Chapter 17: The Expected Value and Standard Error

Size: px
Start display at page:

Download "Chapter 17: The Expected Value and Standard Error"

Transcription

1 Chapter 17: The Expected Value and Standard Error Think about drawing 25 times, with replacement, from the box: Here s one set of 25 draws: , sum = 66 Here s another: , sum = 61 and another: , sum = 62

2 What can we say about the sum of 25 draws from the box? The expected value for the sum of the draws is EVsum = (number of draws) (ave box ) The standard error for the sum of the draws is SEsum = ( number of draws) (SD box ) ave box and SD box are the average and the SD of the tickets in the box.

3 Example: The sum of 25 draws from the box: ave box = 3 SD box = 2 CALCULATOR! so EVsum = (number of draws) (ave box ) = (25)(3) = 75 and SEsum = number of draws (SD box ) = ( 25 ) (2) = 10

4 What do the EVsum and SEsum tell us? Suppose we take many, many samples of size 25 and look at the sum of the draws: sum = sum = sum = sum = sum = sum = sum = sum = sum = sum = 80 Many samples of 25 draws

5 The first 10 sums: , ave = 73.3, SD = sums have ave = 73.3, SD = sums have ave = 74.5, SD of sums have ave = 75.0, SD of sums have ave = 75.0, SD of EVsum tells us the average of the SUM OF THE DRAWS SEsum tells us the SD of the SUM OF THE DRAWS. If we repeat the 25 draws MANY times, we expect the SUM OF THE DRAWS to be around EVsum, with an SD of SEsum.

6 Example 1. Consider the sum of 100 draws from the box What is the expected value of the sum? Its SE?

7 Example 2. (See Chapter 16, Example 5) You play a game in which you roll a die 10 times and get paid the amount shown on the die (each time). How much do you expect to win? Give or take how much?

8 Example 3. (See Chapter 16, Example 6) You play a game in which you roll a die 10 times. Each time a 6 occurs, you win $10, otherwise you lose $1. How much do you expect to win? Give or take how much?

9 Example 4. (See Chapter 16, Example 8) A multiplechoice test has 20 questions, each with 4 possible choices. Each correct answer is worth 5 points, and for each incorrect answer you lose 2 points. If you guess all the answers, what do you expect your test score to be? Give or take how much?

10 Example 5. A true/false test has 20 questions. Each correct answer scores 5 points; each wrong answer scores 0 points. If you guess all the answers, what is your expected score? Give-or-take how much?

11 Example 6. Bet $1 on times in roulette. How much do you expect to win? Give-or-take how much?

12 Example 7. Bet $100 on 17 once in roulette. What are the expected value and the SE for the amount you win?

13 Example 8. Bet $1 on red 100 times in roulette. How much money do you expect to win? Give-or-take how much?

14 Example 9. A child plays a game of chance in which she has a 30% chance of scoring 5 points, a 50% chance of scoring 3 points and a 20% chance of scoring 1 point. If she plays the game 20 times, what is her expected score? Give or take how much?

15 Classifying and Counting If we want to count how many times something happens, the box has 0 s and 1 s. The 1 s represent the thing we are counting, and the 0 s represent everything else. Here is the box for rolling a die and counting the total number of spots we get: Here is the box for rolling a die and counting how many 6 s we get:

16 Example 10. Bet $1 on red 100 times in roulette. How many times do you expect to win? Give-or-take how many?

17 The Normal Curve For a large number of draws, the sum of the draws will follow the normal curve with average EVsum and standard deviation SEsum. In particular, 68% of the time the sum of the draws will be between EVsum SEsum and EVsum + SEsum. 95% of the time the sum of the draws will be between EVsum 2 (SEsum) and EVsum + 2 (SEsum). We can use the normal curve to find the chance that the sum of the draws is in a region of interest. We use EVsum and SEsum to get standard units.

18 Note: It does not matter what is in the box. The histogram for the tickets in the box does not have to follow the normal curve. The sum of the draws will follow the normal curve, even if the tickets in the box are 0 s and 1 s! In fact, we don t even need to know what is in the box, we just need to know the average and the SD of the box.

19 Example 11. Roll a die 100 times and count the total number of spots. What is the chance the total number of spots is more than 367?

20 Example 12. Bet $1 on red 100 times in roulette. What is the chance you win more than $10? What is the chance you lose more than $10? What is the chance you come out ahead?

21 Example 13. A multiple-choice test has 20 questions, each with 4 choices. Each correct answer scores 5 points; each wrong answer makes you lose 2 points. If you guess all the answers, what is the chance you score more than 20 points?

22 Sampling without replacement from a LARGE population is just like sampling with replacement. Example 14. A large crop of apples has an average weight of 4.3 oz with an SD of 1.5 oz. I choose 100 apples at random. What s the chance the total weight is less than 25 pounds?

23 Example 15. Suppose 10% of people in a large population are underweight. If we take a random sample of 1000 people from this population, what is the chance that more than 103 will be underweight?

24 Example 16. For HANES women, a histogram of the amount of beef eaten over a 3-day period has an average of 22 grams and an SD of 41 grams. Suppose I choose 200 of these women, at random, and look at the total amount of beef eaten. What s the chance this total exceeds 5000g? Does it matter that the histogram is not like the normal curve? Why?

Discrete Random Variables Day 1

Discrete Random Variables Day 1 Discrete Random Variables Day 1 What is a Random Variable? Every probability problem is equivalent to drawing something from a bag (perhaps more than once) Like Flipping a coin 3 times is equivalent to

More information

There is no class tomorrow! Have a good weekend! Scores will be posted in Compass early Friday morning J

There is no class tomorrow! Have a good weekend! Scores will be posted in Compass early Friday morning J STATISTICS 100 EXAM 3 Fall 2016 PRINT NAME (Last name) (First name) *NETID CIRCLE SECTION: L1 12:30pm L2 3:30pm Online MWF 12pm Write answers in appropriate blanks. When no blanks are provided CIRCLE your

More information

a) Getting 10 +/- 2 head in 20 tosses is the same probability as getting +/- heads in 320 tosses

a) Getting 10 +/- 2 head in 20 tosses is the same probability as getting +/- heads in 320 tosses Question 1 pertains to tossing a fair coin (8 pts.) Fill in the blanks with the correct numbers to make the 2 scenarios equally likely: a) Getting 10 +/- 2 head in 20 tosses is the same probability as

More information

Math 147 Lecture Notes: Lecture 21

Math 147 Lecture Notes: Lecture 21 Math 147 Lecture Notes: Lecture 21 Walter Carlip March, 2018 The Probability of an Event is greater or less, according to the number of Chances by which it may happen, compared with the whole number of

More information

Midterm 2 Practice Problems

Midterm 2 Practice Problems Midterm 2 Practice Problems May 13, 2012 Note that these questions are not intended to form a practice exam. They don t necessarily cover all of the material, or weight the material as I would. They are

More information

LISTING THE WAYS. getting a total of 7 spots? possible ways for 2 dice to fall: then you win. But if you roll. 1 q 1 w 1 e 1 r 1 t 1 y

LISTING THE WAYS. getting a total of 7 spots? possible ways for 2 dice to fall: then you win. But if you roll. 1 q 1 w 1 e 1 r 1 t 1 y LISTING THE WAYS A pair of dice are to be thrown getting a total of 7 spots? There are What is the chance of possible ways for 2 dice to fall: 1 q 1 w 1 e 1 r 1 t 1 y 2 q 2 w 2 e 2 r 2 t 2 y 3 q 3 w 3

More information

Expected Value, continued

Expected Value, continued Expected Value, continued Data from Tuesday On Tuesday each person rolled a die until obtaining each number at least once, and counted the number of rolls it took. Each person did this twice. The data

More information

Casino Lab AP Statistics

Casino Lab AP Statistics Casino Lab AP Statistics Casino games are governed by the laws of probability (and those enacted by politicians, too). The same laws (probabilistic, not political) rule the entire known universe. If the

More information

Statistics Laboratory 7

Statistics Laboratory 7 Pass the Pigs TM Statistics 104 - Laboratory 7 On last weeks lab we looked at probabilities associated with outcomes of the game Pass the Pigs TM. This week we will look at random variables associated

More information

Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Stat 20: Intro to Probability and Statistics Lecture 17: Using the Normal Curve with Box Models Tessa L. Childers-Day UC Berkeley 23 July 2014 By the end of this lecture... You will be able to: Draw and

More information

Module 5: Probability and Randomness Practice exercises

Module 5: Probability and Randomness Practice exercises Module 5: Probability and Randomness Practice exercises PART 1: Introduction to probability EXAMPLE 1: Classify each of the following statements as an example of exact (theoretical) probability, relative

More information

Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5 {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}

Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5 {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)} Section 8: Random Variables and probability distributions of discrete random variables In the previous sections we saw that when we have numerical data, we can calculate descriptive statistics such as

More information

Chapter 7 Homework Problems. 1. If a carefully made die is rolled once, it is reasonable to assign probability 1/6 to each of the six faces.

Chapter 7 Homework Problems. 1. If a carefully made die is rolled once, it is reasonable to assign probability 1/6 to each of the six faces. Chapter 7 Homework Problems 1. If a carefully made die is rolled once, it is reasonable to assign probability 1/6 to each of the six faces. A. What is the probability of rolling a number less than 3. B.

More information

Ex 1: A coin is flipped. Heads, you win $1. Tails, you lose $1. What is the expected value of this game?

Ex 1: A coin is flipped. Heads, you win $1. Tails, you lose $1. What is the expected value of this game? AFM Unit 7 Day 5 Notes Expected Value and Fairness Name Date Expected Value: the weighted average of possible values of a random variable, with weights given by their respective theoretical probabilities.

More information

6. a) Determine the probability distribution. b) Determine the expected sum of two dice. c) Repeat parts a) and b) for the sum of

6. a) Determine the probability distribution. b) Determine the expected sum of two dice. c) Repeat parts a) and b) for the sum of d) generating a random number between 1 and 20 with a calculator e) guessing a person s age f) cutting a card from a well-shuffled deck g) rolling a number with two dice 3. Given the following probability

More information

The Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.)

The Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.) The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you $ that if you give me $, I ll give you $2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If

More information

Make better decisions. Learn the rules of the game before you play.

Make better decisions. Learn the rules of the game before you play. BLACKJACK BLACKJACK Blackjack, also known as 21, is a popular casino card game in which players compare their hand of cards with that of the dealer. To win at Blackjack, a player must create a hand with

More information

Statistics 1040 Summer 2009 Exam III

Statistics 1040 Summer 2009 Exam III Statistics 1040 Summer 2009 Exam III 1. For the following basic probability questions. Give the RULE used in the appropriate blank (BEFORE the question), for each of the following situations, using one

More information

Exam III Review Problems

Exam III Review Problems c Kathryn Bollinger and Benjamin Aurispa, November 10, 2011 1 Exam III Review Problems Fall 2011 Note: Not every topic is covered in this review. Please also take a look at the previous Week-in-Reviews

More information

STATION 1: ROULETTE. Name of Guesser Tally of Wins Tally of Losses # of Wins #1 #2

STATION 1: ROULETTE. Name of Guesser Tally of Wins Tally of Losses # of Wins #1 #2 Casino Lab 2017 -- ICM The House Always Wins! Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away

More information

November 11, Chapter 8: Probability: The Mathematics of Chance

November 11, Chapter 8: Probability: The Mathematics of Chance Chapter 8: Probability: The Mathematics of Chance November 11, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Probability Rules Probability Rules Rule 1.

More information

CSC/MTH 231 Discrete Structures II Spring, Homework 5

CSC/MTH 231 Discrete Structures II Spring, Homework 5 CSC/MTH 231 Discrete Structures II Spring, 2010 Homework 5 Name 1. A six sided die D (with sides numbered 1, 2, 3, 4, 5, 6) is thrown once. a. What is the probability that a 3 is thrown? b. What is the

More information

SIC BO ON THE MULTI TERMINALS

SIC BO ON THE MULTI TERMINALS How to play SIC BO ON THE MULTI TERMINALS LET S PLAY SIC BO Sic Bo is a Chinese dice game with a history dating back centuries. Originally played using painted bricks, modern Sic Bo has evolved into the

More information

Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? Section 6.1 #16 What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1

More information

Moore, IPS 6e Chapter 05

Moore, IPS 6e Chapter 05 Page 1 of 9 Moore, IPS 6e Chapter 05 Quizzes prepared by Dr. Patricia Humphrey, Georgia Southern University Suppose that you are a student worker in the Statistics Department and they agree to pay you

More information

Random Variables. A Random Variable is a rule that assigns a number to each outcome of an experiment.

Random Variables. A Random Variable is a rule that assigns a number to each outcome of an experiment. Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,

More information

Random Variables. Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5. (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) }

Random Variables. Outcome X (1, 1) 2 (2, 1) 3 (3, 1) 4 (4, 1) 5. (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) } Random Variables When we perform an experiment, we are often interested in recording various pieces of numerical data for each trial. For example, when a patient visits the doctor s office, their height,

More information

STAT 311 (Spring 2016) Worksheet W8W: Bernoulli, Binomial due: 3/21

STAT 311 (Spring 2016) Worksheet W8W: Bernoulli, Binomial due: 3/21 Name: Group 1) For each of the following situations, determine i) Is the distribution a Bernoulli, why or why not? If it is a Bernoulli distribution then ii) What is a failure and what is a success? iii)

More information

Presentation by Toy Designers: Max Ashley

Presentation by Toy Designers: Max Ashley A new game for your toy company Presentation by Toy Designers: Shawntee Max Ashley As game designers, we believe that the new game for your company should: Be equally likely, giving each player an equal

More information

If a fair coin is tossed 10 times, what will we see? 24.61% 20.51% 20.51% 11.72% 11.72% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098%

If a fair coin is tossed 10 times, what will we see? 24.61% 20.51% 20.51% 11.72% 11.72% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098% Coin tosses If a fair coin is tossed 10 times, what will we see? 30% 25% 24.61% 20% 15% 10% Probability 20.51% 20.51% 11.72% 11.72% 5% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098% 0 1 2 3 4 5 6 7 8 9 10 Number

More information

Math Steven Noble. November 24th. Steven Noble Math 3790

Math Steven Noble. November 24th. Steven Noble Math 3790 Math 3790 Steven Noble November 24th The Rules of Craps In the game of craps you roll two dice then, if the total is 7 or 11, you win, if the total is 2, 3, or 12, you lose, In the other cases (when the

More information

Math March 12, Test 2 Solutions

Math March 12, Test 2 Solutions Math 447 - March 2, 203 - Test 2 Solutions Name: Read these instructions carefully: The points assigned are not meant to be a guide to the difficulty of the problems. If the question is multiple choice,

More information

The game of poker. Gambling and probability. Poker probability: royal flush. Poker probability: four of a kind

The game of poker. Gambling and probability. Poker probability: royal flush. Poker probability: four of a kind The game of poker Gambling and probability CS231 Dianna Xu 1 You are given 5 cards (this is 5-card stud poker) The goal is to obtain the best hand you can The possible poker hands are (in increasing order):

More information

Here are two situations involving chance:

Here are two situations involving chance: Obstacle Courses 1. Introduction. Here are two situations involving chance: (i) Someone rolls a die three times. (People usually roll dice in pairs, so dice is more common than die, the singular form.)

More information

or More Events Activities D2.1 Open and Shut Case D2.2 Fruit Machines D2.3 Birthdays Notes for Solutions (1 page)

or More Events Activities D2.1 Open and Shut Case D2.2 Fruit Machines D2.3 Birthdays Notes for Solutions (1 page) D2 Probability of Two or More Events Activities Activities D2.1 Open and Shut Case D2.2 Fruit Machines D2.3 Birthdays Notes for Solutions (1 page) ACTIVITY D2.1 Open and Shut Case In a Game Show in America,

More information

1. Decide whether the possible resulting events are equally likely. Explain. Possible resulting events

1. Decide whether the possible resulting events are equally likely. Explain. Possible resulting events Applications. Decide whether the possible resulting events are equally likely. Explain. Action Possible resulting events a. You roll a number You roll an even number, or you roll an cube. odd number. b.

More information

1. Determine whether the following experiments are binomial.

1. Determine whether the following experiments are binomial. Math 141 Exam 3 Review Problem Set Note: Not every topic is covered in this review. It is more heavily weighted on 8.4-8.6. Please also take a look at the previous Week in Reviews for more practice problems

More information

Name: Practice Exam 3B. April 16, 2015

Name: Practice Exam 3B. April 16, 2015 Department of Mathematics University of Notre Dame Math 10120 Finite Math Spring 2015 Name: Instructors: Garbett & Migliore Practice Exam 3B April 16, 2015 This exam is in two parts on 12 pages and contains

More information

Lecture 21/Chapter 18 When Intuition Differs from Relative Frequency

Lecture 21/Chapter 18 When Intuition Differs from Relative Frequency Lecture 21/Chapter 18 When Intuition Differs from Relative Frequency Birthday Problem and Coincidences Gambler s Fallacy Confusion of the Inverse Expected Value: Short Run vs. Long Run Psychological Influences

More information

Review Questions on Ch4 and Ch5

Review Questions on Ch4 and Ch5 Review Questions on Ch4 and Ch5 1. Find the mean of the distribution shown. x 1 2 P(x) 0.40 0.60 A) 1.60 B) 0.87 C) 1.33 D) 1.09 2. A married couple has three children, find the probability they are all

More information

Exam #1. Good luck! Page 1 of 7

Exam #1. Good luck! Page 1 of 7 Exam # Total: 00 points Date: July, 008 Time: :00 :0 You have hour and 0 minutes to finish the exam. Please read the question carefully and assign your time smartly. Please PRINIT your name on each page

More information

University of California, Berkeley, Statistics 20, Lecture 1. Michael Lugo, Fall Exam 2. November 3, 2010, 10:10 am - 11:00 am

University of California, Berkeley, Statistics 20, Lecture 1. Michael Lugo, Fall Exam 2. November 3, 2010, 10:10 am - 11:00 am University of California, Berkeley, Statistics 20, Lecture 1 Michael Lugo, Fall 2010 Exam 2 November 3, 2010, 10:10 am - 11:00 am Name: Signature: Student ID: Section (circle one): 101 (Joyce Chen, TR

More information

Math 106 Lecture 3 Probability - Basic Terms Combinatorics and Probability - 1 Odds, Payoffs Rolling a die (virtually)

Math 106 Lecture 3 Probability - Basic Terms Combinatorics and Probability - 1 Odds, Payoffs Rolling a die (virtually) Math 106 Lecture 3 Probability - Basic Terms Combinatorics and Probability - 1 Odds, Payoffs Rolling a die (virtually) m j winter, 00 1 Description We roll a six-sided die and look to see whether the face

More information

Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.

Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Math 166 Fall 2008 c Heather Ramsey Page 1 Math 166 - Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 3.2 - Measures of Central Tendency

More information

Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.

Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Math 166 Fall 2008 c Heather Ramsey Page 1 Math 166 - Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 3.2 - Measures of Central Tendency

More information

Compute P(X 4) = Chapter 8 Homework Problems Compiled by Joe Kahlig

Compute P(X 4) = Chapter 8 Homework Problems Compiled by Joe Kahlig 141H homework problems, 10C-copyright Joe Kahlig Chapter 8, Page 1 Chapter 8 Homework Problems Compiled by Joe Kahlig Section 8.1 1. Classify the random variable as finite discrete, infinite discrete,

More information

1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 100 calculators is tested.

1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 100 calculators is tested. 1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 0 calculators is tested. Write down the expected number of faulty calculators in the sample. Find

More information

Math 4653, Section 001 Elementary Probability Fall Week 3 Worksheet

Math 4653, Section 001 Elementary Probability Fall Week 3 Worksheet Week 3 Worksheet Ice Breaker Question: What is the first thing you bought with your own money? 1. If S n = binomial(n, p), find var(s n ). 2. Suppose a lottery ticket has probability p of being a winning

More information

MAT Midterm Review

MAT Midterm Review MAT 120 - Midterm Review Name Identify the population and the sample. 1) When 1094 American households were surveyed, it was found that 67% of them owned two cars. Identify whether the statement describes

More information

Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results:

Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results: Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results: Outcome Frequency 1 8 2 8 3 12 4 7 5 15 8 7 8 8 13 9 9 10 12 (a) What is the experimental probability

More information

* How many total outcomes are there if you are rolling two dice? (this is assuming that the dice are different, i.e. 1, 6 isn t the same as a 6, 1)

* How many total outcomes are there if you are rolling two dice? (this is assuming that the dice are different, i.e. 1, 6 isn t the same as a 6, 1) Compound probability and predictions Objective: Student will learn counting techniques * Go over HW -Review counting tree -All possible outcomes is called a sample space Go through Problem on P. 12, #2

More information

1) What is the total area under the curve? 1) 2) What is the mean of the distribution? 2)

1) What is the total area under the curve? 1) 2) What is the mean of the distribution? 2) Math 1090 Test 2 Review Worksheet Ch5 and Ch 6 Name Use the following distribution to answer the question. 1) What is the total area under the curve? 1) 2) What is the mean of the distribution? 2) 3) Estimate

More information

An Adaptive-Learning Analysis of the Dice Game Hog Rounds

An Adaptive-Learning Analysis of the Dice Game Hog Rounds An Adaptive-Learning Analysis of the Dice Game Hog Rounds Lucy Longo August 11, 2011 Lucy Longo (UCI) Hog Rounds August 11, 2011 1 / 16 Introduction Overview The rules of Hog Rounds Adaptive-learning Modeling

More information

Suppose Y is a random variable with probability distribution function f(y). The mathematical expectation, or expected value, E(Y) is defined as:

Suppose Y is a random variable with probability distribution function f(y). The mathematical expectation, or expected value, E(Y) is defined as: Suppose Y is a random variable with probability distribution function f(y). The mathematical expectation, or expected value, E(Y) is defined as: E n ( Y) y f( ) µ i i y i The sum is taken over all values

More information

Basic Probability & Statistics Exam 2 { Part I { Sections (Chapter 4, Chapter 5) March 19, 2009

Basic Probability & Statistics Exam 2 { Part I { Sections (Chapter 4, Chapter 5) March 19, 2009 NAME: INSTRUCTOR: Dr. Bathi Kasturiarachi Math 30011 Spring 2009 Basic Probability & Statistics Exam 2 { Part I { Sections (Chapter 4, Chapter 5) March 19, 2009 Read through the entire test before beginning.

More information

Probabilities and Probability Distributions

Probabilities and Probability Distributions Probabilities and Probability Distributions George H Olson, PhD Doctoral Program in Educational Leadership Appalachian State University May 2012 Contents Basic Probability Theory Independent vs. Dependent

More information

Math 4610, Problems to be Worked in Class

Math 4610, Problems to be Worked in Class Math 4610, Problems to be Worked in Class Bring this handout to class always! You will need it. If you wish to use an expanded version of this handout with space to write solutions, you can download one

More information

HOMEWORK 3 Due: next class 2/3

HOMEWORK 3 Due: next class 2/3 HOMEWORK 3 Due: next class 2/3 1. Suppose the scores on an achievement test follow an approximately symmetric mound-shaped distribution with mean 500, min = 350, and max = 650. Which of the following is

More information

North Seattle Community College Winter ELEMENTARY STATISTICS 2617 MATH Section 05, Practice Questions for Test 2 Chapter 3 and 4

North Seattle Community College Winter ELEMENTARY STATISTICS 2617 MATH Section 05, Practice Questions for Test 2 Chapter 3 and 4 North Seattle Community College Winter 2012 ELEMENTARY STATISTICS 2617 MATH 109 - Section 05, Practice Questions for Test 2 Chapter 3 and 4 1. Classify each statement as an example of empirical probability,

More information

5. Aprimenumberisanumberthatisdivisibleonlyby1anditself. Theprimenumbers less than 100 are listed below.

5. Aprimenumberisanumberthatisdivisibleonlyby1anditself. Theprimenumbers less than 100 are listed below. 1. (a) Let x 1,x 2,...,x n be a given data set with mean X. Now let y i = x i + c, for i =1, 2,...,n be a new data set with mean Ȳ,wherecisaconstant. What will be the value of Ȳ compared to X? (b) Let

More information

MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #2 - FALL DR. DAVID BRIDGE

MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #2 - FALL DR. DAVID BRIDGE MATH 2053 - CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #2 - FALL 2009 - DR. DAVID BRIDGE MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the

More information

c. If you roll the die six times what are your chances of getting at least one d. roll.

c. If you roll the die six times what are your chances of getting at least one d. roll. 1. Find the area under the normal curve: a. To the right of 1.25 (100-78.87)/2=10.565 b. To the left of -0.40 (100-31.08)/2=34.46 c. To the left of 0.80 (100-57.63)/2=21.185 d. Between 0.40 and 1.30 for

More information

18.S34 (FALL, 2007) PROBLEMS ON PROBABILITY

18.S34 (FALL, 2007) PROBLEMS ON PROBABILITY 18.S34 (FALL, 2007) PROBLEMS ON PROBABILITY 1. Three closed boxes lie on a table. One box (you don t know which) contains a $1000 bill. The others are empty. After paying an entry fee, you play the following

More information

#3. Let A, B and C be three sets. Draw a Venn Diagram and use shading to show the set: PLEASE REDRAW YOUR FINAL ANSWER AND CIRCLE IT!

#3. Let A, B and C be three sets. Draw a Venn Diagram and use shading to show the set: PLEASE REDRAW YOUR FINAL ANSWER AND CIRCLE IT! Math 111 Practice Final For #1 and #2. Let U = { 1, 2, 3, 4, 5, 6, 7, 8} M = {1, 3, 5 } N = {1, 2, 4, 6 } P = {1, 5, 8 } List the members of each of the following sets, using set braces. #1. (M U P) N

More information

, x {1, 2, k}, where k > 0. (a) Write down P(X = 2). (1) (b) Show that k = 3. (4) Find E(X). (2) (Total 7 marks)

, x {1, 2, k}, where k > 0. (a) Write down P(X = 2). (1) (b) Show that k = 3. (4) Find E(X). (2) (Total 7 marks) 1. The probability distribution of a discrete random variable X is given by 2 x P(X = x) = 14, x {1, 2, k}, where k > 0. Write down P(X = 2). (1) Show that k = 3. Find E(X). (Total 7 marks) 2. In a game

More information

Math : Probabilities

Math : Probabilities 20 20. Probability EP-Program - Strisuksa School - Roi-et Math : Probabilities Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 200 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou

More information

Due Friday February 17th before noon in the TA drop box, basement, AP&M. HOMEWORK 3 : HAND IN ONLY QUESTIONS: 2, 4, 8, 11, 13, 15, 21, 24, 27

Due Friday February 17th before noon in the TA drop box, basement, AP&M. HOMEWORK 3 : HAND IN ONLY QUESTIONS: 2, 4, 8, 11, 13, 15, 21, 24, 27 Exercise Sheet 3 jacques@ucsd.edu Due Friday February 17th before noon in the TA drop box, basement, AP&M. HOMEWORK 3 : HAND IN ONLY QUESTIONS: 2, 4, 8, 11, 13, 15, 21, 24, 27 1. A six-sided die is tossed.

More information

Probability Essential Math 12 Mr. Morin

Probability Essential Math 12 Mr. Morin Probability Essential Math 12 Mr. Morin Name: Slot: Introduction Probability and Odds Single Event Probability and Odds Two and Multiple Event Experimental and Theoretical Probability Expected Value (Expected

More information

Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples

Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples Section 6.1 An Introduction to Discrete Probability Page references correspond to locations of Extra Examples icons in the textbook.

More information

Something to Think About

Something to Think About Probability Facts Something to Think About Name Ohio Lottery information: one picks 6 numbers from the set {1,2,3,...49,50}. The state then randomly picks 6 numbers. If you match all 6, you win. The number

More information

2. The value of the middle term in a ranked data set is called: A) the mean B) the standard deviation C) the mode D) the median

2. The value of the middle term in a ranked data set is called: A) the mean B) the standard deviation C) the mode D) the median 1. An outlier is a value that is: A) very small or very large relative to the majority of the values in a data set B) either 100 units smaller or 100 units larger relative to the majority of the values

More information

If a fair coin is tossed 10 times, what will we see? 24.61% 20.51% 20.51% 11.72% 11.72% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098%

If a fair coin is tossed 10 times, what will we see? 24.61% 20.51% 20.51% 11.72% 11.72% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098% Coin tosses If a fair coin is tossed 10 times, what will we see? 30% 25% 24.61% 20% 15% 10% Probability 20.51% 20.51% 11.72% 11.72% 5% 4.39% 4.39% 0.98% 0.98% 0.098% 0.098% 0 1 2 3 4 5 6 7 8 9 10 Number

More information

Probability Paradoxes

Probability Paradoxes Probability Paradoxes Washington University Math Circle February 20, 2011 1 Introduction We re all familiar with the idea of probability, even if we haven t studied it. That is what makes probability so

More information

Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance Chapter 8: Probability: The Mathematics of Chance Free-Response 1. A spinner with regions numbered 1 to 4 is spun and a coin is tossed. Both the number spun and whether the coin lands heads or tails is

More information

Name: Final Exam May 7, 2014

Name: Final Exam May 7, 2014 MATH 10120 Finite Mathematics Final Exam May 7, 2014 Name: Be sure that you have all 16 pages of the exam. The exam lasts for 2 hrs. There are 30 multiple choice questions, each worth 5 points. You may

More information

Guide. Odds. Understanding. The THE HOUSE ADVANTAGE

Guide. Odds. Understanding. The THE HOUSE ADVANTAGE THE HOUSE ADVANTAGE A Guide The Odds to Understanding AMERICAN GAMING ASSOCIATION 1299 Pennsylvania Avenue, NW Suite 1175 Washington, DC 20004 202-552-2675 www.americangaming.org 2005 American Gaming Association.

More information

Mrs. Daniel- AP Stats Chapter 6 MC Practice

Mrs. Daniel- AP Stats Chapter 6 MC Practice Mrs. Daniel- AP Stats Chapter 6 MC Practice Name: Exercises 1 and 2 refer to the following setting. Choose an American household at random and let the random variable X be the number of cars (including

More information

23 Applications of Probability to Combinatorics

23 Applications of Probability to Combinatorics November 17, 2017 23 Applications of Probability to Combinatorics William T. Trotter trotter@math.gatech.edu Foreword Disclaimer Many of our examples will deal with games of chance and the notion of gambling.

More information

Basic Probability Ideas. Experiment - a situation involving chance or probability that leads to results called outcomes.

Basic Probability Ideas. Experiment - a situation involving chance or probability that leads to results called outcomes. Basic Probability Ideas Experiment - a situation involving chance or probability that leads to results called outcomes. Random Experiment the process of observing the outcome of a chance event Simulation

More information

Probability with Set Operations. MATH 107: Finite Mathematics University of Louisville. March 17, Complicated Probability, 17th century style

Probability with Set Operations. MATH 107: Finite Mathematics University of Louisville. March 17, Complicated Probability, 17th century style Probability with Set Operations MATH 107: Finite Mathematics University of Louisville March 17, 2014 Complicated Probability, 17th century style 2 / 14 Antoine Gombaud, Chevalier de Méré, was fond of gambling

More information

Name: Exam 01 (Midterm Part 2 take home, open everything)

Name: Exam 01 (Midterm Part 2 take home, open everything) Name: Exam 01 (Midterm Part 2 take home, open everything) To help you budget your time, questions are marked with *s. One * indicates a straightforward question testing foundational knowledge. Two ** indicate

More information

Mini-Lecture 6.1 Discrete Random Variables

Mini-Lecture 6.1 Discrete Random Variables Mini-Lecture 6.1 Discrete Random Variables Objectives 1. Distinguish between discrete and continuous random variables 2. Identify discrete probability distributions 3. Construct probability histograms

More information

EE 126 Fall 2006 Midterm #1 Thursday October 6, 7 8:30pm DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO

EE 126 Fall 2006 Midterm #1 Thursday October 6, 7 8:30pm DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO EE 16 Fall 006 Midterm #1 Thursday October 6, 7 8:30pm DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO You have 90 minutes to complete the quiz. Write your solutions in the exam booklet. We will

More information

4.3 Rules of Probability

4.3 Rules of Probability 4.3 Rules of Probability If a probability distribution is not uniform, to find the probability of a given event, add up the probabilities of all the individual outcomes that make up the event. Example:

More information

BAYESIAN STATISTICAL CONCEPTS

BAYESIAN STATISTICAL CONCEPTS BAYESIAN STATISTICAL CONCEPTS A gentle introduction Alex Etz @alxetz ß Twitter (no e in alex) alexanderetz.com ß Blog November 5 th 2015 Why do we do statistics? Deal with uncertainty Will it rain today?

More information

Cycle Roulette The World s Best Roulette System By Mike Goodman

Cycle Roulette The World s Best Roulette System By Mike Goodman Cycle Roulette The World s Best Roulette System By Mike Goodman In my forty years around gambling, this is the only roulette system I ve seen almost infallible. There will be times that you will loose

More information

HARD 1 HARD 2. Split the numbers above into three groups of three numbers each, so that the product of the numbers in each group is equal.

HARD 1 HARD 2. Split the numbers above into three groups of three numbers each, so that the product of the numbers in each group is equal. HARD 1 3 4 5 6 7 8 28 30 35 Split the numbers above into three groups of three numbers each, so that the product of the numbers in each group is equal. Answer: (3, 8, 35), (4, 7, 30) and (5, 6, 28). Solution:

More information

Student activity sheet Gambling in Australia quick quiz

Student activity sheet Gambling in Australia quick quiz Student activity sheet Gambling in Australia quick quiz Read the following statements, then circle if you think the statement is true or if you think it is false. 1 On average people in North America spend

More information

Week in Review #5 ( , 3.1)

Week in Review #5 ( , 3.1) Math 166 Week-in-Review - S. Nite 10/6/2012 Page 1 of 5 Week in Review #5 (2.3-2.4, 3.1) n( E) In general, the probability of an event is P ( E) =. n( S) Distinguishable Permutations Given a set of n objects

More information

OUTSIDE IOWA, CALL

OUTSIDE IOWA, CALL WWW.1800BETSOFF.ORG OUTSIDE IOWA, CALL 1-800-522-4700 IOWA DEPARTMENT OF PUBLIC HEALTH, GAMBLING TREATMENT PROGRAM PROMOTING AND PROTECTING THE HEALTH OF IOWANS Printing is made possible with money from

More information

Probability: Anticipating Patterns

Probability: Anticipating Patterns Probability: Anticipating Patterns Anticipating Patterns: Exploring random phenomena using probability and simulation (20% 30%) Probability is the tool used for anticipating what the distribution of data

More information

The student will explain and evaluate the financial impact and consequences of gambling.

The student will explain and evaluate the financial impact and consequences of gambling. What Are the Odds? Standard 12 The student will explain and evaluate the financial impact and consequences of gambling. Lesson Objectives Recognize gambling as a form of risk. Calculate the probabilities

More information

Math 10 Homework 2 ANSWER KEY. Name: Lecturer: Instructions

Math 10 Homework 2 ANSWER KEY. Name: Lecturer: Instructions Math 10 Homework 2 ANSWER KEY Name: Lecturer: Instructions Type your answers and paste images directly into this document. Answers are usually short, with 1-3 sentences. Print out and hand in homework

More information

MATH 1115, Mathematics for Commerce WINTER 2011 Toby Kenney Homework Sheet 6 Model Solutions

MATH 1115, Mathematics for Commerce WINTER 2011 Toby Kenney Homework Sheet 6 Model Solutions MATH, Mathematics for Commerce WINTER 0 Toby Kenney Homework Sheet Model Solutions. A company has two machines for producing a product. The first machine produces defective products % of the time. The

More information

Probability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37

Probability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37 Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete

More information

KS specimen papers

KS specimen papers KS4 2016 specimen papers OCR H3 specimen 14 A straight line goes through the points (p, q) and (r, s), where p + 2 = r q + 4 = s. Find the gradient of the line. AQA F3 H3 specimen 21 When x² = 16 the only

More information

Chapter 11. Sampling Distributions. BPS - 5th Ed. Chapter 11 1

Chapter 11. Sampling Distributions. BPS - 5th Ed. Chapter 11 1 Chapter 11 Sampling Distributions BPS - 5th Ed. Chapter 11 1 Sampling Terminology Parameter fixed, unknown number that describes the population Statistic known value calculated from a sample a statistic

More information

A Mathematical Analysis of Oregon Lottery Keno

A Mathematical Analysis of Oregon Lottery Keno Introduction A Mathematical Analysis of Oregon Lottery Keno 2017 Ted Gruber This report provides a detailed mathematical analysis of the keno game offered through the Oregon Lottery (http://www.oregonlottery.org/games/draw-games/keno),

More information

Introduction to Auction Theory: Or How it Sometimes

Introduction to Auction Theory: Or How it Sometimes Introduction to Auction Theory: Or How it Sometimes Pays to Lose Yichuan Wang March 7, 20 Motivation: Get students to think about counter intuitive results in auctions Supplies: Dice (ideally per student)

More information

Expected Value(Due by EOC Nov. 1)

Expected Value(Due by EOC Nov. 1) Expected Value(Due by EOC Nov. ) Just Give Him The Slip.. a) Suppose you have a bag with slips of paper in it. Some of the slips have a on them, and the rest have a 7. If the expected value of the number

More information