIABLES MATH CIRCLE (ADVANCED) 3/3/2013. 3 k) ( 52 3 )
RANDOM VARIABLES MATH CIRCLE (ADVANCED) //0 0) a) Suppose you flip a fair coin times. i) What is the probability you get 0 heads? ii) head? iii) heads? iv) heads? For = 0,,,, P ( Heads) = ( ) b) Suppose
More informationEconomics 1011a: Intermediate Microeconomics
Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore
More informationStatistics Class 10 2/29/2012
Statistics Class 10 2/29/2012 Quiz 8 When playing roulette at the Bellagio casino in Las Vegas, a gambler is trying to decide whether to bet $5 on the number 13 or to bet $5 that the outcome is any one
More informationDecision & Risk Analysis Lecture 6. Risk and Utility
Risk and Utility Risk  Introduction Payoff Game 1 $14.50 0.5 0.5 $30  $1 EMV 30*0.5+(1)*0.5= 14.5 Game 2 Which game will you play? Which game is risky? $50.00 Figure 13.1 0.5 0.5 $2,000  $1,900 EMV
More informationChapter 5. Discrete Probability Distributions
Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable
More informationSolution. Solution. (a) Sum of probabilities = 1 (Verify) (b) (see graph) Chapter 4 (Sections 4.34.4) Homework Solutions. Section 4.
Math 115 N. Psomas Chapter 4 (Sections 4.34.4) Homework s Section 4.3 4.53 Discrete or continuous. In each of the following situations decide if the random variable is discrete or continuous and give
More information6.042/18.062J Mathematics for Computer Science. Expected Value I
6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you
More informationTABLE OF CONTENTS. ROULETTE FREE System #1  2 ROULETTE FREE System #2  4  5
IMPORTANT: This document contains 100% FREE gambling systems designed specifically for ROULETTE, and any casino game that involves even money bets such as BLACKJACK, CRAPS & POKER. Please note although
More informationChapter 5. Section 5.1: Central Tendency. Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data.
Chapter 5 Section 5.1: Central Tendency Mode: the number or numbers that occur most often. Median: the number at the midpoint of a ranked data. Example 1: The test scores for a test were: 78, 81, 82, 76,
More informationRussell Hunter Publishing Inc
Russell Hunter Street Smart Roulette Video Course Russell Hunter Publishing Inc Street Smart Roulette Video Guide 2015 Russell Hunter and Russell Hunter Publishing. All Rights Reserved All rights reserved.
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 information14.4. Expected Value Objectives. Expected Value
. Expected Value Objectives. Understand the meaning of expected value. 2. Calculate the expected value of lotteries and games of chance.. Use expected value to solve applied problems. Life and Health Insurers
More informationMaking $200 a Day is Easy!
Making $200 a Day is Easy! Firstly, I'd just like to say thank you for purchasing this information. I do not charge a huge amount for it so I hope that you will find it useful. Please note that if you
More informationDiscrete Math in Computer Science Homework 7 Solutions (Max Points: 80)
Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you
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 informationLecture 5: Mathematical Expectation
Lecture 5: Mathematical Expectation Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics İstanbul Kültür University Outline 1 Introduction 2 The Expected Value of a Random
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 informationChapter 6 Continuous Probability Distributions
Continuous Probability Distributions Learning Objectives 1. Understand the difference between how probabilities are computed for discrete and continuous random variables. 2. Know how to compute probability
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 informationA WHISTLE BLOWER S GUIDE FROM HARINGEY RESIDENT AND GAMBLING INDUSTRY EXPERT DEREK WEBB OF PRIME TABLE GAMES
HARINGEY COUNCIL OVERVIEW AND SCRUTINY COMMITTEE INVESTIGATION OF BETTING SHOP CLUSTERING A WHISTLE BLOWER S GUIDE FROM HARINGEY RESIDENT AND GAMBLING INDUSTRY EXPERT DEREK WEBB OF PRIME TABLE GAMES Introduction
More informationECE302 Spring 2006 HW4 Solutions February 6, 2006 1
ECE302 Spring 2006 HW4 Solutions February 6, 2006 1 Solutions to HW4 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in
More informationEasy Casino Profits. Congratulations!!
Easy Casino Profits The Easy Way To Beat The Online Casinos Everytime! www.easycasinoprofits.com Disclaimer The authors of this ebook do not promote illegal, underage gambling or gambling to those living
More informationLecture 25: Money Management Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 25: Money Management Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Money Management Techniques The trading
More informationStandard 12: The student will explain and evaluate the financial impact and consequences of gambling.
TEACHER GUIDE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Priority Academic Student Skills Personal Financial
More informationFinancial Markets. Itay Goldstein. Wharton School, University of Pennsylvania
Financial Markets Itay Goldstein Wharton School, University of Pennsylvania 1 Trading and Price Formation This line of the literature analyzes the formation of prices in financial markets in a setting
More informationThe St Petersburg paradox
The St Petersburg paradox COMPSCI 36: Computational Cognitive Science Dan Navarro & Amy Perfors University of Adelaide Abstract This note provides a somewhat cleaner discussion of the St Petersburg paradox
More informationGreg Fletcher. Sharpshooter Roulette. Video Guide. Russell Hunter Publishing, Inc.
Greg Fletcher Sharpshooter Roulette Video Guide Russell Hunter Publishing, Inc. Sharpshooter Roulette Video Guide 2015 Greg Fletcher and Russell Hunter Publishing. All Rights Reserved All rights reserved.
More informationTerm Project: Roulette
Term Project: Roulette DCY Student January 13, 2006 1. Introduction The roulette is a popular gambling game found in all major casinos. In contrast to many other gambling games such as black jack, poker,
More informationMagic Bomb (VER.AMERICAN ALPHA)
Magic Bomb USER MENU (VER.AMERICAN ALPHA) PARTS SIDE SOLDER SIDE PARTS SIDE SOLDER SIDE This pin is normal low. When it enable is +5V. This pin is connected with the solder side 24th pin of connector 36
More informationWe rst consider the game from the player's point of view: Suppose you have picked a number and placed your bet. The probability of winning is
Roulette: On an American roulette wheel here are 38 compartments where the ball can land. They are numbered 136, and there are two compartments labeled 0 and 00. Half of the compartments numbered 136
More informationIntroduction to Game Theory IIIii. Payoffs: Probability and Expected Utility
Introduction to Game Theory IIIii Payoffs: Probability and Expected Utility Lecture Summary 1. Introduction 2. Probability Theory 3. Expected Values and Expected Utility. 1. Introduction We continue further
More informationChapter 14 From Randomness to Probability
226 Part IV Randomness and Probability Chapter 14 From Randomness to Probability 1. Roulette. If a roulette wheel is to be considered truly random, then each outcome is equally likely to occur, and knowing
More informationOrange High School. Year 7, 2015. Mathematics Assignment 2
Full name: Class teacher: Due date: Orange High School Year 7, 05 Mathematics Assignment Instructions All work must be your own. You are encouraged to use the internet but you need to rewrite your findings
More informationthe GentInG GUIde to slots
the GentInG GUIde to slots reward yourself every time GenTinG rewards Welcome to the world of slots Slots are a familiar sight in most casinos these days but, as with any casino game, it pays to take a
More informationBetting rules and information theory
Betting rules and information theory Giulio Bottazzi LEM and CAFED Scuola Superiore Sant Anna September, 2013 Outline Simple betting in favorable games The Central Limit Theorem Optimal rules The Game
More informationContents. Introduction...1. The Lingo...3. Know the Different Types of Machines...12. Know the Machine You re Playing...35
Contents Introduction...1 The Lingo...3 Know the Different Types of Machines...12 Know the Machine You re Playing...35 General Considerations...46 The Rest of The Story...55 Introduction Casinos today
More information