Section : Combinations and Permutations
|
|
- Roland Walsh
- 6 years ago
- Views:
Transcription
1 Section : 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 (strings of letters) of two distinct letters can be formed from the letters {a, b, c}. Factorial 1
2 Combinations (Order does not matter!) Combination Formula: The number of combinations of n objects taken r at a time is Exercise 1. a) C(100, 2) C(n, r) = n! (n r)!r! b) C(5, 5) Exercise 2. The board of directors of a corporation has 10 members. In how many ways can they choose a committee of 3 board members to negotiate a merger? Exercise 3. How many different selections of two books can be made from a set of nine books? 2
3 Permutations (Order matters!) Permutation Formula: The number of permutations of n objects taken r at a time is P (n, r) = n(n 1)(n 2)(n 3) (n r + 1) OR P (n, r) = n! (n r)! Exercise 4. Eight horses are entered in a race in which a first, second, and third prize will be awarded. Assuming no ties, how many different outcomes are possible? Exercise 5. A club has 10 members. In how many ways can they choose a slate of four officers, consisting of a president, vice president, secretary, and treasurer? Exercise 6. How many ways can you arrange 5 of 10 books on a shelf? Exercise 7. Suppose that you own 10 sweaters and are going on a trip. How many ways can you select six of them to leave at home? 3
4 Exercise 8. Of the 20 applicants for a job, 4 will be selected for intensive interviews. In how many ways can the selection be made? Exercise 9. A poker hand consists of 5 cards selected from a deck of 52 cards. a) How many different poker hands are there? b) How many different poker hands consist entirely of aces and kings? c) How many different poker hands consist entirely of clubs? d) How many different poker hands consist entirely of red cards? 4
5 Exercise 10. In how many ways can five mathematics books and four novels be placed on a bookshelf if the mathematics books must stay together? Solving a Permutation Problem with Like Objects Exercise 11. How many different passwords can be made using all the letters in the word Mississippi? Exercise 12. (You Try!) How many different passwords can be made using all the letters in the word Massachusetts? 5
6 Exercise 13. A committee has four male and five female members. In how many ways can a subcommittee consisting of two males and two females be selected? Exercise 14. An urn contains 25 numbered balls, of which 15 are red and 10 are white. A sample of 3 balls is to be selected. a) How many different samples are possible? b) How many samples contain all red balls? c) How many samples contain 1 red balls and 2 white balls? d) How many samples contain at least 2 red balls? 6
7 Exercise 15. (You Try!) A four-person crew for the international space station is to be chosen from a candidate pool of 10 Americans and 12 Russians. How many different crews are possible if there must be at least two Russians? Section 11.3: Basic Concepts of Probability Definition 1. A sample space is a set of all possible outcomes of an experiment. Exercise 16. An experiment consists of flipping a coin once. Find the sample space. Exercise 17. An experiment consist of flipping a coin twice. Find the sample space. Exercise 18. An experiment consists of flipping a coin three times. Find the sample space. 7
8 Exercise 19. An experiment consists of rolling a single die. Find the sample space. Exercise 20. An experiment consists of rolling a green die and a red die. Find the sample space. Definition 2. An event is a subset of a sample space of an experiment. Exercise 21. Suppose and experiment consists of tossing a coin three times and observing the sequence of heads and tails. Determine the event E = exactly two heads. Exercise 22. Suppose that we have two urns - call them urn I and urn II - each containing red balls and white balls. An experiment consists of selecting an urn and then selecting a ball from that urn and noting its color. a) What is a suitable sample space for this experiment? 8
9 b) Describe the event urn I is selected as a subset of the sample space. Definition 3. Experiments in which each outcome has the same probability are said to be experiments with equally likely outcomes. Exercise 23. Experiment consists of flipping a coin two times. Find probability of every outcome in the sample space. Definition 4. If an experiment with sample space S has equally likely outcomes, then for any event E the probability of E is given by P (E) = n(e) n(s) where n(e) and n(s) denote the number of elements in E and S, respectively. Note: Probability is always a number from 0 to 1. Impossible events always have probability 0 and certain events have probability 1. Exercise 24. A single die is rolled. Find the probability of getting a) A 2. 9
10 b) A number less than 5. Exercise 25. Roll a single die. What is the probability that it lands on an odd number? Complement Rule: P (E) = 1 P (E ) Exercise 26. Of the next 32 trials on the docket in a county court, 5 are homicides, 12 are drug offenses, 6 are assaults, and 9 are property crimes. If jurors are assigned to trials randomly, a) what s the probability that a given juror won t get a homicide case? 10
11 b) what s the probability that a juror gets assigned to a case that isn t a drug offense? Exercise 27. (Birthday Problem) A group of five people is to be selected at random. What is the probability that two or more of them have the same birthday? (Assume that each of the 365 days in a year is an equally likely birthday) Exercise 28. A basketball team has four players. What is the probability that at least two of them were born on the same day of the week? Exercise 29. In a random sample of 500 people, 210 had type O blood, 223 had type A, 51 had type B, and 16 had type AB. Set up a frequency distribution and find the probability that a randomly selected person from the general population has 11
12 a) Type O blood. b) Type A or B blood. c) Neither type A nor type O blood. d) A blood type other than AB. 12
Probability 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 informationChapter 11: Probability and Counting Techniques
Chapter 11: Probability and Counting Techniques Diana Pell Section 11.1: The Fundamental Counting Principle Exercise 1. How many different two-letter words (including nonsense words) can be formed when
More informationChapter 11: Probability and Counting Techniques
Chapter 11: Probability and Counting Techniques Diana Pell Section 11.3: Basic Concepts of Probability Definition 1. A sample space is a set of all possible outcomes of an experiment. Exercise 1. An experiment
More information1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?
1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,
More informationMATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG
MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, Inclusion-Exclusion, and Complement. (a An office building contains 7 floors and has 7 offices
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 informationSection 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
Section 6.1 #16 What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1
More informationFundamental Counting Principle
Lesson 88 Probability with Combinatorics HL2 Math - Santowski Fundamental Counting Principle Fundamental Counting Principle can be used determine the number of possible outcomes when there are two or more
More informationDiscrete Structures Lecture Permutations and Combinations
Introduction Good morning. Many counting problems can be solved by finding the number of ways to arrange a specified number of distinct elements of a set of a particular size, where the order of these
More informationExercises Exercises. 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}?
Exercises Exercises 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}? 3. How many permutations of {a, b, c, d, e, f, g} end with
More informationSection The Multiplication Principle and Permutations
Section 2.1 - The Multiplication Principle and Permutations Example 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small, medium, and large). How many different
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 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 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 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 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 informationA Probability Work Sheet
A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair six-sided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we
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 informationUnit 9: Probability Assignments
Unit 9: Probability Assignments #1: Basic Probability In each of exercises 1 & 2, find the probability that the spinner shown would land on (a) red, (b) yellow, (c) blue. 1. 2. Y B B Y B R Y Y B R 3. Suppose
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 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 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 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 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 informationSection 5.4 Permutations and Combinations
Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to
More informationSection 5.4 Permutations and Combinations
Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! = n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to
More informationIf a regular six-sided die is rolled, the possible outcomes can be listed as {1, 2, 3, 4, 5, 6} there are 6 outcomes.
Section 11.1: The Counting Principle 1. Combinatorics is the study of counting the different outcomes of some task. For example If a coin is flipped, the side facing upward will be a head or a tail the
More informationCounting and Probability Math 2320
Counting and Probability Math 2320 For a finite set A, the number of elements of A is denoted by A. We have two important rules for counting. 1. Union rule: Let A and B be two finite sets. Then A B = A
More informationExam 2 Review (Sections Covered: 3.1, 3.3, , 7.1) 1. Write a system of linear inequalities that describes the shaded region.
Exam 2 Review (Sections Covered: 3.1, 3.3, 6.1-6.4, 7.1) 1. Write a system of linear inequalities that describes the shaded region. 5x + 2y 30 x + 2y 12 x 0 y 0 2. Write a system of linear inequalities
More informationObjectives: Permutations. Fundamental Counting Principle. Fundamental Counting Principle. Fundamental Counting Principle
and Objectives:! apply fundamental counting principle! compute permutations! compute combinations HL2 Math - Santowski! distinguish permutations vs combinations can be used determine the number of possible
More informationWeek 1: Probability models and counting
Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model
More informationChapter 3: Elements of Chance: Probability Methods
Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 3-4 2014-2015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,
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 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 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 informationTEST A CHAPTER 11, PROBABILITY
TEST A CHAPTER 11, PROBABILITY 1. Two fair dice are rolled. Find the probability that the sum turning up is 9, given that the first die turns up an even number. 2. Two fair dice are rolled. Find the probability
More informationEECS 203 Spring 2016 Lecture 15 Page 1 of 6
EECS 203 Spring 2016 Lecture 15 Page 1 of 6 Counting We ve been working on counting for the last two lectures. We re going to continue on counting and probability for about 1.5 more lectures (including
More informationCSC/MTH 231 Discrete Structures II Spring, Homework 5
CSC/MTH 231 Discrete Structures II Spring, 2010 Homework 5 Name 1. A six sided die D (with sides numbered 1, 2, 3, 4, 5, 6) is thrown once. a. What is the probability that a 3 is thrown? b. What is the
More informationPermutations. and. Combinations
Permutations and Combinations Fundamental Counting Principle Fundamental Counting Principle states that if an event has m possible outcomes and another independent event has n possible outcomes, then there
More informationMath 1116 Probability Lecture Monday Wednesday 10:10 11:30
Math 1116 Probability Lecture Monday Wednesday 10:10 11:30 Course Web Page http://www.math.ohio state.edu/~maharry/ Chapter 15 Chances, Probabilities and Odds Objectives To describe an appropriate sample
More informationSection 6.4 Permutations and Combinations: Part 1
Section 6.4 Permutations and Combinations: Part 1 Permutations 1. How many ways can you arrange three people in a line? 2. Five people are waiting to take a picture. How many ways can you arrange three
More information4.4: The Counting Rules
4.4: The Counting Rules The counting rules can be used to discover the number of possible for a sequence of events. Fundamental Counting Rule In a sequence of n events in which the first one has k 1 possibilities
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 informationBlock 1 - Sets and Basic Combinatorics. Main Topics in Block 1:
Block 1 - Sets and Basic Combinatorics Main Topics in Block 1: A short revision of some set theory Sets and subsets. Venn diagrams to represent sets. Describing sets using rules of inclusion. Set operations.
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 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 informationThe probability set-up
CHAPTER 2 The probability set-up 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 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 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 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 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 informationEx 1: A coin is flipped. Heads, you win $1. Tails, you lose $1. What is the expected value of this game?
AFM Unit 7 Day 5 Notes Expected Value and Fairness Name Date Expected Value: the weighted average of possible values of a random variable, with weights given by their respective theoretical probabilities.
More informationMAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability. Preliminary Concepts, Formulas, and Terminology
MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability Preliminary Concepts, Formulas, and Terminology Meanings of Basic Arithmetic Operations in Mathematics Addition: Generally
More informationIndependent Events. If we were to flip a coin, each time we flip that coin the chance of it landing on heads or tails will always remain the same.
Independent Events Independent events are events that you can do repeated trials and each trial doesn t have an effect on the outcome of the next trial. If we were to flip a coin, each time we flip that
More informationMixed Counting Problems
We have studied a number of counting principles and techniques since the beginning of the course and when we tackle a counting problem, we may have to use one or a combination of these principles. The
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 informationCISC 1400 Discrete Structures
CISC 1400 Discrete Structures Chapter 6 Counting CISC1400 Yanjun Li 1 1 New York Lottery New York Mega-million Jackpot Pick 5 numbers from 1 56, plus a mega ball number from 1 46, you could win biggest
More informationClass XII Chapter 13 Probability Maths. Exercise 13.1
Exercise 13.1 Question 1: Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E F) = 0.2, find P (E F) and P(F E). It is given that P(E) = 0.6, P(F) = 0.3, and P(E F) = 0.2 Question 2:
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 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 informationPermutations: The number of arrangements of n objects taken r at a time is. P (n, r) = n (n 1) (n r + 1) =
Section 6.6: Mixed Counting Problems We have studied a number of counting principles and techniques since the beginning of the course and when we tackle a counting problem, we may have to use one or a
More informationName: Class: Date: ID: A
Class: Date: Chapter 0 review. A lunch menu consists of different kinds of sandwiches, different kinds of soup, and 6 different drinks. How many choices are there for ordering a sandwich, a bowl of soup,
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 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 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 informationWeek in Review #5 ( , 3.1)
Math 166 Week-in-Review - S. Nite 10/6/2012 Page 1 of 5 Week in Review #5 (2.3-2.4, 3.1) n( E) In general, the probability of an event is P ( E) =. n( S) Distinguishable Permutations Given a set of n objects
More informationContemporary Mathematics Math 1030 Sample Exam I Chapters Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific
Contemporary Mathematics Math 1030 Sample Exam I Chapters 13-15 Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the left-hand margin.
More informationExamples: Experiment Sample space
Intro to Probability: A cynical person once said, The only two sure things are death and taxes. This philosophy no doubt arose because so much in people s lives is affected by chance. From the time a person
More informationFinite Mathematics MAT 141: Chapter 8 Notes
Finite Mathematics MAT 4: Chapter 8 Notes Counting Principles; More David J. Gisch The Multiplication Principle; Permutations Multiplication Principle Multiplication Principle You can think of the multiplication
More informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and
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 informationSTATISTICAL COUNTING TECHNIQUES
STATISTICAL COUNTING TECHNIQUES I. Counting Principle The counting principle states that if there are n 1 ways of performing the first experiment, n 2 ways of performing the second experiment, n 3 ways
More informationName: Exam 1. September 14, 2017
Department of Mathematics University of Notre Dame Math 10120 Finite Math Fall 2017 Name: Instructors: Basit & Migliore Exam 1 September 14, 2017 This exam is in two parts on 9 pages and contains 14 problems
More informationBasic Probability. Let! = # 8 # < 13, # N -,., and / are the subsets of! such that - = multiples of four. = factors of 24 / = square numbers
Basic Probability Let! = # 8 # < 13, # N -,., and / are the subsets of! such that - = multiples of four. = factors of 24 / = square numbers (a) List the elements of!. (b) (i) Draw a Venn diagram to show
More informationThe probability set-up
CHAPTER The probability set-up.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 space
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 informationSTAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes
STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes Pengyuan (Penelope) Wang May 25, 2011 Review We have discussed counting techniques in Chapter 1. (Principle
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 informationn(s)=the number of ways an event can occur, assuming all ways are equally likely to occur. p(e) = n(e) n(s)
The following story, taken from the book by Polya, Patterns of Plausible Inference, Vol. II, Princeton Univ. Press, 1954, p.101, is also quoted in the book by Szekely, Classical paradoxes of probability
More informationBefore giving a formal definition of probability, we explain some terms related to probability.
probability 22 INTRODUCTION In our day-to-day 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 informationSlide 1 Math 1520, Lecture 15
Slide 1 Math 1520, Lecture 15 Formulas and applications for the number of permutations and the number of combinations of sets of elements are considered today. These are two very powerful techniques for
More informationMath141_Fall_2012 ( Business Mathematics 1) Week 7. Dr. Marco A. Roque Sol Department of Mathematics Texas A&M University
( Business Mathematics 1) Week 7 Dr. Marco A. Roque Department of Mathematics Texas A&M University In this sections we will consider two types of arrangements, namely, permutations and combinations a.
More information1. How to identify the sample space of a probability experiment and how to identify simple events
Statistics Chapter 3 Name: 3.1 Basic Concepts of Probability Learning objectives: 1. How to identify the sample space of a probability experiment and how to identify simple events 2. How to use the Fundamental
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 3-1 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 well-defined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase
More informationFoundations of Computing Discrete Mathematics Solutions to exercises for week 12
Foundations of Computing Discrete Mathematics Solutions to exercises for week 12 Agata Murawska (agmu@itu.dk) November 13, 2013 Exercise (6.1.2). A multiple-choice test contains 10 questions. There are
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 informationACTIVITY 6.7 Selecting and Rearranging Things
ACTIVITY 6.7 SELECTING AND REARRANGING THINGS 757 OBJECTIVES ACTIVITY 6.7 Selecting and Rearranging Things 1. Determine the number of permutations. 2. Determine the number of combinations. 3. Recognize
More informationJUST THE MATHS UNIT NUMBER PROBABILITY 2 (Permutations and combinations) A.J.Hobson
JUST THE MATHS UNIT NUMBER 19.2 PROBABILITY 2 (Permutations and combinations) by A.J.Hobson 19.2.1 Introduction 19.2.2 Rules of permutations and combinations 19.2.3 Permutations of sets with some objects
More informationMost of the time we deal with theoretical probability. Experimental probability uses actual data that has been collected.
AFM Unit 7 Day 3 Notes Theoretical vs. Experimental Probability Name Date Definitions: Experiment: process that gives a definite result Outcomes: results Sample space: set of all possible outcomes Event:
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 informationMAT104: Fundamentals of Mathematics II Counting Techniques Class Exercises Solutions
MAT104: Fundamentals of Mathematics II Counting Techniques Class Exercises Solutions 1. Appetizers: Salads: Entrées: Desserts: 2. Letters: (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U,
More informationProbability Warm-Up 1 (Skills Review)
Probability Warm-Up 1 (Skills Review) Directions Solve to the best of your ability. (1) Graph the line y = 3x 2. (2) 4 3 = (3) 4 9 + 6 7 = (4) Solve for x: 4 5 x 8 = 12? (5) Solve for x: 4(x 6) 3 = 12?
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 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 informationTree and Venn Diagrams
OpenStax-CNX module: m46944 1 Tree and Venn Diagrams OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 Sometimes, when the probability
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 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 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 information