# Lab 11. Simulations. The Concept

Size: px
Start display at page:

Transcription

1 Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that can be used to simulate a wide range of probability problems. The Concept Probabilities are long-run frequencies. When we say something like the probability that event A will occur is 38%, what we mean is that there is an experiment, and one of the possible outcomes of this experiment is the event A. If we were to repeat this experiment infinitely many times, the event A would occur in 38% of them. For example, when we say that the probability of getting heads when we flip a coin is 50%, we mean that if we flip the coin infinitely many times, half of the flips would land heads. No one has the patience to flip a coin infinitely many times, even if they could live long enough. The best we can hope for is to see the outcome of a large number of flips. Fortunately, computers make it possible to repeat an experiment many times, very quickly. Rather than flip a real coin

2 times, which takes quite a bit of effort, we can program a computer to do the same in the blink of an eye. Or faster. Suppose we program a computer to do virtual coin flips. Suppose everyone in the room flips a virtual coin 1000 times. Will everyone get the same number of heads? Almost certainly not. (But it s possible!) If everyone gets a different amount, how can we use our simulation to figure out a probability? The answer is that the best we can do is get an approximation. Probabilities determined from simulations or from actual experiments are called experimental probabilities or synonymously, empirical probabilities. An experimental probability differs from person to person and from attempt to attempt. The important thing to keep in mind is that the more trials you do, the more likely you are to get an experimental probability that is close to the theoretical probability. To sum things up, your experimental probability is an approximation of the theoretical probability and the larger the sample size, the better the approximation. In other words, if everyone in the room flips a real coin 20 times, the percentage of heads will vary quite a bit and might stray quite a ways (relatively speaking) from 50%. But, if everyone flips a real coin 1000 times, each person s percentage of heads will be closer to 50%. To do our simulations, we need a simulation machine. Our machine is called a box model. It is a mental abstraction, but it is also a routine programmed into Stata. This routine allows you to program the computer and order it to carry out certain types of simulations. What s a box model? A box model is a box with slips of paper in it, like you might see in a raffle. The slips of paper have numbers on them. We reach into the box, pull out a slip and record the number. We could repeat this many times to get an idea of the approximate probability of seeing certain values. Let s make a box model for estimating the probability of getting a 3 when rolling a fair die. The first step is to decide what values go on the tickets. 72

3 We need to make sure that all possible outcomes are represented. Therefore, we will need one ticket with a 1, one with a 2, another with a 3, and so on up to 6. The second step is to decide how many of each ticket. For a fair die, each outcome is equally likely, so each value in the box should have equal representation. Therefore, we should have the same number of tickets for each value. The simplest way to do this is to have each value appear exactly once. If we wanted to though, we could have 100 tickets for each value. Once the box model is created, we are ready to use it. For this thought experiment, imagine mixing up the tickets and then reaching into the box to draw one out. If it s a 3, we make a note of this. If not, we do nothing. No matter what, we put the ticket back, shuffle the box, and repeat. We repeat this many times, maybe When we re done, our experimental probability is the number of 3 s divided by the number of trials. Note that there s another way of drawing out tickets. We could, if we wanted to, draw a ticket out, record its number, and then throw it away. This is called sampling without replacement. It is useful for experiments in which outcomes can only occur once, such as selecting lottery numbers. Using Stata to make and run box models Before we get started, we need to do a couple things. *** Important *** First, you need to change your working directory. Click on the File menu and select Set Working Folder. The default is your Documents folder. This is the correct location, so click on the Choose button. Second, to ensure we truly are taking random samples, we want to randomly set a seed for Stata to start from.. set seed YOUR STUDENT ID NUMBER 73

4 We are not going to use a real box model. Instead, we will use a virtual box model. A model box model, if you will. Our virtual box model exists within some Stata code. Don t worry, you don t have to write any code. You just have to execute the program that will then guide you to build a box model. The program that does this is not part of the usual Stata, 1 and so if your computer does not already have this installed, you need to install it. If you are doing this in the Statistics lab, then it has probably already been installed. Type. bxmodel Does anything happen? If you get an error (unrecognized command), then you need to follow the instructions below to install it. Otherwise, hit cancel and skip past the installation steps. Commands needed for installing bxmodel. net from net install bxmodel Using the bxmodel program requires three steps. Create the box with values and frequencies Choose the type of sampling and number of repetitions Analyze the output To create the box, you will need to use the editor to create a data set with two columns or variables. The first variable represents the different values the experiment can take and the second variable represents the number of tickets each value should have in the box. 1 Full directions for using the box model are available at 74

5 . edit To correctly use the bxmodel program, the name of the first variable must be value and the name of the second variable must be n. To simulate the roll of a fair die, put the numbers 1 through 6 in the value column. Each value should only appear once in our box, so put a 1 beside each value in the n column. Click on Preserve and then close the editor. Finally, choose Save as under the File menu and save under a memorable name, such as, fairdie.dta. Now you are ready to run the box model. In the command window, type. bxmodel A dialogue box will appear. Enter file name: The name of the file where you saved your box model. Number: How many tickets you want drawn out of the box. Type of Box Model: The possible model types available. (We have briefly discussed the sampling with replacement and without replacement options. The birthday options will be discussed later.) Repetitions: The number of times you want to repeat the experiment or how many samples you want to draw. The program does not save the raw results. Instead, for each repetition or sample, it saves the sum of the tickets in the sample, the average of the tickets in the sample, and the standard deviation of the tickets in the sample. In other words, if you set up a box model and told it to do 100 repetitions, you would end up with 100 different sum totals, 100 different averages, and 100 different standard deviations. Getting back to our dice problem... Question 1: What is the theoretical probability of rolling a 3 on a fair die? 75

6 Remember, our goal is to simulate rolling a fair die. We want to roll the die one time, record the result, and repeat. For this experiment, we want to do this 100 times. The bxmodel input should be the file we created previously, the fact that we only want to roll the die once (or analogously, we want to select one ticket from the box), sampling with replacement (the number 6 is not removed from a die once it has been rolled), and the number of times we want to repeat the experiment. Enter file name: fairdie.dta Number: 1 Type of Box Model: Sum, mean, sd - w/ replacement Repetitions: 100 Look at the output that the bxmodel gives you.. list You should see three variables, sum, mean, and sd. The variables sum and mean contain the same information. This is because we only rolled the die once or drew only one ticket. The sd variable is full of missing values. You can not compute a standard deviation for a sample of size one. We have 100 observations, each representing a different roll of the die. To summarize the output of the bxmodel program, we can use the tabulate command.. tabulate sum Question 2: What is the experimental probability of rolling a 3? How far off is this from what you expected? Does this seem reasonable? The empirical probability density function should be similar to the theoretical probability density function. In this case, the probability of each outcome 76

7 is 1/6, and so we expect each of the 6 outcomes (1,2,3,4,5,6) to occur about 1/6th of the time. We can verify this by looking at a graph of the empirical probability density function.. histogram sum, discrete width(1) xlabel(1 2 to 6) yline(.16667) Question 3: What is the empirical probability density function you obtained for rolling a fair die? Empirical probabilities will change with every experiment we do. To exam this phenomenon, we will repeat the above experiment 10 times. Note: Because of a quirk with the bxmodel program, we have to remove the file that the program created before it will correctly run again. You have to do this every time you want to run the bxmodel program.. erase dta Question 4: Repeat the above experiment (100 repetitions of rolling a single die) 10 times. Each time, calculate (i) the experimental probability of rolling a 3 and (ii) the difference between the experimental probability and the theoretical probability. As you have seen, variation occurs in each repetition of the experiment. This is a direct result of it being a random process. We can change the experiment in a number of ways. For example, we can roll the die many more times. In a moment, we will repeat the experiment, but this time rolling the die 10,000 times! Question 5: You calculated the difference between the experimental probability and the theoretical probability of rolling a 3. How should the differences you compute for the 10,000 rolls compare to the differences you computed for the experiment with 100 rolls? 77

8 Question 6: Now do 10,000 repetitions of rolling a single die. Repeat 10 times. Each time calculate (i) the experimental probability of rolling a 3 and (ii) the difference between the experimental probability and the theoretical probability. To play the popular casino games of craps, you roll two dice. The outcome is based on the sum of the dice. The game places special significance on an outcome of 7. (If a 7 occurs on the first throw, this is usually good for the player. Otherwise 7 s are usually bad news.) Question 7: What is the theoretical probability of rolling a 7? Simulate the experiment of throwing two dice together 1000 times. What is your experimental probability of rolling a 7? How does this differ from the theoretical probability? In the game of craps, the payoffs (the money the casino gives you if you win) are proportional to the probability of certain outcomes. Inversely proportional, of course! The less likely the outcome, the greater the payoff. Question 8: According to the experiment you just did, which outcomes have the biggest payoffs? Does this match with your theoretical expectations? In the card game poker, you are dealt a hand, that consists of 5 cards. To vastly oversimplify the game, the more rare your hand, the better of an advantage you have over your opponents. One rare hand is to have all of your cards the same suit. (There are four suits: hearts, diamonds, clubs, and spades.) Create a new box with values and frequencies and use it to estimate the probability that all 5 cards will be spades. Hint: There are 52 cards in a deck and 13 in each suit. Assign the value 1 to spades and 0 to all the other suits. A hand consists of 5 cards dealt 78

9 without replacement. First, do the experiment with 1000 repetitions. Try this two or three times. Then try a few times with 10,000 repetitions. Question 9: What experimental probabilities did you find in your experiment? Based on your results, can you guess what the theoretical probability is? Why are so many repetitions necessary? Question 10: Calculate the theoretical probability. Was your experimental probability close to this? The Birthday Model In a classroom of 20 people, what are the chances that at least two will have the same birthday? We can use the bxmodel program to estimate probabilities such as this. The first step is to imagine that each day has a number. (The second step is to ignore leap year day.) Next, we make a crucial assumption: that birthdays are uniformly distributed. 2 In other words, a person is just as likely to have his or her birthday on one day as any other. We can imagine that each person goes through life with a number between 1 and 365 assigned to them. When we assemble 20 people in the classroom, it s as if we just randomly selected, with replacement, 20 numbers from a box that has the numbers in it. A single repetition consists of drawing 20 tickets from the box, with replacement. But this time, it won t help us to record the sum or mean of the tickets. Instead, we want to know how many of the tickets are identical. If 2 Is this assumption true? Check out the Stata Coding and Birthdays Lab. 79

10 two of the twenty people have the same number, then they have the same birthday! The first step is to create a data file with two variables. The variable labeled value will have the numbers 1 through 365, and the variable labeled n will consist of a string of 1 s. (Each value appears in the box exactly once.) You may be thinking that it will be rather tedious to type all 730 numbers into the Stata editor. Luckily, Stata provides us with a better way. Type the following commands:. clear. set obs 365. generate value = n. generate n = 1 The first command clears the memory. The second command tells Stata to save space for 365 observations. Next, the generate command creates a new variable named value that is given all of the values from 1 to n (where n is equal to the number of observations, 365 in this case.) The final command creates a new variable called n and gives it a value of 1 for all observations. Save this data set as bday.dta Now to answer the question proposed at the beginning of this section, run the bxmodel program, using the following input: Enter file name: bday.dta Number: 20 Type of Box Model: Birthday - w/ replacement Repetitions: 1000 Look at the data:. list This time there are only two variables, match and unique. From glancing at the data, we can see that match seems to equal either 0 or 1, whereas 80

11 unique has values like 19, 20, and 18. For each repetition, match is 1 if any two or more tickets in the 20 selected match, 0 if there are no matches. The variable unique counts the number of tickets in those selected that are unique. If there are two people with the same birthday, match will be 1 and unique will be 19 (since there are 19 unique numbers.) If 3 people share the same birthday, match will still be 1, but unique will be 18. If there are two pairs with the same birthday (i.e., two people were born on, say, day 34, and two more were born on day 165), then match will be 1 and unique will be 18. If none of the 20 shares a birthday, match will be 0 and unique will be 20. We can estimate the probability of a match occurring by counting how many of the repetitions had a match. The experimental probability is the number of repetitions for which match is 1, divided by the number of repetitions. Stata calculates this as the mean of match.. summarize match Question 11: In a classroom of 20 people, what are the chances that at least two people will have the same birthday? Question 12: How many people are needed in a classroom for the chance that at least two have the same birthday to be over 90% The set up for the birthday problem can be used to solve other types of problems. A children s cereal is having a special promotion. Each box of cereal has one of seven tokens inside. If you collect all 7 tokens, you send them in and receive a Big Prize! Assume the tokens are evenly distributed, so that if you select a box at random, tokens have an equal chance of appearing. Create a box to simulate this promotion. Question 13: Suppose you buy 7 cereal boxes. What is the probability you will get all 7 tokens and win the Big Prize? 81

12 Question 14: How many boxes must you buy for the probability of getting all 7 tokens to be at least 50%? 82

### MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

### Stat 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

### Minimax 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 Microeconomics-Charles W Upton Zero Sum Games

### Question: 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

### Math 728 Lesson Plan

Math 728 Lesson Plan Tatsiana Maskalevich January 27, 2011 Topic: Probability involving sampling without replacement and dependent trials. Grade Level: 8-12 Objective: Compute the probability of winning

### Ch. 13.2: Mathematical Expectation

Ch. 13.2: Mathematical Expectation Random Variables Very often, we are interested in sample spaces in which the outcomes are distinct real numbers. For example, in the experiment of rolling two dice, we

### AP 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

### Section 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

### Mathematical 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

### 6th Grade Lesson Plan: Probably Probability

6th Grade Lesson Plan: Probably Probability Overview This series of lessons was designed to meet the needs of gifted children for extension beyond the standard curriculum with the greatest ease of use

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

### Section 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

### Section 6-5 Sample Spaces and Probability

492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)

### (b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball.

Examples for Chapter 3 Probability Math 1040-1 Section 3.1 1. Draw a tree diagram for each of the following situations. State the size of the sample space. (a) You flip a coin three times. (b) You draw

### Probabilistic 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

### Probability 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

### PROBABILITY SECOND EDITION

PROBABILITY SECOND EDITION Table of Contents How to Use This Series........................................... v Foreword..................................................... vi Basics 1. Probability All

### Chapter 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.

### AMS 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

### That s Not Fair! ASSESSMENT #HSMA20. Benchmark Grades: 9-12

That s Not Fair! ASSESSMENT # Benchmark Grades: 9-12 Summary: Students consider the difference between fair and unfair games, using probability to analyze games. The probability will be used to find ways

### Bayesian 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

### There are a number of superb online resources as well that provide excellent blackjack information as well. We recommend the following web sites:

3. Once you have mastered basic strategy, you are ready to begin learning to count cards. By counting cards and using this information to properly vary your bets and plays, you can get a statistical edge

### Chapter 4 - Practice Problems 1

Chapter 4 - Practice Problems SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) Compare the relative frequency formula

### In 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

### Basic 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.

### The 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

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Practice Test Chapter 9 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the odds. ) Two dice are rolled. What are the odds against a sum

### Math Games For Skills and Concepts

Math Games p.1 Math Games For Skills and Concepts Original material 2001-2006, John Golden, GVSU permission granted for educational use Other material copyright: Investigations in Number, Data and Space,

### STATISTICS 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

### Lesson Plans for (9 th Grade Main Lesson) Possibility & Probability (including Permutations and Combinations)

Lesson Plans for (9 th Grade Main Lesson) Possibility & Probability (including Permutations and Combinations) Note: At my school, there is only room for one math main lesson block in ninth grade. Therefore,

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### The mathematical branch of probability has its

ACTIVITIES for students Matthew A. Carlton and Mary V. Mortlock Teaching Probability and Statistics through Game Shows The mathematical branch of probability has its origins in games and gambling. And

### Loteria Workshop is a great way to bring Mexican art and gaming tradition to your computer!

Loteria Workshop Loteria Workshop is a great way to bring Mexican art and gaming tradition to your computer! The Loteria game of chance - background Loteria is something like Bingo, using pictures instead

### It is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important

PROBABILLITY 271 PROBABILITY CHAPTER 15 It is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important subject of human knowledge.

### A 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

### Ready, Set, Go! Math Games for Serious Minds

Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 -

### Homework 20: Compound Probability

Homework 20: Compound Probability Definition The probability of an event is defined to be the ratio of times that you expect the event to occur after many trials: number of equally likely outcomes resulting

### Fundamentals of Probability

Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible

### 6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0.

Name: Date:. For each of the following scenarios, determine the appropriate distribution for the random variable X. A) A fair die is rolled seven times. Let X = the number of times we see an even number.

### The Procedures of Monte Carlo Simulation (and Resampling)

154 Resampling: The New Statistics CHAPTER 10 The Procedures of Monte Carlo Simulation (and Resampling) A Definition and General Procedure for Monte Carlo Simulation Summary Until now, the steps to follow

### Chapter 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

### Solution. Solution. (a) Sum of probabilities = 1 (Verify) (b) (see graph) Chapter 4 (Sections 4.3-4.4) Homework Solutions. Section 4.

Math 115 N. Psomas Chapter 4 (Sections 4.3-4.4) 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

### MA 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

### Feb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172-179)

Feb 7 Homework Solutions Math 151, Winter 2012 Chapter Problems (pages 172-179) Problem 3 Three dice are rolled. By assuming that each of the 6 3 216 possible outcomes is equally likely, find the probabilities

### Probability --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..

### Expected 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

### Statistics 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

### Using games to support. Win-Win Math Games. by Marilyn Burns

4 Win-Win Math Games by Marilyn Burns photos: bob adler Games can motivate students, capture their interest, and are a great way to get in that paperand-pencil practice. Using games to support students

### Math 202-0 Quizzes Winter 2009

Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile

### THE WINNING ROULETTE SYSTEM.

THE WINNING ROULETTE SYSTEM. Please note that all information is provided as is and no guarantees are given whatsoever as to the amount of profit you will make if you use this system. Neither the seller

### Section 6.1 Discrete Random variables Probability Distribution

Section 6.1 Discrete Random variables Probability Distribution Definitions a) Random variable is a variable whose values are determined by chance. b) Discrete Probability distribution consists of the values

### Problem of the Month: Fair Games

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

### Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology Madras

Operations and Supply Chain Management Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology Madras Lecture - 41 Value of Information In this lecture, we look at the Value

### Mathematical 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

### Colored Hats and Logic Puzzles

Colored Hats and Logic Puzzles Alex Zorn January 21, 2013 1 Introduction In this talk we ll discuss a collection of logic puzzles/games in which a number of people are given colored hats, and they try

### Chapter 7 Probability. Example of a random circumstance. Random Circumstance. What does probability mean?? Goals in this chapter

Homework (due Wed, Oct 27) Chapter 7: #17, 27, 28 Announcements: Midterm exams keys on web. (For a few hours the answer to MC#1 was incorrect on Version A.) No grade disputes now. Will have a chance to

### Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space

### The Binomial Probability Distribution

The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

### Texas Hold em. From highest to lowest, the possible five card hands in poker are ranked as follows:

Texas Hold em Poker is one of the most popular card games, especially among betting games. While poker is played in a multitude of variations, Texas Hold em is the version played most often at casinos

### Unit 19: Probability Models

Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,

### Probability definitions

Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a data-generating

### Introduction to Discrete Probability. Terminology. Probability definition. 22c:19, section 6.x Hantao Zhang

Introduction to Discrete Probability 22c:19, section 6.x Hantao Zhang 1 Terminology Experiment A repeatable procedure that yields one of a given set of outcomes Rolling a die, for example Sample space

### CS 341 Software Design Homework 5 Identifying Classes, UML Diagrams Due: Oct. 22, 11:30 PM

CS 341 Software Design Homework 5 Identifying Classes, UML Diagrams Due: Oct. 22, 11:30 PM Objectives To gain experience doing object-oriented design To gain experience developing UML diagrams A Word about

### Week 5: Expected value and Betting systems

Week 5: Expected value and Betting systems Random variable A random variable represents a measurement in a random experiment. We usually denote random variable with capital letter X, Y,. If S is the sample

### Video Poker in South Carolina: A Mathematical Study

Video Poker in South Carolina: A Mathematical Study by Joel V. Brawley and Todd D. Mateer Since its debut in South Carolina in 1986, video poker has become a game of great popularity as well as a game

### Introductory 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

### Definition and Calculus of Probability

In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the

### Easy 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

### Practical 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

### AP: LAB 8: THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics

Ms. Foglia Date AP: LAB 8: THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance,

### Chapter 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

### Assessment Management

Facts Using Doubles Objective To provide opportunities for children to explore and practice doubles-plus-1 and doubles-plus-2 facts, as well as review strategies for solving other addition facts. www.everydaymathonline.com

### Thursday, October 18, 2001 Page: 1 STAT 305. Solutions

Thursday, October 18, 2001 Page: 1 1. Page 226 numbers 2 3. STAT 305 Solutions S has eight states Notice that the first two letters in state n +1 must match the last two letters in state n because they

### Standard 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

### ST 371 (IV): Discrete Random Variables

ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible

### Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific

Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the left-hand margin. You

### Coin Flip Questions. Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT.

Coin Flip Questions Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. 1 How many ways can you get exactly 1 head? 2 How many ways can you get exactly 2 heads?

### Lecture 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).

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### Session 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

### Sample Questions for Mastery #5

Name: Class: Date: Sample Questions for Mastery #5 Multiple Choice Identify the choice that best completes the statement or answers the question.. For which of the following binomial experiments could

### Hooray for the Hundreds Chart!!

Hooray for the Hundreds Chart!! The hundreds chart consists of a grid of numbers from 1 to 100, with each row containing a group of 10 numbers. As a result, children using this chart can count across rows

### OA4-13 Rounding on a Number Line Pages 80 81

OA4-13 Rounding on a Number Line Pages 80 81 STANDARDS 3.NBT.A.1, 4.NBT.A.3 Goals Students will round to the closest ten, except when the number is exactly halfway between a multiple of ten. PRIOR KNOWLEDGE

### Five daily lessons. Page 8 Page 8. Page 12. Year 2

Unit 2 Place value and ordering Year 1 Spring term Unit Objectives Year 1 Read and write numerals from 0 to at least 20. Begin to know what each digit in a two-digit number represents. Partition a 'teens'

### Martin J. Silverthorne. Triple Win. Roulette. A Powerful New Way to Win \$4,000 a Day Playing Roulette! Silverthorne Publications, Inc.

Martin J. Silverthorne Triple Win Roulette A Powerful New Way to Win \$4,000 a Day Playing Roulette! Silverthorne Publications, Inc. By Martin J. Silverthorne COPYRIGHT 2011 Silverthorne Publications, Inc.

### V. 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

### The game of roulette is played by throwing a small ball onto a rotating wheel with thirty seven numbered sectors.

LIVE ROULETTE The game of roulette is played by throwing a small ball onto a rotating wheel with thirty seven numbered sectors. The ball stops on one of these sectors. The aim of roulette is to predict

### Concepts of Probability

Concepts of Probability Trial question: we are given a die. How can we determine the probability that any given throw results in a six? Try doing many tosses: Plot cumulative proportion of sixes Also look

### STAT 35A HW2 Solutions

STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },

### Random variables, probability distributions, binomial random variable

Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

### LAB : THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics

Period Date LAB : THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance,

### Question of the Day. Key Concepts. Vocabulary. Mathematical Ideas. QuestionofDay

QuestionofDay Question of the Day What is the probability that in a family with two children, both are boys? What is the probability that in a family with two children, both are boys, if we already know

### \$2 4 40 + ( \$1) = 40

THE EXPECTED VALUE FOR THE SUM OF THE DRAWS In the game of Keno there are 80 balls, numbered 1 through 80. On each play, the casino chooses 20 balls at random without replacement. Suppose you bet on the

### 6.042/18.062J 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

### Chapter 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

### 2. 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

### TeachingEnglish Lesson plans. Conversation Lesson News. Topic: News

Conversation Lesson News Topic: News Aims: - To develop fluency through a range of speaking activities - To introduce related vocabulary Level: Intermediate (can be adapted in either direction) Introduction