Probability definitions
|
|
- Sandra Polly Russell
- 7 years ago
- Views:
Transcription
1 Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a data-generating process Examples: toss a coin (one or more times), roll 2 dice, select 5 cards from a deck, interview 100 people for market research, observe the reaction of 50 patients to a new drug. 3. Sample space = set of all possible outcomes of an experiment. Example: if two dice (a red die and a green die) are tossed, any outcome is described by the number on the red die and the number on the green die. Note that the outcome is a low level description of what happened. Let the number on the red die be indicated by bold italics, so that (3, 2) indicates a 3 on the red die and a 2 on the green die. In the experiment of tossing a red die and a green die, we can list the sample space in a table: 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 4. Event = any collection or subset of outcomes in the sample space. Events are simple if they contain exactly one outcome, and are compound if they contain more than one. Example: We can define the event A = both dice show the same number, and calculate P(A) = 6 / 36. (If you xerox or copy the sheet, shade in the appropriate areas) Define event B = sum of two numbers at least 10. P (B) = 6 / Random variables associate a number with an event. If we define the random variable X as the sum of the two numbers on the dice, we can say P (X = 6) = 5/36 6. Union of two events, denoted by as in A B, constructs a new event both dice show the same number, or the sum of the two numbers is at least 10. This should be read as A or B. Find the probability of A or B in the example above. Does P(A or B) = P(A) + P(B)? Why not? 7. Intersection of two events, denoted by as in A B, constructs a new event - both dice show the same number, which is greater than or equal to 10. This should be read as A and B. Find the probability of A and B in the example above. Does P (A and B) = P(A) times P(B)? Why not? 8. Complement of an event A, denoted by a prime or superscript c, as in A or A c, indicates those outcomes not in the event A. What is the probability of A in the example above? Note that P(A ) = P(A) This relationship often makes calculations much simpler, especially when the problem includes phrases such as at least or at most. Example (Chevalier de la Mere): what is the probability that at least one six turns up in 4 tosses of a die? [Hint: it is a little more than half, De la Mere found that out by extensive experiment.] What is the probability that at least one double six turns up in 24 tosses of two dice? [The chance of double six is 1/36, but we compensate by having 6 times as many tosses, so de la Mere thought the probability should be the same. Is it? Hint: de la Mere lost a lot of money by believing this]
2 Statistics 1040 Dr. McGahagan Probability problems Simple occurences: Event Get a tail in a single toss of a fair coin Roll a 3 on a normal, 6 sided die Probability Draw a heart from a deck or cards Child born on a Saturday or Sunday More complicated events, for which it will be helpful to use random variables to keep track of outcomes (for example, one might want the random variable X = number of heads in 3 tosses of a fair coin) Event Probability Toss 3 heads in 3 tosses of a coin Roll boxcars (6 on each of 2 dice) Be dealt a flush (5 cards of same suit) in a standard 5 card deal. Toss exactly 2 heads and two tails IN THAT ORDER in 4 tosses of a coin Toss exactly 2 heads and two tails IN ANY ORDER in 4 tosses of a coin Roll either boxcars or snake-eyes (2 sixes or 2 ones in a roll of 2 dice.
3 Addition and multiplication rules the full story In some of the above problems, we used the addition rule in the form P(A) or P(B) = P(A) + P(B) and the multiplication rule in the form P(A and B) = P(A) P(B) We must extend the rules to take account of 1. events that are not mutually exclusive that is, events which can both happen at the same time. Suppose you have been dealt a poker hand of Jack,Queen, King and Ace of hearts along with the 2 of clubs. You are interested in the change of coming up with a winning hand if you discard the 2 and draw another card. What are the chances of getting either a 10 or a heart? We can assume that either a flush or a straight will win but there is the chance here of getting a royal flush with the 10 of hearts. 2. Events that are not independent that is, in which the probability of one event affects the probability of another. What is the chance of drawing two hearts in a row? Hint: NOT 13/52 times 13/52. Why? Suppose the chance that, for the US as a whole, the chance that a family s first car is domestic is.75, denoted as P(D1) =.75, and the chance that the second car is domestic is.4, denoted as P(D2) =.4 What is the chance that both cars are domestic? If the two events are independent, we could apply the multiplication rule P(D1 and D2) = P(D1) x P(D2) = 0.75 x 0.4 = 0.30 But it may be that purchasers show buyer loyalty, that is, those who purchased a domestic car for their first car are more likely than the average to buy a domestic car for their second car. Assume that P(D2 given D1) = 0.6 also written P(D1 D2) = 0.6 and calculate P(D1 and D2). Hint: look back at the two hearts in a row problem. P (D1 and D2) = P(D1) x P (D2 D1) =.75 x 0.6 = 0.45
4 A visual presentation of a similar problem: D1 = a family's first car is domestic; F1 = probability first car is foreign D2 = a family's second car is domestic; F2 = probability second car is foreign Given that P(D1) = 0.75 and P(D2) = 0.4, and assuming there are 100 total cars and that P(D1 and D2) = 0.35 fill in the following table: A few numbers have been filled in to get you started. Be sure you understand how they were arrived at, and how they reflect the statements above. Car 2 is domestic Car 2 is foreign ROW TOTALS Car 1 is domestic Car 1 is foreign COLUMN totals After doing so, calculate all the joint probabilities: P(D1 and D2) = 0.35 P(D1 and F2) = P(F1 and D2) = P(F1 and F2) = And all the conditional probabilities P(D2 D1) = P(D2 F1) = P(F2 D1) = P(F2 F1) = What is the probability that (for two car families described in the table above): a. Both a family's cars are domestic? b. Both a family's cars are foreign? c. A family has one domestic and one foreign car? d. We know that the Smith family has at least one domestic car. What is the chance that they also have a foreign car? e. We know that the Jones family has at least one foreign car. What is the chance that they also have a domestic car?
5 Answers: Car 2 is domestic Car 2 is foreign ROW TOTALS Car 1 is domestic Car 1 is foreign COLUMN totals The JOINT PROBABILITIES are easy: The table gives the NUMBERS of families in each category -- there are 35 families with both car 1 and car 2 being domestic, so the probability that any two-car family chosen at random having two domestic cars is 35 / 100 = 0.35 or 35 percent. P (D1 and D2 ) = 0.35 (answer for [a] on last page) P (D1 and F2) = 0.40 P (F1 and D2) = 0.05 P (F1 and F2) = 0.20 (answer for [b] on last page) For [c] on the previous page, note that families will have one domestic and one foreign car if their two cars are in the square D1 and F2 OR in the square F1 and D2. The OR is telling us to ADD the joint probabilities P(D1 and F2) + P (F1 and D2) = = 0.45 The ROW and COLUMN totals give the MARGINAL PROBABILITIES, since they are written in the margin of the detailed table (don't think of marginal cost!). P(D1) = 0.75 P (F1) = 0.40 P (D2) = 0.25 P (F2) = 0.60 CONDITIONAL PROBABILITIES can be read off the table: If you are GIVEN that D1 is domestic, you know you are in the first ROW of the table -- the information given means the family is one of the 75 for whom the first car is domestic. You can mentally reduce the entire table to: Car 2 is domestic Car 2 is foreign ROW TOTALS Car 1 is domestic Hence the probability that their second car is foreign is: P (F2 D1) = 40 / 75 = If we are GIVEN that the second car is domestic, the probability that the first car is foreign is P (F1 D2) = 5 / 40 = (mentally reduce the entire table to the first column) If we are given (as in part [d]) that the Smith family has at least one domestic car, we delete the F1 - F2 square from the table, leaving 80 families, so chance of also having a foreign car is 45 / 80 = For the Jones family in part [e], with at least one foreign car, delete the D1-D2 square; their chance of also having a domestic car is 20 / 65 =
6 Bayes's Theorem Suppose that 80 percent of the taxicabs in town are owned by the Yellow Cab Company and 20 percent are owned by the Blue Cab Company, and that they are painted accordingly. A taxicab driver was arrested in a bank robbery. A witness claims that a cab was used as the getaway car, and thinks that the cab was blue, although he admits that the light was poor. As a result of repeated tests in similar lighting conditions, the defense finds that the witness is 75 percent accurate -- that he correctly identifies a blue cab as blue 75 percent of the time, but incorrectly identifies a blue cab as yellow 25 percent of the time. We should: (a) reject the witness testimony as not perfectly accurate, and treat the probability of the cab being blue as 20 percent. (b) accept the witness as having a 75 percent chance of being right and the cab being blue (c) treat the probability of the cab being blue as somewhere between 20 and 75, but closer to 20 (that is, more than 20 but less than 47.5) (d) treat the probability of the cab being blue as somewhere between 20 and 75, but closer to 75 (that is, more than 47.5 but less than 75) Answer: The answer will be between the two extremes -- although not perfectly accurate, the witness is right more often than not, and his claim that the cab is blue raises the chance to more than 20 percent. But it is NOT 75 percent: the test establishes the chance the witness SAYS the cab is blue, GIVEN THAT it is in fact blue at 75 percent, but we are interested in the "inverse probability" that the cab is REALLY blue, GIVEN that the witness SAYS it is blue. P (Says B B) = 0.75 P (Says B B) = P (B and SAYS B) / P (B) from the definition of conditional probability P (B Says B) = P (B and SAYS B) / P (Says B) has the same numerator, but a different denominator. It is a simple application of the multiplication rule for dependent events to compute the numerator: P (B and Says B) = P (B) * P (Says B B) = 0.20 * (.75) = 0.15 Make a table and fill in the other possibilites: P (B and Says Y) = P (B) * P (Says Y B) = 0.20 * (.25) = 0.05 P (Y and Says B) = P (Y) * P (Says B Y) = 0.80 * (.25) = 0.20 P (Y and Says Y) = P (Y) * P (Says Y Y) = 0.80 * (.75) = 0.60 Says B Says Y Row totals IS B IS Y Col. totals [Grand total = 1.00 or 100 percent] We know the witness said B, so we know that the first column is the only one that counts. Note that despite his 75 percent accuracy, the fact that there are more yellow cabs means that our witness makes more mistakes than correct identifications. P (B Says B) = 15 / 35 = 3/7 = = percent
Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314
Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space
More informationStatistics 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
More informationDefinition and Calculus of Probability
In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the
More informationSection 6.2 Definition of Probability
Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will
More informationAP 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
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 informationBasic 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
More informationSection 6-5 Sample Spaces and Probability
492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More informationBayesian Tutorial (Sheet Updated 20 March)
Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that
More informationMath 3C Homework 3 Solutions
Math 3C Homework 3 s Ilhwan Jo and Akemi Kashiwada ilhwanjo@math.ucla.edu, akashiwada@ucla.edu Assignment: Section 2.3 Problems 2, 7, 8, 9,, 3, 5, 8, 2, 22, 29, 3, 32 2. You draw three cards from a standard
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
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 information1 Combinations, Permutations, and Elementary Probability
1 Combinations, Permutations, and Elementary Probability Roughly speaking, Permutations are ways of grouping things where the order is important. Combinations are ways of grouping things where the order
More information2.5 Conditional Probabilities and 2-Way Tables
2.5 Conditional Probabilities and 2-Way Tables Learning Objectives Understand how to calculate conditional probabilities Understand how to calculate probabilities using a contingency or 2-way table It
More informationThe study of probability has increased in popularity over the years because of its wide range of practical applications.
6.7. Probability. The study of probability has increased in popularity over the years because of its wide range of practical applications. In probability, each repetition of an experiment is called a trial,
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 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 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 informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6-sided dice. What s the probability of rolling at least one 6? There is a 1
More informationPROBABILITY. SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and 1.
PROBABILITY SIMPLE PROBABILITY SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and. There are two categories of simple probabilities. THEORETICAL
More information(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball.
Examples for Chapter 3 Probability Math 1040-1 Section 3.1 1. Draw a tree diagram for each of the following situations. State the size of the sample space. (a) You flip a coin three times. (b) You draw
More informationSTAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia
STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that
More informationMath 141. Lecture 2: More Probability! Albyn Jones 1. jones@reed.edu www.people.reed.edu/ jones/courses/141. 1 Library 304. Albyn Jones Math 141
Math 141 Lecture 2: More Probability! Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Outline Law of total probability Bayes Theorem the Multiplication Rule, again Recall
More informationCurriculum Design for Mathematic Lesson Probability
Curriculum Design for Mathematic Lesson Probability This curriculum design is for the 8th grade students who are going to learn Probability and trying to show the easiest way for them to go into this class.
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 informationLecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University
Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments
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 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 informationDiscrete 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,
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 information8.3 Probability Applications of Counting Principles
8. Probability Applications of Counting Principles In this section, we will see how we can apply the counting principles from the previous two sections in solving probability problems. Many of the probability
More informationExam. 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,
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 information2. 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
More informationChapter 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
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 informationMath Games For Skills and Concepts
Math Games p.1 Math Games For Skills and Concepts Original material 2001-2006, John Golden, GVSU permission granted for educational use Other material copyright: Investigations in Number, Data and Space,
More informationProbability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to
More informationIt is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important
PROBABILLITY 271 PROBABILITY CHAPTER 15 It is remarkable that a science, which began with the consideration of games of chance, should be elevated to the rank of the most important subject of human knowledge.
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 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 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 informationCombinatorial Proofs
Combinatorial Proofs Two Counting Principles Some proofs concerning finite sets involve counting the number of elements of the sets, so we will look at the basics of counting. Addition Principle: If A
More informationProbability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1 www.math12.com
Probability --QUESTIONS-- Principles of Math - Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..
More informationFundamentals of Probability
Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible
More informationChapter 4: Probability and Counting Rules
Chapter 4: Probability and Counting Rules Learning Objectives Upon successful completion of Chapter 4, you will be able to: Determine sample spaces and find the probability of an event using classical
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 informationProbability & Probability Distributions
Probability & Probability Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Probability & Probability Distributions p. 1/61 Probability & Probability Distributions Elementary Probability Theory Definitions
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 informationHoover High School Math League. Counting and Probability
Hoover High School Math League Counting and Probability Problems. At a sandwich shop there are 2 kinds of bread, 5 kinds of cold cuts, 3 kinds of cheese, and 2 kinds of dressing. How many different sandwiches
More informationSTAT 35A HW2 Solutions
STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },
More informationProbability. 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
More informationA Few Basics of Probability
A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study
More informationReady, Set, Go! Math Games for Serious Minds
Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 -
More informationVideo Poker in South Carolina: A Mathematical Study
Video Poker in South Carolina: A Mathematical Study by Joel V. Brawley and Todd D. Mateer Since its debut in South Carolina in 1986, video poker has become a game of great popularity as well as a game
More informationIntroductory 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
More informationThat s Not Fair! ASSESSMENT #HSMA20. Benchmark Grades: 9-12
That s Not Fair! ASSESSMENT # Benchmark Grades: 9-12 Summary: Students consider the difference between fair and unfair games, using probability to analyze games. The probability will be used to find ways
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationProbabilities. 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
More informationProbability. 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
More informationProbability and Venn diagrams UNCORRECTED PAGE PROOFS
Probability and Venn diagrams 12 This chapter deals with further ideas in chance. At the end of this chapter you should be able to: identify complementary events and use the sum of probabilities to solve
More informationThe Calculus of Probability
The Calculus of Probability Let A and B be events in a sample space S. Partition rule: P(A) = P(A B) + P(A B ) Example: Roll a pair of fair dice P(Total of 10) = P(Total of 10 and double) + P(Total of
More informationHaving a coin come up heads or tails is a variable on a nominal scale. Heads is a different category from tails.
Chi-square Goodness of Fit Test The chi-square test is designed to test differences whether one frequency is different from another frequency. The chi-square test is designed for use with data on a nominal
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 informationMATHEMATICS 154, SPRING 2010 PROBABILITY THEORY Outline #3 (Combinatorics, bridge, poker)
Last modified: February, 00 References: MATHEMATICS 5, SPRING 00 PROBABILITY THEORY Outline # (Combinatorics, bridge, poker) PRP(Probability and Random Processes, by Grimmett and Stirzaker), Section.7.
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 informationActivity- The Energy Choices Game
Activity- The Energy Choices Game Purpose Energy is a critical resource that is used in all aspects of our daily lives. The world s supply of nonrenewable resources is limited and our continued use of
More informationChapter 4 - Practice Problems 2
Chapter - Practice Problems 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. 1) If you flip a coin three times, the
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 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 informationConditional Probability, Hypothesis Testing, and the Monty Hall Problem
Conditional Probability, Hypothesis Testing, and the Monty Hall Problem Ernie Croot September 17, 2008 On more than one occasion I have heard the comment Probability does not exist in the real world, and
More informationHomework 8 Solutions
CSE 21 - Winter 2014 Homework Homework 8 Solutions 1 Of 330 male and 270 female employees at the Flagstaff Mall, 210 of the men and 180 of the women are on flex-time (flexible working hours). Given that
More informationConditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of conditional probability and independence
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 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 informationFactory example D MA ND 0 MB D ND
Bayes Theorem Now we look at how we can use information about conditional probabilities to calculate reverse conditional probabilities i.e., how we calculate ( A B when we know ( B A (and some other things.
More informationStandard 12: The student will explain and evaluate the financial impact and consequences of gambling.
TEACHER GUIDE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Priority Academic Student Skills Personal Financial
More informationProbability and Compound Events Examples
Probability and Compound Events Examples 1. A compound event consists of two or more simple events. ossing a die is a simple event. ossing two dice is a compound event. he probability of a compound event
More informationChapter 13 & 14 - Probability PART
Chapter 13 & 14 - Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14 - Probability 1 / 91 Why Should We Learn Probability Theory? Dr. Joseph
More informationHooray for the Hundreds Chart!!
Hooray for the Hundreds Chart!! The hundreds chart consists of a grid of numbers from 1 to 100, with each row containing a group of 10 numbers. As a result, children using this chart can count across rows
More informationSTT 200 LECTURE 1, SECTION 2,4 RECITATION 7 (10/16/2012)
STT 200 LECTURE 1, SECTION 2,4 RECITATION 7 (10/16/2012) TA: Zhen (Alan) Zhang zhangz19@stt.msu.edu Office hour: (C500 WH) 1:45 2:45PM Tuesday (office tel.: 432-3342) Help-room: (A102 WH) 11:20AM-12:30PM,
More informationStatistics and Data Analysis B01.1305
Statistics and Data Analysis B01.1305 Professor William Greene Phone: 212.998.0876 Office: KMC 7-78 Home page: www.stern.nyu.edu/~wgreene Email: wgreene@stern.nyu.edu Course web page: www.stern.nyu.edu/~wgreene/statistics/outline.htm
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 informationChapter 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
More informationHow To Solve The Social Studies Test
Math 00 Homework #0 Solutions. Section.: ab. For each map below, determine the number of southerly paths from point to point. Solution: We just have to use the same process as we did in building Pascal
More informationWHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
More informationREPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.
REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game
More informationProbability and Expected Value
Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are
More 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 informationChapter 16: law of averages
Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................
More informationSolutions: Problems for Chapter 3. Solutions: Problems for Chapter 3
Problem A: You are dealt five cards from a standard deck. Are you more likely to be dealt two pairs or three of a kind? experiment: choose 5 cards at random from a standard deck Ω = {5-combinations of
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 informationSection 7C: The Law of Large Numbers
Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half
More informationCrude: The Oil Game 1
Crude: The Oil Game 1 Contents Game Components... 3 Introduction And Object Of The Game... 4 Setting Up The Game... 4 Pick A Starting Player... 6 The Special Purchase Round... 6 Sequence Of Play... 7 The
More informationQuestion of the Day. Key Concepts. Vocabulary. Mathematical Ideas. QuestionofDay
QuestionofDay Question of the Day What is the probability that in a family with two children, both are boys? What is the probability that in a family with two children, both are boys, if we already know
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 informationMathematical goals. Starting points. Materials required. Time needed
Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about
More informationReview for Test 2. Chapters 4, 5 and 6
Review for Test 2 Chapters 4, 5 and 6 1. You roll a fair six-sided die. Find the probability of each event: a. Event A: rolling a 3 1/6 b. Event B: rolling a 7 0 c. Event C: rolling a number less than
More information2. Three dice are tossed. Find the probability of a) a sum of 4; or b) a sum greater than 4 (may use complement)
Probability Homework Section P4 1. A two-person committee is chosen at random from a group of four men and three women. Find the probability that the committee contains at least one man. 2. Three dice
More information