Chapter 5  Elementary Probability Theory


 Sabina Howard
 3 years ago
 Views:
Transcription
1 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 was Pierre Simon de Laplace, who is often credited with being the father of probability theory. In the twentieth century a coherent mathematical theory of probability was developed through people such as Chebyshev, Markov, and Kolmogorov. Probability The study of probability is concerned with random phenomena. Even though we cannot be certain whether a given result will occur, we often can obtain a good measure of its likelihood, or probability. In the study of probability, any observation, or measurement, of a random phenomenon is an experiment. The possible results of the experiment are called outcomes, and the set of all possible outcomes is called the sample space. Usually we are interested in some particular collection of the possible outcomes. Any such subset of the sample space is called an event. Example: Tossing a Coin If a single fair coin is tossed, find the probability that it will land heads up. Definition: A phenomenon is Random if individual outcomes are uncertain cannot be determined beforehand but there is a regular distribution of outcomes in a large number of repetitions of the experiment. Probability is the relative frequency of an outcome over a long series of repetitions of the event. Probability is a numerical measure between 0 and 1 that describes the likelihood that an event will occur 1 P a g e
2 Probability Probability is a numerical measure that indicates the likelihood of an event. All probabilities are between 0 and 1, inclusive. A probability of 0 means the event is impossible. A probability of 1 means the event is certain to occur. Events with probabilities near 1 are likely to occur. Events can be named with capital letters: A, B, C P(A) means the probability of A occurring. P(A) is read P of A 0 P(A) 1 P(A) = 1, the event is certain to occur P(A) = 0, the event is certain not to occur Theoretical Probability Formula If all outcomes in a sample space S are equally likely, and E is an event within that sample space, then the theoretical probability of the event E is given by number of favorable outcomes ne ( ) PE ( ). total number of outcomes ns ( ) Example  Among a sample of 50 dog owners, 23 feed their dogs Mighty Mutt dry dog food. Estimate the probability that a dog owner selected at random feeds their dogs Mighty Mutt dry food. a). 23/50 b). 27/50 c). 1/23 d). 23/27 Example: Flipping a Cup A cup is flipped 100 times. It lands on its side 84 times, on its bottom 6 times, and on its top 10 times. Find the probability that it will land on its top. 2 P a g e
3 Empirical Probability Formula If E is an event that may happen when an experiment is performed, then the empirical probability of event E is given by number of times event E occurred PE ( ). number of times the experiment was performed Example: Card Hands There are 2,598,960 possible hands in poker. If there are 36 possible ways to have a straight flush, find the probability of being dealt a straight flush. Example: Gender of a Student A school has 820 male students and 835 female students. If a student from the school is selected at random, what is the probability that the student would be a female? Law of Large Numbers As an experiment is repeated more and more times, the proportion of outcomes favorable to any particular event will tend to come closer and closer to the theoretical probability of that event. Example: Toss a coin repeatedly. The relative frequency gets closer and closer to P(head) = 0.50 Relative Frequency n = number of heads f = number of flips P a g e
4 Probability in Genetics Gregor Mendel, an Austrian monk used the idea of randomness to establish the study of genetics. To study the flower color of certain pea plants he found that: Pure red crossed with pure white produces red. Mendel theorized that red is dominant (symbolized by R), while white is recessive (symbolized by r). The pure red parent carried only genes for red (R), and the pure white parent carried only genes for white (r). Every offspring receives one gene from each parent which leads to the tables below. Every second generation is red because R is dominant. Example: Probability of Flower Color Referring to the 2 nd to 3 rd generation table (previous slide), determine the probability that a third generation will be a) red b) white Base the probability on the sample space of equally likely outcomes: S = {RR, Rr, rr, rr}. 4 P a g e
5 Intersection of Sets The intersection of sets A and B, written A B, is the set of elements common in both A and B, or Union of Sets The union of sets A and B, written A B, is the set of elements belonging to wither of the sets, or Disjoint Sets Two sets have no elements in common, sets A and B are disjoint if A B = ø. Meaning of A and B  Intersection of Sets AND The Venn diagram to the left shows two sets A and B the intersection of the two sets is the gray portion,, U is the universe Meaning of A or B Union of Sets OR The Venn diagram to the left shows two sets A and B the union of the two sets is the gray portion,, U is the universe 5 P a g e
6 Events Involving Not and Or Properties of Probability Let E be an event from the sample space S. That is, E is a subset of S. Then the following properties hold PE ( ) 1 (The probability of an event is between 0 and 1, inclusive.) 2. P( ) 0 (The probability of an impossible event is 0.) 3. PS ( ) 1 (The probability of a certain event is 1.) Example: Rolling a Die When a single fair die is rolled, find the probability of each event. a) the number 3 is rolled b) a number other than 3 is rolled c) the number 7 is rolled d) a number less than 7 is rolled Events Involving Not The table on the next slide shows the correspondences that are the basis for the probability rules developed in this section. For example, the probability of an event not happening involves the complement and subtraction. 6 P a g e
7 Correspondences Set Theory Logic Arithmetic Operation or Connective (Symbol) Operation or Connective (Symbol) Operation or Connective (Symbol) Complement Not Subtraction Union Or Addition Intersection And Multiplication Probability of a Complement The probability that an event E will not occur is equal to one minus the probability that it will occur. P(not E) P( S) P( E) 1 PE ( ) So we have and P( E) P E 1 P( E) 1 P( E ). Example: Complement When a single card is drawn from a standard 52card deck, what is the probability that is will not be an ace? 7 P a g e
8 Events Involving Or Probability of one event or another should involve the union and addition. Mutually Exclusive Events Two events A and B are mutually exclusive events if they have no outcomes in common. (Mutually exclusive events cannot occur simultaneously.) Addition Rule of Probability (for A or B) If A and B are any two events, then P( A or B) P( A) P( B) P( A and B). If A and B are mutually exclusive, then P( A or B) P( A) P( B). Example: Probability Involving Or When a single card is drawn from a standard 52card deck, what is the probability that it will be a king or a diamond? Example: Probability Involving Or If a single die is rolled, what is the probability of a 2 or odd? 8 P a g e
9 Probability versus Statistics Probability is the field of study that makes statements about what will occur when a sample is drawn from a known population. Statistics is the field of study that describes how samples are to be obtained and how inferences are to be made about unknown populations. Conditional Probability; Events Involving And Conditional Probability Sometimes the probability of an event must be computed using the knowledge that some other event has happened (or is happening, or will happen the timing is not important). This type of probability is called conditional probability. The probability of event B, computed on the assumption that event A has happened, is called the conditional probability of B, given A, and is denoted P(B A). Example: Selecting From a Set of Numbers From the sample space S = {2, 3, 4, 5, 6, 7, 8, 9}, a single number is to be selected randomly. Given the events A: selected number is odd, and B selected number is a multiple of 3. find each probability. a) P(B) b) P(A and B) c) P(B A) 9 P a g e
10 Conditional Probability Formula The conditional probability of B, given A, and is given by P( A B) P( A and B) P( B A). P( A) P( A) Example: Probability in a Family Given a family with two children, find the probability that both are boys, given that at least one is a boy. Mutually Exclusive Events Two events are mutually exclusive if they cannot occur at the same time. Mutually Exclusive = Disjoint If A and B are mutually exclusive, then P(A and B) = 0 Addition Rules If A and B are mutually exclusive, then P(A or B) = P(A) + P(B). If A and B are not mutually exclusive, then P(A or B) = P(A) + P(B) P(A and B). 10 P a g e
11 Independent Events Two events A and B are called independent events if knowledge about the occurrence of one of them has no effect on the probability of the other one, that is, if P(B A) = P(B), or equivalently P(A B) = P(A). P(A B) denotes the probability that event A will occur given that event B has occurred. This is called conditional probability. Read Probability of A given B. Two event s A and B are called dependent events if the occurrence of event B has changed the probability that event A will occur, that is Example: Checking for Independence A single card is to be drawn from a standard 52card deck. Given the events A: the selected card is an ace B: the selected card is red a) Find P(B). b) Find P(B A). c) Determine whether events A and B are independent. Multiplication Rules 11 P a g e
12 Example: Multiplication, Independent Events Suppose you are going to throw two fair die. What is the probability of getting a 5 on each die? Example: Selecting From an Jar of Balls, Dependent Events Jeff draws balls from the jar below. He draws two balls without replacement. Find the probability that he draws a red ball and then a blue ball, in that order. 4 red 3 blue 2 yellow 12 P a g e
13 Example: Selecting From an Jar of Balls, Independent Example Jeff draws balls from the jar below. He draws two balls, this time with replacement. Find the probability that he gets a red and then a blue ball, in that order. 4 red 3 blue 2 yellow Critical Thinking Pay attention to translating events described by common English phrases into events described using and, or, complement, or given. Rules and definitions of probabilities have extensive applications in everyday lives. 13 P a g e
14 Multiplication Rule for Counting This rule extends to outcomes involving three, four, or more series of events. Example  A coin is tossed and a sixsided die is rolled. How many outcomes are possible? a). 8 b). 10 c). 12 d). 18 Tree Diagrams Displays the outcomes of an experiment consisting of a sequence of activities. The total number of branches equals the total number of outcomes. Each unique outcome is represented by following a branch from start to finish. Tree Diagrams with Probability We can also label each branch of the tree with its respective probability. To obtain the probability of the events, we can multiply the probabilities as we work down a particular branch. 14 P a g e
15 Example  Place five balls in an urn: three red and two blue. Select a ball, note the color, and, without replacing the first ball, select a second ball. The Factorial For any counting number n, the quantity n factorial is given by Example Evaluating Expressions Containing Factorials Evaluate each expression. a) 3! b) 6! c) (63)! d) 6!3! Arrangements of n Distinct objects n! Te total number of different ways to arrange n distinct objects is n! Example Arranging Essays Erika Berg has seven essays to include in her English 1A folder. In how many different orders can she arrange them? 15 P a g e
16 Permutations  arrangements are often called permutations, the number of permutations of n distinct things taken r at a time is denoted np r Since the number of objects being arranged cannot exceed the total number available, we assume for our purposes here that r n. Applying the fundamental counting principle to arrangements of this type gives: np r = Permutations are to evaluate the number of arrangements of n things taken r at a time, where repetitions are not allowed, and the order of the items is important. Example  For a group of seven people, how many ways can four of them be seated in four chairs? a). 35 b). 3 c). 28 d). 840 Alternative Notations are P(n,r) and For Example 4P 2 means the number of permutations of 4 distinct things taken 2 at a time. Using a graphing calculator we can perform this calculation directly as follows for a TI83: For 10P 6 enter in 10 then hit MATH Scroll over to PRB and scroll down to 2 (npr) hit enter then enter in 6 then hit enter. 10P 6 = P a g e
17 Example Using the Factorial Formula for Permutations Evaluate each permutation a) 4P 2 b) 8P 5 Note that 5P 5 is equal to 5! It is true for all whole numbers n that np n = n! Combinations  the number of combinations of n things taken r at a time (that is the number of size r subsets, given a set of size n) is written nc r Since there are n things available and we are choosing r of them, we can read nc r as n choose r. The formula for evaluating numbers of combinations nc r Permutations are to evaluate the number of arrangements of n things taken r at a time, where repetitions are not allowed, and the order of the items is important. Combinations are the number of combinations of n things taken r at a time (that is the number of size r subsets, given a set of size n), where repetitions are not allowed, and the order is not important. Factorial Formula for Combinations The number of combinations, or subsets, of n, distinct things taken r at a time, where r n, can be calculated as nc r Alternative Notations are C(n,r) and and 17 P a g e
18 Using a graphing calculator we can perform this calculation directly as follows for a TI83: For 14C 6 enter in 14 then hit MATH Scroll over to PRB and scroll down to 2 (ncr) hit enter then enter in 6 then hit enter. 14C 6 = 3003 Example Evaluate each combination a) 9C 7 b) 24C 18 Example  Among eleven people, how many ways can eight of them be chosen to be seated? a). 6,652,800 b). 165 c). 3 d) P a g e
Probability. 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 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 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 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 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 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 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 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 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 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 informationThe study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability
The study of probability is concerned with the likelihood of events occurring Like combinatorics, the origins of probability theory can be traced back to the study of gambling games Still a popular branch
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 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 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 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 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 informationProbability Rules. 2) The probability, P, of any event ranges from which of the following?
Name: WORKSHEET : Date: Answer the following questions. 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if. 2) The probability,
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 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 informationSection : Combinations and Permutations
Section 11.111.2: Combinations and Permutations Diana Pell A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words
More informationWEEK 7 REVIEW. Multiplication Principle (6.3) Combinations and Permutations (6.4) Experiments, Sample Spaces and Events (7.1)
WEEK 7 REVIEW Multiplication Principle (6.3) Combinations and Permutations (6.4) Experiments, Sample Spaces and Events (7.) Definition of Probability (7.2) WEEK 87.3, 7.4 and Test Review THE MULTIPLICATION
More informationMath Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.
Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166  Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1  Experiments, Sample Spaces,
More informationMath Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.
Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166  Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1  Experiments, Sample Spaces,
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 informationCHAPTER 8 Additional Probability Topics
CHAPTER 8 Additional Probability Topics 8.1. Conditional Probability Conditional probability arises in probability experiments when the person performing the experiment is given some extra information
More informationLecture 6 Probability
Lecture 6 Probability Example: When you toss a coin, there are only two possible outcomes, heads and tails. What if we toss a coin two times? Figure below shows the results of tossing a coin 5000 times
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 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 informationProbability Test Review Math 2. a. What is? b. What is? c. ( ) d. ( )
Probability Test Review Math 2 Name 1. Use the following venn diagram to answer the question: Event A: Odd Numbers Event B: Numbers greater than 10 a. What is? b. What is? c. ( ) d. ( ) 2. In Jason's homeroom
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 informationBusiness 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 information5 Elementary Probability Theory
5 Elementary Probability Theory 5.1 What is Probability? The Basics We begin by defining some terms. Random Experiment: any activity with a random (unpredictable) result that can be measured. Trial: one
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 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 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 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 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 Theory. Mohamed I. Riffi. Islamic University of Gaza
Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 1 Probability Properties of probability Counting techniques 1 Chapter 1 Probability Probability Theorem P(φ)
More informationCHAPTER 7 Probability
CHAPTER 7 Probability 7.1. Sets A set is a welldefined collection of distinct objects. Welldefined means that we can determine whether an object is an element of a set or not. Distinct means that we can
More informationINDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2
INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2 WARM UP Students in a mathematics class pick a card from a standard deck of 52 cards, record the suit, and return the card to the deck. The results
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 informationDiscrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #23: Discrete Probability Based on materials developed by Dr. Adam Lee The study of probability is
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 informationBefore giving a formal definition of probability, we explain some terms related to probability.
probability 22 INTRODUCTION In our daytoday life, we come across statements such as: (i) It may rain today. (ii) Probably Rajesh will top his class. (iii) I doubt she will pass the test. (iv) It is unlikely
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 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 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 informationName: Class: Date: 6. An event occurs, on average, every 6 out of 17 times during a simulation. The experimental probability of this event is 11
Class: Date: Sample Mastery # Multiple Choice Identify the choice that best completes the statement or answers the question.. One repetition of an experiment is known as a(n) random variable expected value
More informationChapter 3: PROBABILITY
Chapter 3 Math 3201 1 3.1 Exploring Probability: P(event) = Chapter 3: PROBABILITY number of outcomes favourable to the event total number of outcomes in the sample space An event is any collection of
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 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 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 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 informationGeorgia Department of Education Georgia Standards of Excellence Framework GSE Geometry Unit 6
How Odd? Standards Addressed in this Task MGSE912.S.CP.1 Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (or, and, not). MGSE912.S.CP.7
More informationProbability  Grade 10 *
OpenStaxCNX module: m32623 1 Probability  Grade 10 * Rory Adams Free High School Science Texts Project Sarah Blyth Heather Williams This work is produced by OpenStaxCNX and licensed under the Creative
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 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 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 informationMath 166: Topics in Contemporary Mathematics II
Math 166: Topics in Contemporary Mathematics II Xin Ma Texas A&M University September 30, 2017 Xin Ma (TAMU) Math 166 September 30, 2017 1 / 11 Last Time Factorials For any natural number n, we define
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 information4.1 What is Probability?
4.1 What is Probability? between 0 and 1 to indicate the likelihood of an event. We use event is to occur. 1 use three major methods: 1) Intuition 3) Equally Likely Outcomes Intuition  prediction based
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 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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Statistics Homework Ch 5 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability
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 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 information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is
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 informationIf you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics
If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics probability that you get neither? Class Notes The Addition Rule (for OR events) and Complements
More informationCS 237: Probability in Computing
CS 237: Probability in Computing Wayne Snyder Computer Science Department Boston University Lecture 5: o Independence reviewed; Bayes' Rule o Counting principles and combinatorics; o Counting considered
More informationProbability. Dr. Zhang Fordham Univ.
Probability! Dr. Zhang Fordham Univ. 1 Probability: outline Introduction! Experiment, event, sample space! Probability of events! Calculate Probability! Through counting! Sum rule and general sum rule!
More information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,
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 informationChapter 5 Probability
Chapter 5 Probability Math150 What s the likelihood of something occurring? Can we answer questions about probabilities using data or experiments? For instance: 1) If my parking meter expires, I will probably
More informationProbability. Probabilty Impossibe Unlikely Equally Likely Likely Certain
PROBABILITY Probability The likelihood or chance of an event occurring If an event is IMPOSSIBLE its probability is ZERO If an event is CERTAIN its probability is ONE So all probabilities lie between 0
More informationNwheatleyschaller s The Next Step...Conditional Probability
CK12 FOUNDATION Nwheatleyschaller s The Next Step...Conditional Probability Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) Meery To access a customizable version of
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 informationKey Concepts. Theoretical Probability. Terminology. Lesson 111
Key Concepts Theoretical Probability Lesson  Objective Teach students the terminology used in probability theory, and how to make calculations pertaining to experiments where all outcomes are equally
More informationM146  Chapter 5 Handouts. Chapter 5
Chapter 5 Objectives of chapter: Understand probability values. Know how to determine probability values. Use rules of counting. Section 51 Probability Rules What is probability? It s the of the occurrence
More informationSection 6.1 #16. Question: What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit?
Section 6.1 #16 What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1
More informationTree and Venn Diagrams
OpenStaxCNX module: m46944 1 Tree and Venn Diagrams OpenStax College This work is produced by OpenStaxCNX and licensed under the Creative Commons Attribution License 4.0 Sometimes, when the probability
More informationAlgebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations
Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations Objective(s): Vocabulary: I. Fundamental Counting Principle: Two Events: Three or more Events: II. Permutation: (top of p. 684)
More informationNovember 8, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 8, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Crystallographic notation The first symbol
More informationObjective 1: Simple Probability
Objective : Simple Probability To find the probability of event E, P(E) number of ways event E can occur total number of outcomes in sample space Example : In a pet store, there are 5 puppies, 22 kittens,
More informationThe probability setup
CHAPTER 2 The probability setup 2.1. Introduction and basic theory We will have a sample space, denoted S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample
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 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 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 informationChapter 1: Sets and Probability
Chapter 1: Sets and Probability Section 1.31.5 Recap: Sample Spaces and Events An is an activity that has observable results. An is the result of an experiment. Example 1 Examples of experiments: Flipping
More informationExample 1. An urn contains 100 marbles: 60 blue marbles and 40 red marbles. A marble is drawn from the urn, what is the probability that the marble
Example 1. An urn contains 100 marbles: 60 blue marbles and 40 red marbles. A marble is drawn from the urn, what is the probability that the marble is blue? Assumption: Each marble is just as likely to
More informationIntroduction to Probability and Statistics I Lecture 7 and 8
Introduction to Probability and Statistics I Lecture 7 and 8 Basic Probability and Counting Methods Computing theoretical probabilities:counting methods Great for gambling! Fun to compute! If outcomes
More informationChapter 4: Probability
Student Outcomes for this Chapter Section 4.1: Contingency Tables Students will be able to: Relate Venn diagrams and contingency tables Calculate percentages from a contingency table Calculate and empirical
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 informationMATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG
MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, InclusionExclusion, and Complement. (a An office building contains 7 floors and has 7 offices
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 informationElementary Statistics. Basic Probability & Odds
Basic Probability & Odds What is a Probability? Probability is a branch of mathematics that deals with calculating the likelihood of a given event to happen or not, which is expressed as a number between
More informationThe point value of each problem is in the lefthand margin. You must show your work to receive any credit, except on problems 1 & 2. Work neatly.
Introduction to Statistics Math 1040 Sample Exam II Chapters 57 4 Problem Pages 4 Formula/Table Pages Time Limit: 90 Minutes 1 No Scratch Paper Calculator Allowed: Scientific Name: The point value of
More informationProbability of Independent and Dependent Events. CCM2 Unit 6: Probability
Probability of Independent and Dependent Events CCM2 Unit 6: Probability Independent and Dependent Events Independent Events: two events are said to be independent when one event has no affect on the probability
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 informationUnit 14 Probability. Target 3 Calculate the probability of independent and dependent events (compound) AND/THEN statements
Target 1 Calculate the probability of an event Unit 14 Probability Target 2 Calculate a sample space 14.2a Tree Diagrams, Factorials, and Permutations 14.2b Combinations Target 3 Calculate the probability
More information