# Lecture 13. Understanding Probability and Long-Term Expectations

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Lecture 13 Understanding Probability and Long-Term Expectations

2 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). So toss a coin twice. Do it! Did you get one head & one tail? What s it all mean?

3 Probability as long term relative frequency

4 What about Probability? The foundation of Probability theory lies in problems associated with gambling and games of chance The Romans used played a game with ASTRAGALI - Heel bones of animals

5 DICE DICE as we know them were invented around 300 BC

6 I lied, cheated and stole to become a millionaire. Now anybody at all can win the lottery and become a millionaire

7 LOTTO 6/42 What are the chances of winning with one selection of 6 numbers? Matches Odds 6 1 in 5,245, in 24, in 555

8 LOTTO 6/42 The average time to win each of the prizes is given by: Match 3 with Bonus 2 Years, 6 Weeks Match 4 2 Years, 8 Months Match Years, 9 Months Match 5 with Bonus 4323 Years, 5 Months Share in Jackpot 25,220 Years

9 Why do people still play the lottery? If you re not in you can t win! You never know your luck until you try! My chances of winning a million are better than my chances of earning a million. The lottery is a tax on the statistically challenged.

10 Thought Question 1: Two very different queries about probability: a. If you flip a coin and do it fairly, what is the probability that it will land heads up? b. What is the probability that you will eventually own a home; that is, how likely do you think it is? (If you already own a home, what is the probability that you will own a different home within the next 5 years?) For which question was it easier to provide a precise answer? Why?

11 Thought Question 2: Explain what it means for someone to say that the probability of his or her eventually owning a home is 70%.

12 Thought Question 3: Explain what s wrong with this partial answer to Thought Question 1: The probability that I will eventually own a home, or of any other particular event happening, is 1/2 because either it will happen or it won t.

13 Thought Question 4: Why do you think insurance companies charge young men more than they do older men for automobile insurance, but charge older men more for life insurance?

14 Thought Question 5: How much would you be willing to pay for a ticket to a contest in which there was a 1% chance that you would win \$500 and a 99% chance that you would win nothing? Explain your answer.

15 16.1 Probability What does the word probability mean? Two distinct interpretations: For the probability of winning a lottery based on buying a single ticket -- we can quantify the chances exactly. For the probability that we will eventually buy a home -- we are basing our assessment on personal beliefs about how life will evolve for us.

16 16.2 The Relative-Frequency Interpretation Relative-frequency interpretation: applies to situations that can be repeated over and over again. Examples: Buying a weekly lottery ticket and observing whether it is a winner. Commuting to work daily and observing whether a certain traffic signal is red when we encounter it. Testing individuals in a population and observing whether they carry a gene for a certain disease. Observing births and noting if baby is male or female.

17 Idea of Long-Run Relative Frequency Probability = proportion of time it occurs over the long run Long-run relative frequency of males born in the United States is about (Information Please Almanac, 1991, p. 815) Possible results for relative frequency of male births: Proportion of male births jumps around at first but starts to settle down just above.51 in the long run.

18 Determining the Probability of an Outcome Method 1: Make an Assumption about the Physical World Example: assume coins made such that they are equally likely to land with heads or tails up when flipped. probability of a flipped coin showing heads up is ½. Method 2: Observe the Relative Frequency Example: observe the relative frequency of male births in a given city over the course of a year. In 1987 there were a total of 3,809,394 live births in the U.S., of which 1,951,153 where males. probability of male birth is 1,951,153/3,809,394 =

19 Summary of Relative-Frequency Interpretation of Probability Can be applied when situation can be repeated numerous times and outcome observed each time. Relative frequency should settle down to constant value over long run, which is the probability. Does not apply to situations where outcome one time is influenced by or influences outcome the next time. Cannot be used to determine whether outcome will occur on a single occasion but can be used to predict long-term proportion of times the outcome will occur.

20 16.3 The Personal-Probability Interpretation Person probability: the degree to which a given individual believes the event will happen. They must be between 0 and 1 and be coherent. Examples: Probability of finding a parking space downtown on Saturday. Probability that a particular candidate for a position would fit the job best.

21 16.4 Applying Some Simple Probability Rules Rule 1: If there are only two possible outcomes in an uncertain situation, then their probabilities must add to 1. Example 1: If probability of a single birth resulting in a boy is 0.51, then the probability of it resulting in a girl is 0.49.

22 Simple Probability Rules Rule 2: If two outcomes cannot happen simultaneously, they are said to be mutually exclusive. The probability of one or the other of two mutually exclusive outcomes happening is the sum of their individual probabilities. Example 5: If you think probability of getting an A in your statistics class is 50% and probability of getting a B is 30%, then probability of getting either an A or a B is 80%. Thus, probability of getting C or less is 20% (using Rule 1).

23 Simple Probability Rules Rule 3: If two events do not influence each other, and if knowledge about one doesn t help with knowledge of the probability of the other, the events are said to be independent of each other. If two events are independent, the probability that they both happen is found by multiplying their individual probabilities. Example 7: Woman will have two children. Assume outcome of 2 nd birth independent of 1 st and probability birth results in boy is Then probability of a boy followed by a girl is (0.51)(0.49) = About a 25% chance a woman will have a boy and then a girl.

24 Simple Probability Rules Rule 4: If the ways in which one event can occur are a subset of those in which another event can occur, then the probability of the subset event cannot be higher than the probability of the one for which it is a subset. Example 10: Suppose you are 18 and speculating about your future. You decide the probability you will eventually get married and have children is 75%. By Rule 4, probability that you will eventually get married is at least 75%.

25 16.5 When Will It Happen? Probability an outcome will occur on any given instance is p. Probability the outcome will not occur is (1 p). Outcome each time is independent of outcome all other times. Probability it doesn t occur on 1 st try but does occur on 2 nd try is (1 p)p.

26 Example 11: Number of Births to First Girls Probability of a birth resulting in a boy is about.51, and the probability of a birth resulting in a girl is about.49. Suppose couple will continue having children until have a girl. Assuming outcomes of births are independent of each other, probabilities of having the first girl on the first, second, third, fifth, and seventh tries are shown below.

27 Accumulated Probability Probability of 1 st occurrence not happening by occasion n is (1 p) n. Probability 1 st occurrence has happened by occasion n is [1 (1 p) n ]. Example 12: Getting Infected with HIV Suppose probability of infected is 1/500 = Probability of not infected is So probability of infection by 2 nd encounter is [1 ( ) 2 ] =

28 16.6 Long-Term Gains, Losses, and Expectations Long-Term Outcomes Can Be Predicted Example 14: Insurance Policies A simple case: All customers charged \$500/year, 10% submit a claim in any given year and claims always for \$1500. How much can the company expect to make per customer? Average Gain =.90 (\$500).10 (\$1000) = \$350 per customer

29 Expected Value (EV) EV = expected value = A 1 p 1 + A 2 p 2 + A 3 p A k p k where A 1, A 2, A 3,, A k are the possible amounts and p 1, p 2, p 3,, p k are the associated probabilities. Example 14, continued: A 1 p 1 + A 2 p 2 = (.10)( \$1000) + (.90)(\$500) = \$350 = expected value of company s profit per customer. Key: Expected value is the average value per measurement over the long run and not necessarily a typical value for any one occasion or person.

30 Example 15: California Decco Lottery Game Decco: player chooses one card from each of four suits. A winning card drawn from each suit. Prizes awarded based on the number of matches. Cost to play = \$1. EV = (\$4999)(1/28,561) + (\$49)(1/595) + (\$4)(.0303) + ( \$1)(.726) = \$0.35 Over many repetitions, players will lose an average of 35 cents per play. The Lottery Commission will only pay out about 65 cents for each \$1 ticket sold.

31 Expected Value as Mean Number If measurement taken over a large group of individuals, the expected value can be interpreted as the mean value per individual. Example: Suppose 40% of people in a population smoke a pack of cigarettes a day (20 cigarettes) and remaining 60% smoke none. Expected number of cigarettes smoked per day by one person: EV = (0.40)(20 cigarettes) + (0.60)(0 cigarettes) = 8 cigarettes On average 8 cigarettes are smoked per person per day. Here again, the EV is not a value we actually expect to measure on any one individual.

32 Example: Consider a lottery game where you pay \$1 to play and have the following chances of winning certain prizes. What is the expected value of this game? Prize Net Gain Probability \$0.00 -\$ \$1.00 \$ \$10.00 \$ \$ \$ Answer: EV= (-\$1)(0.7) + (\$0)(0.25) +(\$9)(0.049) + (99)(0.001) = -\$0.16 This result tells us that over many repetitions of the game, you will lose an average of 16 cents per game.

33 Roulette Gamble What is the EV of betting \$1 on black 13 in Roulette?

34 Roulette Gamble What is the EV of betting \$1 on black 13 in Roulette? Outcome Probability of \$ Payoff occurrence 13 1/38 \$35 Not 13 37/38 \$-1

35 Roulette Gamble What is the EV of betting \$1 on black 13 in Roulette? (\$35 * 1/38) + (\$-1 * 37/38) = \$ Each time you place a bet on the roulette table you should expect to lose 5.26%.

36 Case Study 16.1: Birthdays and Death Days Is There a Connection? Study Details: Data from death certificates of adults in California who died of natural causes between 1979 and Each death classified as to how many weeks after the birthday it occurred. Compared actual numbers of deaths to expected number (seasonally adjusted). Results: Women: biggest peak in Week 0; highest number of women died the week after their birthdays. Men: biggest peak was in Week 51; highest number of men died the week before their birthdays. Probability biggest peak for women in Week 0 AND biggest peak for men in Week 51 = (1/52)(1/52) = Source: Phillips, Van Voorhies, and Ruth, 1992

37 For Those Who Like Formulas

38 We re ready to play some games

39 An Example Experiment: Roll Two Dice Possible Outcomes: Any number from 1 to 6 can appear on each die. There are 36 possible outcomes Each Outcome in the Sample Space is equally probable. So the probability of each outcome is 1/36 What is the probability of the Event - get combined total of 7 on the dice

40 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (6,1 ) (5,2 ) (4,3 ) (3,4 ) (2,5 ) (1,6 )

41 Lets Play a Game html

42 The Monty Hall Problem Let s Make A Deal: a game show back in 90 s. A player is given the choice of three doors. Behind one door is the Grand Prize (a car and a cruise); behind the other two doors, booby prizes (stinking pigs). The player picks a door, and the host peeks behind the doors and opens one of the rest of the doors. There is a booby prize behind the open door. The host offers the player either to stay with the door that was chosen at the beginning, or to switch to the remaining closed door. Which is better: to switch doors or to stay with the original choice? What are the chances of winning in either case?

43 Let s Make a Deal! Assume your are a contestant on this game show. There are three doors: 1 2 3

44 Let s Make a Deal! You have been asked to select one of these three doors

45 Let s Make a Deal! Behind one of the doors is the grand prize! 1

46 Let s Make a Deal! Behind one of the doors is the booby prize! 2

47 Let s Make a Deal! Now assume you have selected Door # 1. Monty Hall (the host) will now open one of the other doors (door # 2 or 3) and show you the prize behind that door (it won t be the grand prize).

48 Let s Make a Deal! He will now ask you whether you would like to keep your original selection (Door #1) or switch to the other unopened door (Door # 2 or 3). Should you switch?

49 Let s Make a Deal! Marilyn vos Savant - When you first choose door No. 1 from among the three, there s a 1/3 chance that the prize is behind that one and a 2/3 chance that it s behind one of the others. But then the host steps in and gives you a clue. If the prize is behind No. 2, the host shows you No. 3; and if the prize is behind is behind No. 3, the host shows you No. 2. So when you switch, you win if the prize is behind No. 2 or No. 3. YOU WIN EITHER WAY! But if you don t switch, you win only if the prize is behind door No. 1.

50 Let s Make a Deal! Grand Prize You choose: Host Opens: You Switch Result Is In: To: or 3 2 or 3 Lose Win Win Win or 3 1 or 3 Lose Win Win Win or 2 1 or 2 Lose

51 Let s Make a Deal! You lose in 3/9 cases - p(lose) = 1/3 You win in 6/9 cases - p(win) = 2/3.

### Quiz CHAPTER 16 NAME: UNDERSTANDING PROBABILITY AND LONG- TERM EXPECTATIONS

Quiz CHAPTER 16 NAME: UNDERSTANDING PROBABILITY AND LONG- TERM EXPECTATIONS 1. Give two examples of ways that we speak about probability in our every day lives. NY REASONABLE ANSWER OK. EXAMPLES: 1) WHAT

### 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

### 36 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

### Elementary 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

### 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

### 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

### 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

### Week 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

### 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

### 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

### 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

### 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

### A probability experiment is a chance process that leads to well-defined outcomes. 3) What is the difference between an outcome and an event?

Ch 4.2 pg.191~(1-10 all), 12 (a, c, e, g), 13, 14, (a, b, c, d, e, h, i, j), 17, 21, 25, 31, 32. 1) What is a probability experiment? A probability experiment is a chance process that leads to well-defined

### 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

### 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

### We rst consider the game from the player's point of view: Suppose you have picked a number and placed your bet. The probability of winning is

Roulette: On an American roulette wheel here are 38 compartments where the ball can land. They are numbered 1-36, and there are two compartments labeled 0 and 00. Half of the compartments numbered 1-36

### The Casino Lab STATION 1: CRAPS

The Casino Lab 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 very wealthy, while others will

### PROBABILITY 14.3. section. The Probability of an Event

4.3 Probability (4-3) 727 4.3 PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques

### 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.

### LET S MAKE A DEAL! ACTIVITY

LET S MAKE A DEAL! ACTIVITY NAME: DATE: SCENARIO: Suppose you are on the game show Let s Make A Deal where Monty Hall (the host) gives you a choice of three doors. Behind one door is a valuable prize.

### 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

### MULTIPLE 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

### Exam. Name. How many distinguishable permutations of letters are possible in the word? 1) CRITICS

Exam Name How many distinguishable permutations of letters are possible in the word? 1) CRITICS 2) GIGGLE An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm,

### 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

### Chapter 16: law of averages

Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................

### Find 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

### The New Mexico Lottery

The New Mexico Lottery 26 February 2014 Lotteries 26 February 2014 1/27 Today we will discuss the various New Mexico Lottery games and look at odds of winning and the expected value of playing the various

### 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

### Basic Probability Theory II

RECAP Basic Probability heory II Dr. om Ilvento FREC 408 We said the approach to establishing probabilities for events is to Define the experiment List the sample points Assign probabilities to the sample

### Standard 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

### Statistics 100A Homework 3 Solutions

Chapter Statistics 00A Homework Solutions Ryan Rosario. Two balls are chosen randomly from an urn containing 8 white, black, and orange balls. Suppose that we win \$ for each black ball selected and we

### Basic concepts in probability. Sue Gordon

Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are

### Elementary 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

### 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.

### 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,

### 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

### Betting 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

### 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

### 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

### Probability: 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

### 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

### Find an expected value involving two events. Find an expected value involving multiple events. Use expected value to make investment decisions.

374 Chapter 8 The Mathematics of Likelihood 8.3 Expected Value Find an expected value involving two events. Find an expected value involving multiple events. Use expected value to make investment decisions.

### 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

### Probabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I

Victor Adamchi Danny Sleator Great Theoretical Ideas In Computer Science Probability Theory I CS 5-25 Spring 200 Lecture Feb. 6, 200 Carnegie Mellon University We will consider chance experiments with

### Determine 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

### 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,

### Chapter 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

### Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

### 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,

### Expected Value. 24 February 2014. Expected Value 24 February 2014 1/19

Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery

### Problem sets for BUEC 333 Part 1: Probability and Statistics

Problem sets for BUEC 333 Part 1: Probability and Statistics I will indicate the relevant exercises for each week at the end of the Wednesday lecture. Numbered exercises are back-of-chapter exercises from

### The Normal Approximation to Probability Histograms. Dice: Throw a single die twice. The Probability Histogram: Area = Probability. Where are we going?

The Normal Approximation to Probability Histograms Where are we going? Probability histograms The normal approximation to binomial histograms The normal approximation to probability histograms of sums

### 14.4. Expected Value Objectives. Expected Value

. Expected Value Objectives. Understand the meaning of expected value. 2. Calculate the expected value of lotteries and games of chance.. Use expected value to solve applied problems. Life and Health Insurers

### ACMS 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

### PROBABILITY C A S I N O L A B

A P S T A T S A Fabulous PROBABILITY C A S I N O L A B AP Statistics Casino Lab 1 AP STATISTICS CASINO LAB: INSTRUCTIONS The purpose of this lab is to allow you to explore the rules of probability in the

### 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

### Mind 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

### Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes

Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all

### 6.3 Probabilities with Large Numbers

6.3 Probabilities with Large Numbers In general, we can t perfectly predict any single outcome when there are numerous things that could happen. But, when we repeatedly observe many observations, we expect

### Lotto Master Formula (v1.3) The Formula Used By Lottery Winners

Lotto Master Formula (v.) The Formula Used By Lottery Winners I. Introduction This book is designed to provide you with all of the knowledge that you will need to be a consistent winner in your local lottery

### AP 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!

### Statistics 100A Homework 2 Solutions

Statistics Homework Solutions Ryan Rosario Chapter 9. retail establishment accepts either the merican Express or the VIS credit card. total of percent of its customers carry an merican Express card, 6

### Contemporary 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

### Probability. a number between 0 and 1 that indicates how likely it is that a specific event or set of events will occur.

Probability Probability Simple experiment Sample space Sample point, or elementary event Event, or event class Mutually exclusive outcomes Independent events a number between 0 and 1 that indicates how

### Lesson 13: Games of Chance and Expected Value

Student Outcomes Students analyze simple games of chance. Students calculate expected payoff for simple games of chance. Students interpret expected payoff in context. esson Notes When students are presented

### PROBABILITY SECOND EDITION

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

### ACMS 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

### Chapter 5 A Survey of Probability Concepts

Chapter 5 A Survey of Probability Concepts True/False 1. Based on a classical approach, the probability of an event is defined as the number of favorable outcomes divided by the total number of possible

### Ch. 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

### Economics 1011a: Intermediate Microeconomics

Lecture 11: Choice Under Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 11: Choice Under Uncertainty Tuesday, October 21, 2008 Last class we wrapped up consumption over time. Today we

### 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

### 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

### The number of phone calls to the attendance office of a high school on any given school day A) continuous B) discrete

Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) State whether the variable is discrete or continuous.

### 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

### TOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE. Topic P2: Sample Space and Assigning Probabilities

TOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE Roulette is one of the most popular casino games. The name roulette is derived from the French word meaning small

### Decision 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

### What is the probability of throwing a fair die and receiving a six? Introduction to Probability. Basic Concepts

Basic Concepts Introduction to Probability A probability experiment is any experiment whose outcomes relies purely on chance (e.g. throwing a die). It has several possible outcomes, collectively called

### Chapter 20: chance error in sampling

Chapter 20: chance error in sampling Context 2 Overview................................................................ 3 Population and parameter..................................................... 4

### Introduction and Overview

Introduction and Overview Probability and Statistics is a topic that is quickly growing, has become a major part of our educational program, and has a substantial role in the NCTM Standards. While covering

### Decision Theory. 36.1 Rational prospecting

36 Decision Theory Decision theory is trivial, apart from computational details (just like playing chess!). You have a choice of various actions, a. The world may be in one of many states x; which one

### STATISTICS 8, FINAL EXAM. Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4

STATISTICS 8, FINAL EXAM NAME: KEY Seat Number: Last six digits of Student ID#: Circle your Discussion Section: 1 2 3 4 Make sure you have 8 pages. You will be provided with a table as well, as a separate

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### Lotto! Online Product Guide

BCLC Lotto! Online Product Guide Resource Manual for Lottery Retailers 29/04/2014 The focus of this document is to provide retailers the tools needed in order to feel knowledgeable when selling and discussing

### Slide 1 Math 1520, Lecture 23. This lecture covers mean, median, mode, odds, and expected value.

Slide 1 Math 1520, Lecture 23 This lecture covers mean, median, mode, odds, and expected value. Slide 2 Mean, Median and Mode Mean, Median and mode are 3 concepts used to get a sense of the central tendencies

### Random Variables. 9. Variation 1. Find the standard deviations of the random variables in Exercise 1.

Random Variables 1. Expected value. Find the expected value of each random variable: a) x 10 20 30 P(X=x) 0.3 0.5 0.2 b) x 2 4 6 8 P(X=x) 0.3 0.4 0.2 0.1 2. Expected value. Find the expected value of each

### Probability OPRE 6301

Probability OPRE 6301 Random Experiment... Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.

### KENO TRAINING MANUAL

KENO TRAINING MANUAL 1 Contents I. How to play Keno Pg 3 1. Keno Menu Pg 3 2. Keno Kwikfacts Pg 3 3. Keno Pg 4 4. Kwikpicks Pg 8 5. Jackpots Pg 10 6. Keno Bonus Pg 10 7. Lucky Last Pg 10 8. Heads or Tails

### R Simulations: Monty Hall problem

R Simulations: Monty Hall problem Monte Carlo Simulations Monty Hall Problem Statistical Analysis Simulation in R Exercise 1: A Gift Giving Puzzle Exercise 2: Gambling Problem R Simulations: Monty Hall

### ! Insurance and Gambling

2009-8-18 0 Insurance and Gambling Eric Hehner Gambling works as follows. You pay some money to the house. Then a random event is observed; it may be the roll of some dice, the draw of some cards, or the

### Chapter 17: Thinking About Chance

Chapter 17: Thinking About Chance Thought Question Flip a coin 20 times and record the results of each flip (H or T). Do we know what will come up before we flip the coin? If we flip a coin many, many

### STRIKE FORCE ROULETTE

STRIKE FORCE ROULETTE Cycles, cycles, cycles... You cannot get away from them in the game of Roulette. Red, black, red, black... Red, red, red, red. Black, black, black... Red, red, black, black... 1st

### How To Increase Your Odds Of Winning Scratch-Off Lottery Tickets!

How To Increase Your Odds Of Winning Scratch-Off Lottery Tickets! Disclaimer: All of the information inside this report reflects my own personal opinion and my own personal experiences. I am in NO way

### 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

### Gaming 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.

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli

### 13.0 Central Limit Theorem

13.0 Central Limit Theorem Discuss Midterm/Answer Questions Box Models Expected Value and Standard Error Central Limit Theorem 1 13.1 Box Models A Box Model describes a process in terms of making repeated

### Chapter 4 Probability

The Big Picture of Statistics Chapter 4 Probability Section 4-2: Fundamentals Section 4-3: Addition Rule Sections 4-4, 4-5: Multiplication Rule Section 4-7: Counting (next time) 2 What is probability?

### Inside the pokies - player guide

Inside the pokies - player guide 3nd Edition - May 2009 References 1, 2, 3 Productivity Commission 1999, Australia s Gambling Industries, Report No. 10, AusInfo, Canberra. 4 Victorian Department of Justice,