Introduction to Probability and Statistics I Lecture 7 and 8


 Augusta Lamb
 3 years ago
 Views:
Transcription
1 Introduction to Probability and Statistics I Lecture 7 and 8 Basic Probability and Counting Methods
2 Computing theoretical probabilities:counting methods Great for gambling! Fun to compute! If outcomes are equally likely to occur P( A) # of ways A can occur total# of outcomes Note: these are called counting methods because we have to count the number of ways A can occur and the number of total possible outcomes
3 Summary of Counting Methods Counting methods for computing probabilities Permutations order matters! Combinations Order doesn t matter With replacement Without replacement Without replacement
4 Permutations Order matters! A permutation is an ordered arrangement of objects With replacement=once an event occurs, it can occur again (after you roll a 6, you can roll a 6 again on the same die) Without replacement=an event cannot repeat (after you draw an ace of spades out of a deck, there is 0 probability of getting it again)
5 Combinations Order doesn t matter Introduction to combination function, or choosing Written as: n C r or n r Spoken: n choose r
6 Combinations How many twocard hands can I draw from a deck when order does not matter (eg, ace of spades followed by ten of clubs is the same as ten of clubs followed by ace of 52 cards 51 cards spades) 52x ! (52 2)!2
7 Summary of Counting Methods Counting methods for computing probabilities Permutations order matters! Combinations Order doesn t matter With replacement: n r Without replacement: n(n1)(n2) (nr+1)= n! ( n r)! Without replacement: n r n! ( n r)! r!
8 Important Terms Random Experiment a process leading to an uncertain outcome Basic Outcome a possible outcome of a random experiment Sample Space the collection of all possible outcomes of a random experiment Event any subset of basic outcomes from the sample space
9 32 Probability Probability the chance that an uncertain event will occur (always between 0 and 1) 1 Certain 0 P(A) 1 For any event A 5 0 Impossible
10 Assessing Probability There are three approaches to assessing the probability of an uncertain event: 1 classical probability N probability of event A A N numberof outcomesthat satisfythe event total numberof outcomesinthe samplespace Assumes all outcomes in the sample space are equally likely to occur
11 Counting the Possible Outcomes Use the Combinations formula to determine the number of combinations of n things taken k at a time C n k k!(n n! k)! where n! = n(n1)(n2) (1) 0! = 1 by definition
12 Assessing Probability Three approaches (continued) 2 relative frequency probability n probability of event A A n numberof events inthe populationthat satisfyevent total numberof events inthe population A the limit of the proportion of times that an event A occurs in a large number of trials, n 3 subjective probability an individual opinion or belief about the probability of occurrence
13 Probability Postulates 1 If A is any event in the sample space S, then 0 P(A) 2 Let A be an event in S, and let O i denote the basic outcomes Then (the notation means that the summation is over all the basic outcomes in A) 1 P(A) P(O i) A 3 P(S) = 1
14 33 Probability Rules The Complement rule: P( A) 1P(A) ie, P(A) P(A) 1 The Addition rule: The probability of the union of two events is P(A B) P(A) P(B) P(A B)
15 A Probability Table Probabilities and joint probabilities for two events A and B are summarized in this table: B B A P(A B) P(A B) P(A) A P( A B) P( A B) P(A) P(B) P(B) P(S) 10
16 Addition Rule Example Consider a standard deck of 52 cards, with four suits: Let event A = card is an Ace Let event B = card is from a red suit
17 Addition Rule Example (continued) P(Red U Ace) = P(Red) + P(Ace)  P(Red Ace) = 26/52 + 4/522/52 = 28/52 Color Type Red Black Total Ace NonAce Total Don t count the two red aces twice!
18 Conditional Probability A conditional probability is the probability of one event, given that another event has occurred: P(A B) P(A B) P(B) The conditional probability of A given that B has occurred P(B A) P(A B) P(A) The conditional probability of B given that A has occurred
19 Conditional Probability Example Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD) 20% of the cars have both What is the probability that a car has a CD player, given that it has AC? ie, we want to find P(CD AC)
20 Conditional Probability Example Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD) 20% of the cars have both CD No CD Total AC No AC Total (continued) P(CD AC) P(CD AC) P(AC)
21 Conditional Probability Example (continued) Given AC, we only consider the top row (70% of the cars) Of these, 20% have a CD player 20% of 70% is 2857% CD No CD Total AC No AC Total P(CD AC) P(CD AC) P(AC)
22 Multiplication Rule Multiplication rule for two events A and B: P(A B) P(A B)P(B) also P(A B) P(B A) P(A)
23 Multiplication Rule Example P(Red Ace) = P(Red Ace)P(Ace) number of cards that are redandace total number of cards Color Type Red Black Total Ace NonAce Total
24 Statistical Independence Two events are statistically independent if and only if: P(A B) P(A)P(B) Events A and B are independent when the probability of one event is not affected by the other event If A and B are independent, then P(A B) P(A) if P(B)>0 P(B A) P(B) if P(A)>0
25 Statistical Independence Example Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD) 20% of the cars have both CD No CD Total AC No AC Total Are the events AC and CD statistically independent?
26 Statistical Independence Example P(AC CD) = 02 CD No CD Total AC No AC Total (continued) P(AC) = 07 P(CD) = 04 P(AC)P(CD) = (07)(04) = 028 P(AC CD) = 02 P(AC)P(CD) = 028 So the two events are not statistically independent
27 34 Bivariate Probabilities Outcomes for bivariate events: B 1 B 2 B k A 1 P(A 1 B 1 ) P(A 1 B 2 ) P(A 1 B k ) A 2 P(A 2 B 1 ) P(A 2 B 2 ) P(A 2 B k ) A h P(A h B 1 ) P(A h B 2 ) P(A h B k )
28 Joint and Marginal Probabilities The probability of a joint event, A B: P(A B) number of outcomessatisfyinga andb total number of elementaryoutcomes Computing a marginal probability: P(A) P(A B 1) P(A B2) P(A B k ) Where B 1, B 2,, B k are k mutually exclusive and collectively exhaustive events
29 Marginal Probability Example P(Ace) P(Ace Red) P(Ace Black) Color Type Red Black Total Ace NonAce Total
30 Using a Tree Diagram Given AC or no AC: All Cars P(AC CD) = 2 P(AC CD) = 5 P(AC CD) = P(AC CD) = 1
31 Odds The odds in favor of a particular event are given by the ratio of the probability of the event divided by the probability of its complement The odds in favor of A are odds P(A) 1P(A) P(A) P(A)
32 Odds: Example Calculate the probability of winning if the odds of winning are 3 to 1: odds 3 1 P(A) 1P(A) Now multiply both sides by 1 P(A) and solve for P(A): 3 x (1 P(A)) = P(A) 3 3P(A) = P(A) 3 = 4P(A) P(A) = 075
33 Overinvolvement Ratio The probability of event A 1 conditional on event B 1 divided by the probability of A 1 conditional on activity B 2 is defined as the overinvolvement ratio: P(A P(A 1 1 B B An overinvolvement ratio greater than 1 implies that event A 1 increases the conditional odds ration in favor of B 1 : P(B1 A1) P(B1) P(B A ) P(B ) ) ) 2
34 35 Bayes Theorem P(E i A) P(A E P(A E i P(A) 1 )P(E ) )P(E i 1 P(A E ) P(A E 2 i )P(E ) )P(E 2 i ) P(A E k )P(E k ) where: E i = i th event of k mutually exclusive and collectively exhaustive events A = new event that might impact P(E i )
35 Bayes Theorem Example A drilling company has estimated a 40% chance of striking oil for their new well A detailed test has been scheduled for more information Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have had detailed tests Given that this well has been scheduled for a detailed test, what is the probability that the well will be successful?
36 Bayes Theorem Example (continued) Let S = successful well U = unsuccessful well P(S) = 4, P(U) = 6 (prior probabilities) Define the detailed test event as D Conditional probabilities: P(D S) = 6 P(D U) = 2 Goal is to find P(S D)
37 Bayes Theorem Example Apply Bayes Theorem: (continued) P(S D) P(D P(D S)P(S) S)P(S) P(D U)P(U) (6)(4) (6)(4) (2)(6) So the revised probability of success (from the original estimate of 4), given that this well has been scheduled for a detailed test, is 667
Business Statistics. Chapter 4 Using Probability and Probability Distributions QMIS 120. Dr. Mohammad Zainal
Department of Quantitative Methods & Information Systems Business Statistics Chapter 4 Using Probability and Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter,
More informationChapter 4 Student Lecture Notes 41
Chapter 4 Student Lecture Notes 41 Basic Business Statistics (9 th Edition) Chapter 4 Basic Probability 2004 PrenticeHall, Inc. Chap 41 Chapter Topics Basic Probability Concepts Sample spaces and events,
More informationTextbook: pp Chapter 2: Probability Concepts and Applications
1 Textbook: pp. 3980 Chapter 2: Probability Concepts and Applications 2 Learning Objectives After completing this chapter, students will be able to: Understand the basic foundations of probability analysis.
More informationContents 2.1 Basic Concepts of Probability Methods of Assigning Probabilities Principle of Counting  Permutation and Combination 39
CHAPTER 2 PROBABILITY Contents 2.1 Basic Concepts of Probability 38 2.2 Probability of an Event 39 2.3 Methods of Assigning Probabilities 39 2.4 Principle of Counting  Permutation and Combination 39 2.5
More informationProbability. Ms. Weinstein Probability & Statistics
Probability Ms. Weinstein Probability & Statistics Definitions Sample Space The sample space, S, of a random phenomenon is the set of all possible outcomes. Event An event is a set of outcomes of a random
More informationChapter 4: Probability and Counting Rules
Chapter 4: Probability and Counting Rules Before we can move from descriptive statistics to inferential statistics, we need to have some understanding of probability: Ch4: Probability and Counting Rules
More informationApplied Statistics I
Applied Statistics I Liang Zhang Department of Mathematics, University of Utah June 12, 2008 Liang Zhang (UofU) Applied Statistics I June 12, 2008 1 / 29 In Probability, our main focus is to determine
More informationStatistics Intermediate Probability
Session 6 oscardavid.barrerarodriguez@sciencespo.fr April 3, 2018 and Sampling from a Population Outline 1 The Monty Hall Paradox Some Concepts: Event Algebra Axioms and Things About that are True Counting
More informationProbability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College
Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical
More informationCHAPTER 9  COUNTING PRINCIPLES AND PROBABILITY
CHAPTER 9  COUNTING PRINCIPLES AND PROBABILITY Probability is the Probability is used in many realworld fields, such as insurance, medical research, law enforcement, and political science. Objectives:
More informationChapter 3: Elements of Chance: Probability Methods
Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 34 20142015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,
More informationCHAPTERS 14 & 15 PROBABILITY STAT 203
CHAPTERS 14 & 15 PROBABILITY STAT 203 Where this fits in 2 Up to now, we ve mostly discussed how to handle data (descriptive statistics) and how to collect data. Regression has been the only form of statistical
More informationChapter 5  Elementary Probability Theory
Chapter 5  Elementary Probability Theory Historical Background Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling
More informationNovember 6, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern
More informationBasic 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 informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More information8.2 Union, Intersection, and Complement of Events; Odds
8.2 Union, Intersection, and Complement of Events; Odds Since we defined an event as a subset of a sample space it is natural to consider set operations like union, intersection or complement in the context
More informationDef: The intersection of A and B is the set of all elements common to both set A and set B
Def: Sample Space the set of all possible outcomes Def: Element an item in the set Ex: The number "3" is an element of the "rolling a die" sample space Main concept write in Interactive Notebook Intersection:
More informationChapter 6: Probability and Simulation. The study of randomness
Chapter 6: Probability and Simulation The study of randomness 6.1 Randomness Probability describes the pattern of chance outcomes. Probability is the basis of inference Meaning, the pattern of chance outcomes
More informationCHAPTER 2 PROBABILITY. 2.1 Sample Space. 2.2 Events
CHAPTER 2 PROBABILITY 2.1 Sample Space A probability model consists of the sample space and the way to assign probabilities. Sample space & sample point The sample space S, is the set of all possible outcomes
More informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More informationWeek 3 Classical Probability, Part I
Week 3 Classical Probability, Part I Week 3 Objectives Proper understanding of common statistical practices such as confidence intervals and hypothesis testing requires some familiarity with probability
More informationClassical Definition of Probability Relative Frequency Definition of Probability Some properties of Probability
PROBABILITY Recall that in a random experiment, the occurrence of an outcome has a chance factor and cannot be predicted with certainty. Since an event is a collection of outcomes, its occurrence cannot
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 4 Probability Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education School of Continuing
More informationSTOR 155 Introductory Statistics. Lecture 10: Randomness and Probability Model
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 10: Randomness and Probability Model 10/6/09 Lecture 10 1 The Monty Hall Problem Let s Make A Deal: a game show
More informationProbability is the likelihood that an event will occur.
Section 3.1 Basic Concepts of is the likelihood that an event will occur. In Chapters 3 and 4, we will discuss basic concepts of probability and find the probability of a given event occurring. Our main
More informationSuch a description is the basis for a probability model. Here is the basic vocabulary we use.
5.2.1 Probability Models When we toss a coin, we can t know the outcome in advance. What do we know? We are willing to say that the outcome will be either heads or tails. We believe that each of these
More informationSection Introduction to Sets
Section 1.1  Introduction to Sets Definition: A set is a welldefined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase
More informationProbability (Devore Chapter Two)
Probability (Devore Chapter Two) 101635101 Probability Winter 20112012 Contents 1 Axiomatic Probability 2 1.1 Outcomes and Events............................... 2 1.2 Rules of Probability................................
More informationAxiomatic Probability
Axiomatic Probability The objective of probability is to assign to each event A a number P(A), called the probability of the event A, which will give a precise measure of the chance thtat A will occur.
More informationProbability Models. Section 6.2
Probability Models Section 6.2 The Language of Probability What is random? Empirical means that it is based on observation rather than theorizing. Probability describes what happens in MANY trials. Example
More informationChapter 5: Probability: What are the Chances? Section 5.2 Probability Rules
+ Chapter 5: Probability: What are the Chances? Section 5.2 + TwoWay Tables and Probability When finding probabilities involving two events, a twoway table can display the sample space in a way that
More informationSTAT 155 Introductory Statistics. Lecture 11: Randomness and Probability Model
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 11: Randomness and Probability Model 10/5/06 Lecture 11 1 The Monty Hall Problem Let s Make A Deal: a game show
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even
More informationEmpirical (or statistical) probability) is based on. The empirical probability of an event E is the frequency of event E.
Probability and Statistics Chapter 3 Notes Section 31 I. Probability Experiments. A. When weather forecasters say There is a 90% chance of rain tomorrow, or a doctor says There is a 35% chance of a successful
More informationTheory of Probability  Brett Bernstein
Theory of Probability  Brett Bernstein Lecture 3 Finishing Basic Probability Review Exercises 1. Model flipping two fair coins using a sample space and a probability measure. Compute the probability of
More informationLecture 4: Chapter 4
Lecture 4: Chapter 4 C C Moxley UAB Mathematics 19 September 16 4.2 Basic Concepts of Probability Procedure Event Simple Event Sample Space 4.2 Basic Concepts of Probability Procedure Event Simple Event
More informationProbability. Engr. Jeffrey T. Dellosa.
Probability Engr. Jeffrey T. Dellosa Email: jtdellosa@gmail.com Outline Probability 2.1 Sample Space 2.2 Events 2.3 Counting Sample Points 2.4 Probability of an Event 2.5 Additive Rules 2.6 Conditional
More informationSection 6.5 Conditional Probability
Section 6.5 Conditional Probability Example 1: An urn contains 5 green marbles and 7 black marbles. Two marbles are drawn in succession and without replacement from the urn. a) What is the probability
More informationProbability  Chapter 4
Probability  Chapter 4 In this chapter, you will learn about probability its meaning, how it is computed, and how to evaluate it in terms of the likelihood of an event actually happening. A cynical person
More informationGrade 7/8 Math Circles February 25/26, Probability
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Probability Grade 7/8 Math Circles February 25/26, 2014 Probability Centre for Education in Mathematics and Computing Probability is the study of how likely
More informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 4 Probability and Counting Rules 2 Objectives Determine sample spaces and find the probability of an event using classical probability or empirical
More informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Week 6 Lecture Notes Discrete Probability Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. Introduction and
More informationLecture 4: Chapter 4
Lecture 4: Chapter 4 C C Moxley UAB Mathematics 17 September 15 4.2 Basic Concepts of Probability Procedure Event Simple Event Sample Space 4.2 Basic Concepts of Probability Procedure Event Simple Event
More informationAPPENDIX 2.3: RULES OF PROBABILITY
The frequentist notion of probability is quite simple and intuitive. Here, we ll describe some rules that govern how probabilities are combined. Not all of these rules will be relevant to the rest of this
More information05 Adding Probabilities. 1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins.
1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins. d. a. Copy the table and add a column to show the experimental probability of the spinner landing on
More informationIntermediate Math Circles November 1, 2017 Probability I
Intermediate Math Circles November 1, 2017 Probability I Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application.
More informationIndependent and Mutually Exclusive Events
Independent and Mutually Exclusive Events By: OpenStaxCollege Independent and mutually exclusive do not mean the same thing. Independent Events Two events are independent if the following are true: P(A
More information1MA01: Probability. Sinéad Ryan. November 12, 2013 TCD
1MA01: Probability Sinéad Ryan TCD November 12, 2013 Definitions and Notation EVENT: a set possible outcomes of an experiment. Eg flipping a coin is the experiment, landing on heads is the event If an
More informationThe 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 informationProbability and Statistics. Copyright Cengage Learning. All rights reserved.
Probability and Statistics Copyright Cengage Learning. All rights reserved. 14.2 Probability Copyright Cengage Learning. All rights reserved. Objectives What Is Probability? Calculating Probability by
More informationRaise your hand if you rode a bus within the past month. Record the number of raised hands.
166 CHAPTER 3 PROBABILITY TOPICS Raise your hand if you rode a bus within the past month. Record the number of raised hands. Raise your hand if you answered "yes" to BOTH of the first two questions. Record
More informationFind the probability of an event by using the definition of probability
LESSON 101 Probability Lesson Objectives Find the probability of an event by using the definition of probability Vocabulary experiment (p. 522) trial (p. 522) outcome (p. 522) sample space (p. 522) event
More informationTotal. STAT/MATH 394 A  Autumn Quarter Midterm. Name: Student ID Number: Directions. Complete all questions.
STAT/MATH 9 A  Autumn Quarter 015  Midterm Name: Student ID Number: Problem 1 5 Total Points Directions. Complete all questions. You may use a scientific calculator during this examination; graphing
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 informationProbability Concepts and Counting Rules
Probability Concepts and Counting Rules Chapter 4 McGrawHill/Irwin Dr. Ateq Ahmed AlGhamedi Department of Statistics P O Box 80203 King Abdulaziz University Jeddah 21589, Saudi Arabia ateq@kau.edu.sa
More informationCounting Methods and Probability
CHAPTER Counting Methods and Probability Many good basketball players can make 90% of their free throws. However, the likelihood of a player making several free throws in a row will be less than 90%. You
More informationLesson 4: Chapter 4 Sections 12
Lesson 4: Chapter 4 Sections 12 Caleb Moxley BSC Mathematics 14 September 15 4.1 Randomness What s randomness? 4.1 Randomness What s randomness? Definition (random) A phenomenon is random if individual
More informationCompound Probability. Set Theory. Basic Definitions
Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic
More informationMath 1313 Section 6.2 Definition of Probability
Math 1313 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
More informationProbability and Randomness. Day 1
Probability and Randomness Day 1 Randomness and Probability The mathematics of chance is called. The probability of any outcome of a chance process is a number between that describes the proportion of
More informationChapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
More information4.1 Sample Spaces and Events
4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an
More informationChapter 15 Probability Rules!
Chapter 15 Probability Rules! 151 What s It About? Chapter 14 introduced students to basic probability concepts. Chapter 15 generalizes and expands the Addition and Multiplication Rules. We discuss conditional
More informationUnit 1 Day 1: Sample Spaces and Subsets. Define: Sample Space. Define: Intersection of two sets (A B) Define: Union of two sets (A B)
Unit 1 Day 1: Sample Spaces and Subsets Students will be able to (SWBAT) describe events as subsets of sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions,
More informationChapter 6: Probability and Simulation. The study of randomness
Chapter 6: Probability and Simulation The study of randomness Introduction Probability is the study of chance. 6.1 focuses on simulation since actual observations are often not feasible. When we produce
More informationStatistics 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 information7.1 Experiments, Sample Spaces, and Events
7.1 Experiments, Sample Spaces, and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment
More informationAnswer each of the following problems. Make sure to show your work.
Answer each of the following problems. Make sure to show your work. 1. A board game requires each player to roll a die. The player with the highest number wins. If a player wants to calculate his or her
More information"Well, statistically speaking, you are for more likely to have an accident at an intersection, so I just make sure that I spend less time there.
6.2 Probability Models There was a statistician who, when driving his car, would always accelerate hard before coming to an intersection, whiz straight through it, and slow down again once he was beyond
More informationNovember 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 informationProbably About Probability p <.05. Probability. What Is Probability? Probability of Events. Greg C Elvers
Probably About p
More informationPROBABILITY. 1. Introduction. Candidates should able to:
PROBABILITY Candidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for the total score when two fair dice are thrown), or by calculation
More informationCSC/MATA67 Tutorial, Week 12
CSC/MATA67 Tutorial, Week 12 November 23, 2017 1 More counting problems A class consists of 15 students of whom 5 are prefects. Q: How many committees of 8 can be formed if each consists of a) exactly
More informationAnswer each of the following problems. Make sure to show your work.
Answer each of the following problems. Make sure to show your work. 1. A board game requires each player to roll a die. The player with the highest number wins. If a player wants to calculate his or her
More informationSTAT 430/510 Probability
STAT 430/510 Probability Hui Nie Lecture 1 May 26th, 2009 Introduction Probability is the study of randomness and uncertainty. In the early days, probability was associated with games of chance, such as
More informationObjective: Determine empirical probability based on specific sample data. (AA21)
Do Now: What is an experiment? List some experiments. What types of things does one take a "chance" on? Mar 1 3:33 PM Date: Probability  Empirical  By Experiment Objective: Determine empirical probability
More informationSTAT 3743: Probability and Statistics
STAT 3743: Probability and Statistics G. Jay Kerns, Youngstown State University Fall 2010 Probability Random experiment: outcome not known in advance Sample space: set of all possible outcomes (S) Probability
More informationChapter 3: Probability (Part 1)
Chapter 3: Probability (Part 1) 3.1: Basic Concepts of Probability and Counting Types of Probability There are at least three different types of probability Subjective Probability is found through people
More informationApplications of Probability
Applications of Probability CK12 Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive
More informationModule 4 Project Maths Development Team Draft (Version 2)
5 Week Modular Course in Statistics & Probability Strand 1 Module 4 Set Theory and Probability It is often said that the three basic rules of probability are: 1. Draw a picture 2. Draw a picture 3. Draw
More informationRANDOM EXPERIMENTS AND EVENTS
Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In daytoday life we see that before commencement of a cricket match two captains go for a toss. Tossing of a coin is an activity and getting
More informationMathematics 'A' level Module MS1: Statistics 1. Probability. The aims of this lesson are to enable you to. calculate and understand probability
Mathematics 'A' level Module MS1: Statistics 1 Lesson Three Aims The aims of this lesson are to enable you to calculate and understand probability apply the laws of probability in a variety of situations
More informationSample Spaces, Events, Probability
Sample Spaces, Events, Probability CS 3130/ECE 3530: Probability and Statistics for Engineers August 28, 2014 Sets A set is a collection of unique objects. Sets A set is a collection of unique objects.
More informationProbability 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 informationProbability Review 41
Probability Review 41 For the following problems, give the probability to four decimals, or give a fraction, or if necessary, use scientific notation. Use P(A) = 1  P(not A) 1) A coin is tossed 6 times.
More information4.3 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 informationGrade 6 Math Circles Fall Oct 14/15 Probability
1 Faculty of Mathematics Waterloo, Ontario Centre for Education in Mathematics and Computing Grade 6 Math Circles Fall 2014  Oct 14/15 Probability Probability is the likelihood of an event occurring.
More information(a) Suppose you flip a coin and roll a die. Are the events obtain a head and roll a 5 dependent or independent events?
Unit 6 Probability Name: Date: Hour: Multiplication Rule of Probability By the end of this lesson, you will be able to Understand Independence Use the Multiplication Rule for independent events Independent
More informationDefine and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)
12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the
More informationA Probability Work Sheet
A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair sixsided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we
More informationP(X is on ) Practice Test  Chapter 13. BASEBALL A baseball team fields 9 players. How many possible batting orders are there for the 9 players?
Point X is chosen at random on. Find the probability of each event. P(X is on ) P(X is on ) BASEBALL A baseball team fields 9 players. How many possible batting orders are there for the 9 players? or 362,880.
More informationChapter 12: Probability & Statistics. Notes #2: Simple Probability and Independent & Dependent Events and Compound Events
Chapter 12: Probability & Statistics Notes #2: Simple Probability and Independent & Dependent Events and Compound Events Theoretical & Experimental Probability 1 2 Probability: How likely an event is to
More informationCSE 312: Foundations of Computing II Quiz Section #2: InclusionExclusion, Pigeonhole, Introduction to Probability (solutions)
CSE 31: Foundations of Computing II Quiz Section #: InclusionExclusion, Pigeonhole, Introduction to Probability (solutions) Review: Main Theorems and Concepts Binomial Theorem: x, y R, n N: (x + y) n
More information, the of all of a probability experiment. consists of outcomes. (b) List the elements of the event consisting of a number that is greater than 4.
41 Sample Spaces and Probability as a general concept can be defined as the chance of an event occurring. In addition to being used in games of chance, probability is used in the fields of,, and forecasting,
More informationMathematical Foundations HW 5 By 11:59pm, 12 Dec, 2015
1 Probability Axioms Let A,B,C be three arbitrary events. Find the probability of exactly one of these events occuring. Sample space S: {ABC, AB, AC, BC, A, B, C, }, and S = 8. P(A or B or C) = 3 8. note:
More informationSection Summary. Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning
Section 7.1 Section Summary Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning Probability of an Event PierreSimon Laplace (17491827) We first study PierreSimon
More informationMathacle. Name: Date:
Quiz Probability 1.) A telemarketer knows from past experience that when she makes a call, the probability that someone will answer the phone is 0.20. What is probability that the next two phone calls
More informationFundamental. If one event can occur m ways and another event can occur n ways, then the number of ways both events can occur is:.
12.1 The Fundamental Counting Principle and Permutations Objectives 1. Use the fundamental counting principle to count the number of ways an event can happen. 2. Use the permutations to count the number
More informationProbability and Counting Techniques
Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each
More information