Solution. Solution. (a) Sum of probabilities = 1 (Verify) (b) (see graph) Chapter 4 (Sections ) Homework Solutions. Section 4.
|
|
- Ralph Pierce
- 9 years ago
- Views:
Transcription
1 Math 115 N. Psomas Chapter 4 (Sections ) Homework s Section Discrete or continuous. In each of the following situations decide if the random variable is discrete or continuous and give a reason for your answer. (a) Your Web page has five different links and a user can click on one of the links or can leave the page. You record the length of time that a user spends on the Web page before clicking one of the links or leaving the page. (b) The number of hits on your Web page. (c) The yearly income of a visitor to your Web page. (a) Continuous (b) Discrete (c) Book says discrete because income is measured to the nearest 100th of a dollar - nearest cent. For most practical purposes though, it can be safely considered continuous Texas hold em. The game of Texas hold em starts with each player receiving two cards. Here is the probability distribution for the number of aces in two-card hands: (a) Verify that this assignment of probabilities satisfies the requirement that the sum of the probabilities for a discrete distribution must be 1. (b) Make a probability histogram for this distribution. (c) What is the probability that a hand contains at least one ace? Show two different ways to calculate this probability. Texas Hold'em No of Aces Probability How to compute probability (48/52)*(47/51) (4/52)*(48/51)+(48/52)*(4/51) (4/52)*(3/51) (a) Sum of probabilities = 1 (Verify) (b) (see graph)
2 (c) P(At least one Ace) = P(1 or 2 aces) = P(1) + P(2) = = OR... P(At least one Ace) = 1 - P(no aces) = = Spell-checking software. Spell-checking software catches nonword errors, which result in a string of letters that is not a word, as when the is typed as teh. When undergraduates are asked to write a 250-word essay (without spell-checking), the number X of nonword errors has the following distribution: (a) Sketch the probability distribution for this random variable. (b) Write the event at least one non-word error in terms of X. What is the probability of this event? (c) Describe the event X 2 in words. What is its probability? What is the probability that X < 2? (a) See graph
3 (b) X > 0, or X 1 P(X > 0) = P(X 1) = P{1, 2, 3, 4} = 1 - P{0} = = 0.9 (c) "At most 2 non-word errors" P("At most 2 non-word errors") = P(X 2) = P{0, 1, 2} = = 0.7 P( X < 2) = P{0, 1} = = Length of human pregnancies. The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. Call the length of a randomly chosen pregnancy Y. (a) Make a sketch of the density curve for this random variable. (b) What is P(Y 280)? (a) See graph
4 (b) P( Y 280 ) = P(Z ( )/16 ) = P(Z.875) = ( )/2 = Note: Here because the z-value.875 happens to fall exactly in the middle of.87 and.88, the two z-values listed in table A, averaging the probabilities listed in table A for z=.87 & z =.88 give a more accurate value for P(Z.875). Alternatively, use TI-83/84's normalcdf(a, b, µ, σ) function to compute P( Y 280 ). normalcdf(0, 280, 266, 16) = Tossing two dice. Some games of chance rely on tossing two dice. Each die has six faces, marked with 1, 2,, 6 spots called pips. The dice used in casinos are carefully balanced so that each face is equally likely to come up. When two dice are tossed, each of the 36 possible pairs of faces is equally likely to come up. The outcome of interest to a gambler is the sum of the pips on the two up-faces. Call this random variable X. (a) Write down all 36 possible pairs of faces. (b) If all pairs have the same probability, what must be the probability of each pair? (c) Write the value of X next to each pair of faces and use this information with the result of (b) to give the probability distribution of X. Draw a probability histogram to display the distribution. (d) One bet available in craps wins if a 7 or an 11 comes up on the next roll of two dice. What is the probability of rolling a 7 or an 11 on the next roll? (e) Several bets in craps lose if a 7 is rolled. If any outcome other than 7 occurs, these bets either win or continue to the next roll. What is the probability that anything other than a 7 is rolled?
5 (a) (1,1) (2,1) (3,1) (4,1) (5,1) (6,1) 2 (1,2) (2,2) (3,2) (4,2) (5,2) (6,2) 3 (1,3) (2,3) (3,3) (4,3) (5,3) (6,3) 4 (1,4) (2,4) (3,4) (4,4) (5,4) (6,4) 5 (1,5) (2,5) (3,5) (4,5) (5,5) (6,5) 6 (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) (b) 1/36 (c) (d) P(7 or 11) = P(7) + P(11) = = Note: Probability values shown on the histogram are rounded. If you use them to answer question (d) the result will be slightly off. In this case (e) P(not a 7) = 1 - P(7) = 1-6/36 = 30/36 =.83333
6 Section Mean of the distribution for the number of aces. In Exercise 4.54 you examined the probability distribution for the number of aces when you are dealt two cards in the game of Texas hold em. Let X represent the number of aces in a randomly selected deal of two cards in this game. Here is the probability distribution for the random variable X: Find μ X, the mean of the probability distribution of X. μ X = (0)(0.8507) + (1)(0.1448) + (2)(0.0045) = Note: This is the average number of aces a player is dealt in the long run Mean of the grade distribution. Example 4.22 gives the distribution of grades (A = 4, B = 3, and so on) in English 210 at North Carolina State University as Find the average (that is, the mean) grade in this course. μ X = (0)(0.05) + (1)(0.04) + (2)(0.20) + (3)(0.40) + (4)(0.31) = Standard deviation of the number of aces. Refer to Exercise Find the standard deviation of the number of aces. No of Aces X P(X) X*P(X) (X-µ) P(X)*(X-µ) µ = σ 2 = σ =
7 Using TI-83/84 Enter the values of X in L1 & the probability for each value in L2 Press STAT > select CALC > Select 1: 1-Var Stats > Press Enter > Type L1,L2 Press Enter 4.78 Standard deviation of the grades. Refer to Exercise Find the standard deviation of the grade distribution. Work as in Exercise 4.77
8 4.80 Find the mean of the sum. Figure 4.12 (page 259) displays the density curve of the sum Y = X 1 + X 2 of two independent random numbers, each uniformly distributed between 0 and 1. (a) The mean of a continuous random variable is the balance point of its density curve. Use this fact to find the mean of Y from Figure (b) Use the same fact to find the means of X 1 and X 2. (They have the density curve pictured in Figure 4.9, (page 254.) Verify that the mean of Y is the sum of the mean of X 1 and the mean of X 2. Figure 4.9 Figure 4.12 (a) By inspection (looking at Figure 4.12), the balance point 1. Therefore µ Y = X1 + X2 = 1. (b) Similarly, the mean for X 1 & X 2 (µ X1 & µ X2 ) is 0.5 µ Y = X1 + X2 = µ X1 + µ X2
9 4.83 Means and variances of sums. The rules for means and variances allow you to find the mean and variance of a sum of random variables without first finding the distribution of the sum, which is usually much harder to do. (a) A single toss of a balanced coin has either 0 or 1 head, each with probability 1/2. What are the mean and standard deviation of the number of heads? (b) Toss a coin four times. Use the rules for means and variances to find the mean and standard deviation of the total number of heads. (c) Example 4.23 (page 251) finds the distribution of the number of heads in four tosses. Find the mean and standard deviation from this distribution. Your results in parts (b) and (c) should agree. (a) Here think of X as the variable that equals the # of heads you get when you toss a coin once. X = 0 if tails shows up and X = 1 if heads shows up. With probabilities: P(0) = 1/2 & P(1) = 1/2 In summary: X 0 1 P(X).5.5 µ X =.5 & σ X =.5 (Use your calculator & work like in exercise 4.77) (b) Here think of Y as the sum of four variables each having the same distribution as the variable X in part (a). i.e. Y = X 1 + X 2 + X 3 + X 4 According to the rules for means and variances: µ y = µ x1 + µ x2 + µ x3 + µ x4 = 4*(.5) = 2 Var(Y) = Var(X 1 ) + Var(X 2 ) + Var(X 3 ) + Var(X 4 ) = 4* (.25) = 1 ; Note: Var(X) = (.5) 2 =.25 σ y = SQRT(1) = 1
10 4.85 A mechanical assembly. A mechanical assembly (Figure 4.15) consists of a rod with a bearing on each end. The three parts are manufactured independently, and all vary a bit from part to part. The length of the rod has mean 12 centimeters (cm) and standard deviation millimeters (mm). The length of a bearing has mean 2 cm and standard deviation mm. What are the mean and standard deviation of the total length of the assembly? Figure 4.15 Sketch of a mechanical assembly, for Exercise (Done in class) Length of entire assembly (L) = Length of Bearing 1 (B 1 )Length of Rod (R) + Length of Bearing 2 (B 2 ) In short, L = B 1 + R + B 2. We are given that: µ R = 12cm, µ B1 = µ B2 = 2cm, σ R = 0.004mm, σ B1 = σ B2 = 0.001mm According to the rules of means & variances µ L = B1+R + B2 = µ B1 + µ R + µ B2 = *2 = 16 cm σ 2 L = B1+R + B2 = σ 2 B1 + σ 2 R + σ 2 B2 = (0.004) 2 + 2*(0.001) 2 = σ L = B1+R + B2 = SQRT( ) = mm 4.93 Life insurance. According to the current Commissioners Standard Ordinary mortality table, adopted by state insurance regulators in December 2002, a 25-year-old man has these probabilities of dying during the next five years: (a) What is the probability that the man does not die in the next five years? (b) An online insurance site offers a term insurance policy that will pay $100,000 if a 25- year-old man dies within the next five years. The cost is $175 per year. So the insurance
11 company will take in $875 from this policy if the man does not die within five years. If he does die, the company must pay $100,000. Its loss depends on how many premiums were paid, as follows: What is the insurance company s mean cash intake from such polices? (a) P(25 year old man will be alive on his 30th birthday) = = 1 - ( ) = (b) Gain probability distribution for the insurance company: Age at death Ins. Gain (X) - $99,825 - $99,650 - $99,475 - $99,300 - $99,125 $875 P(X) µ X = (- $99,825)*( ) + (- $99,650)*( ) +...+($875)*( ) = Computations (Easier done with a calculator. Enter Ins. Gain in L1, & Probability in L2 and work as in exercise 4.77) Age at death Probability Ins. Gain(x) X*P(X) $99, $99, $99, $99, $99, $ µ = $ Risk for one versus thousands of life insurance policies. It would be quite risky for you to insure the life of a 25-year-old friend under the terms of Exercise There is a high probability that your friend would live and you would gain $875 in premiums. But if he were to die, you would lose almost $100,000. Explain carefully why selling insurance is not risky for an insurance company that insures many thousands of 25-year-old men.
Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 4.4 Homework
Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 4.4 Homework 4.65 You buy a hot stock for $1000. The stock either gains 30% or loses 25% each day, each with probability.
More informationX X AP Statistics Solutions to Packet 7 X Random Variables Discrete and Continuous Random Variables Means and Variances of Random Variables
AP Statistics Solutions to Packet 7 Random Variables Discrete and Continuous Random Variables Means and Variances of Random Variables HW #44, 3, 6 8, 3 7 7. THREE CHILDREN A couple plans to have three
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 information4.1 4.2 Probability Distribution for Discrete Random Variables
4.1 4.2 Probability Distribution for Discrete Random Variables Key concepts: discrete random variable, probability distribution, expected value, variance, and standard deviation of a discrete random variable.
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 informationUnit 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,
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 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 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 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 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 real-valued function defined on the sample space of some experiment. For instance,
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 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 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 informationSTT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random
More informationStatistics 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 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 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 informationMind on Statistics. Chapter 8
Mind on Statistics Chapter 8 Sections 8.1-8.2 Questions 1 to 4: For each situation, decide if the random variable described is a discrete random variable or a continuous random variable. 1. Random variable
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 informationWeek 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
More informationExam 3 Review/WIR 9 These problems will be started in class on April 7 and continued on April 8 at the WIR.
Exam 3 Review/WIR 9 These problems will be started in class on April 7 and continued on April 8 at the WIR. 1. Urn A contains 6 white marbles and 4 red marbles. Urn B contains 3 red marbles and two white
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 informationJoint Exam 1/P Sample Exam 1
Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question
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 informationChapter 16: law of averages
Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................
More informationLab 11. Simulations. The Concept
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
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 informationb. What is the probability of an event that is certain to occur? ANSWER: P(certain to occur) = 1.0
MTH 157 Sample Test 2 ANSWERS Student Row Seat M157ST2a Chapters 3 & 4 Dr. Claude S. Moore Score SHOW ALL NECESSARY WORK. Be Neat and Organized. Good Luck. 1. In a statistics class, 12 students own their
More informationSTOCHASTIC MODELING. Math3425 Spring 2012, HKUST. Kani Chen (Instructor)
STOCHASTIC MODELING Math3425 Spring 212, HKUST Kani Chen (Instructor) Chapters 1-2. Review of Probability Concepts Through Examples We review some basic concepts about probability space through examples,
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 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 informationSolution (Done in class)
MATH 115 CHAPTER 4 HOMEWORK Sections 4.1-4.2 N. PSOMAS 4.6 Winning at craps. The game of craps starts with a come-out roll where the shooter rolls a pair of dice. If the total is 7 or 11, the shooter wins
More informationStats on the TI 83 and TI 84 Calculator
Stats on the TI 83 and TI 84 Calculator Entering the sample values STAT button Left bracket { Right bracket } Store (STO) List L1 Comma Enter Example: Sample data are {5, 10, 15, 20} 1. Press 2 ND and
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 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 informationThe 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
More informationCh. 13.3: More about Probability
Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the
More informationDetermine the empirical probability that a person selected at random from the 1000 surveyed uses Mastercard.
Math 120 Practice Exam II Name You must show work for credit. 1) A pair of fair dice is rolled 50 times and the sum of the dots on the faces is noted. Outcome 2 4 5 6 7 8 9 10 11 12 Frequency 6 8 8 1 5
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 informationChapter 5 - Practice Problems 1
Chapter 5 - Practice Problems 1 Identify the given random variable as being discrete or continuous. 1) The number of oil spills occurring off the Alaskan coast 1) A) Continuous B) Discrete 2) The ph level
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
STATISTICS/GRACEY PRACTICE TEST/EXAM 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the given random variable as being discrete or continuous.
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 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 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 informationACMS 10140 Section 02 Elements of Statistics October 28, 2010. Midterm Examination II
ACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Name DO NOT remove this answer page. DO turn in the entire exam. Make sure that you have all ten (10) pages of the examination
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 informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 7, due Wedneday, March 14 Happy Pi Day! (If any errors are spotted, please email them to morrison at math dot berkeley dot edu..5.10 A croissant
More informationMULTIPLE 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
More informationFeb 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
More informationContemporary 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
More informationRandom Variables and Probability
CHAPTER 9 Random Variables and Probability IN THIS CHAPTER Summary: We ve completed the basics of data analysis and we now begin the transition to inference. In order to do inference, we need to use the
More informationFind the indicated probability. 1) If a single fair die is rolled, find the probability of a 4 given that the number rolled is odd.
Math 0 Practice Test 3 Fall 2009 Covers 7.5, 8.-8.3 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. ) If a single
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 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 informationProbability Models.S1 Introduction to Probability
Probability Models.S1 Introduction to Probability Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard The stochastic chapters of this book involve random variability. Decisions are
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 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 informationRandom Variables. Chapter 2. Random Variables 1
Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets
More information$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
More informationTexas 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
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 informationSTA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science
STA 130 (Winter 2016): An Introduction to Statistical Reasoning and Data Science Mondays 2:10 4:00 (GB 220) and Wednesdays 2:10 4:00 (various) Jeffrey Rosenthal Professor of Statistics, University of Toronto
More informationChapter 4. Probability Distributions
Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive
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 informationMAS108 Probability I
1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper
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 information7 CONTINUOUS PROBABILITY DISTRIBUTIONS
7 CONTINUOUS PROBABILITY DISTRIBUTIONS Chapter 7 Continuous Probability Distributions Objectives After studying this chapter you should understand the use of continuous probability distributions and the
More informationMATH 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
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 informationHigh School Statistics and Probability Common Core Sample Test Version 2
High School Statistics and Probability Common Core Sample Test Version 2 Our High School Statistics and Probability sample test covers the twenty most common questions that we see targeted for this level.
More informationWEEK #22: PDFs and CDFs, Measures of Center and Spread
WEEK #22: PDFs and CDFs, Measures of Center and Spread Goals: Explore the effect of independent events in probability calculations. Present a number of ways to represent probability distributions. Textbook
More information1. General...3. 2. Black Jack...5. 3. Double Deck Black Jack...13. 4. Free Bet Black Jack...20. 5. Craps...28. 6. Craps Free Craps...
Table of Contents Sec Title Page # 1. General...3 2. Black Jack...5 3. Double Deck Black Jack...13 4. Free Bet Black Jack...20 5. Craps...28 6. Craps Free Craps...36 7. Roulette...43 8. Poker...49 9. 3-Card
More informationFind the effective rate corresponding to the given nominal rate. Round results to the nearest 0.01 percentage points. 2) 15% compounded semiannually
Exam Name Find the compound amount for the deposit. Round to the nearest cent. 1) $1200 at 4% compounded quarterly for 5 years Find the effective rate corresponding to the given nominal rate. Round results
More informationKnow it all. Table Gaming Guide
Know it all. Table Gaming Guide Winners wanted. Have fun winning at all of your favorite games: Blackjack, Craps, Mini Baccarat, Roulette and the newest slots. Add in seven mouthwatering dining options
More information13:69E-1.10 Blackjack table; card reader device; physical characteristics; inspections
13:69E-1.10 Blackjack table; card reader device; physical characteristics; inspections (a) (t) (No change.) (aa) If a casino licensee offers Free Bet Blackjack pursuant to N.J.A.C. 3:69F- 2.31, the blackjack
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 informationSIXTEEN PLAYERS, TWO DICE, ONE SHOOTER
SIXTEEN PLAYERS, TWO DICE, ONE SHOOTER Craps is the most exciting game at Potawatomi Bingo Casino. One shooter of the dice determines the outcome for up to 15 other players. Craps also features a variety
More informationGaming the Law of Large Numbers
Gaming the Law of Large Numbers Thomas Hoffman and Bart Snapp July 3, 2012 Many of us view mathematics as a rich and wonderfully elaborate game. In turn, games can be used to illustrate mathematical ideas.
More informationECE 316 Probability Theory and Random Processes
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
More information2. Discrete random variables
2. Discrete random variables Statistics and probability: 2-1 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be
More informationSolution Let us regress percentage of games versus total payroll.
Assignment 3, MATH 2560, Due November 16th Question 1: all graphs and calculations have to be done using the computer The following table gives the 1999 payroll (rounded to the nearest million dolars)
More informationExample: Find the expected value of the random variable X. X 2 4 6 7 P(X) 0.3 0.2 0.1 0.4
MATH 110 Test Three Outline of Test Material EXPECTED VALUE (8.5) Super easy ones (when the PDF is already given to you as a table and all you need to do is multiply down the columns and add across) Example:
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 informationEveryday Math Online Games (Grades 1 to 3)
Everyday Math Online Games (Grades 1 to 3) FOR ALL GAMES At any time, click the Hint button to find out what to do next. Click the Skip Directions button to skip the directions and begin playing the game.
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 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 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 informationThursday, 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
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 informationMONEY MANAGEMENT. Guy Bower delves into a topic every trader should endeavour to master - money management.
MONEY MANAGEMENT Guy Bower delves into a topic every trader should endeavour to master - money management. Many of us have read Jack Schwager s Market Wizards books at least once. As you may recall it
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
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 informationSums of Independent Random Variables
Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables
More informationcachecreek.com 14455 Highway 16 Brooks, CA 95606 888-77-CACHE
Baccarat was made famous in the United States when a tuxedoed Agent 007 played at the same tables with his arch rivals in many James Bond films. You don t have to wear a tux or worry about spies when playing
More information3.4 The Normal Distribution
3.4 The Normal Distribution All of the probability distributions we have found so far have been for finite random variables. (We could use rectangles in a histogram.) A probability distribution for a continuous
More informationST 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
More informationX: 0 1 2 3 4 5 6 7 8 9 Probability: 0.061 0.154 0.228 0.229 0.173 0.094 0.041 0.015 0.004 0.001
Tuesday, January 17: 6.1 Discrete Random Variables Read 341 344 What is a random variable? Give some examples. What is a probability distribution? What is a discrete random variable? Give some examples.
More informationThe 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
More informationIntroduction 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
More information