# Math141_Fall_2012 ( Business Mathematics 1) Week 7. Dr. Marco A. Roque Sol Department of Mathematics Texas A&M University

Size: px
Start display at page:

Download "Math141_Fall_2012 ( Business Mathematics 1) Week 7. Dr. Marco A. Roque Sol Department of Mathematics Texas A&M University"

Transcription

1 ( Business Mathematics 1) Week 7 Dr. Marco A. Roque Department of Mathematics Texas A&M University

2 In this sections we will consider two types of arrangements, namely, permutations and combinations a. each permutation consists of three letters a, b, c. Thus, we may think such a sequence as being constructed by filling in each of the three blanks Definition Given a set of distinct objects, a permutation of the set is an arrangement of these objects in a definite order. That the order is important, can be seen from the situation where we have to deal with University Student Identification Numbers, so it is not the same to have the number than even though we have the same digits. 1st letter 2nd letter 3rd letter To pick the first letter we have three options, once the first letter is picked, for the second letter we just have two options and finally after the first two letters have been selected, then for the third letter we just have one option. Thus, by the multiplication principle we have Example Let A = { a, b, c } a. Find the number of permutations of A. b. List all the permutations of A with the aid of a tree diagram That is, we have (3)(2)(1) = 6 possible permutations

3 b. Using a tree diagram For the first place we have nine options, then for the second place we have eight options,.., for the ninth place we just have one option. Then, by the multiplication principle we have (9)(8)(7) (1) = 362, 880 Definition 2) Find the number of ways in which a baseball team consisting of nine people can arrange themselves in a line for a group picture. For any natural number n, the factorial of n denoted by n! is defined as n! = n (n-1)(n-2) 1 OBS In this case we have nine places to fill in. n! = n(n-1)! n! = n(n-1)(n-2)!

4 Using the factorial of a number, we can give a way to calculate the number of permutations of a set with n members. In this way, the number of permutations of n distinct objects taken n at a time, denoted by P(n,n) is given by P(n,n) = n! Therefore by the multiplication principle, there are (n) (n-1) (n-2) (n-r+1) different permutations of n elements taken r at a time, but the above quantity is equivalent to Now, in some cases, given a set of n distinct objects, we would be interested in finding the number of permutations of the set taken r ( n) at a time. In that situation we will have r different places to be fill in (n ) (n 1 )... (n r+1 ) (n r ) (n r 1 ) (n r 2 )...(1) (n ) (n 1 )... (n r+1 )= (n r ) (n r 1 ) (n r 2)... (1 ) (n ) (n 1 )... (n r+ 1 )= n! (n r )! Therefore r Permutations of n distinct objects Thus, the first place has n options, then the second places has (n-1) options, the third place has (n-2),.., and the r-th place has ( n-r+1) options. The number of permutations of n distinct objects taken r at a time, denoted by P(n,r), is given by P (n,r ) = n! (n r )!

5 P (8,4) = 8! (8 4 )! = 8! 4! = 1680 Examples 1) Let A = { a, b, c, d}. find all the permutations of A taken two at a time. Here, n = 4 and r = 2, so 2) Find the number of ways a President,Vicepresident, Secretary, and a Treasurer can be chosen from a committee of eight members. P (4,2 ) = 4! (4 2 )! = 4! 2! = 12 Since the order is important, we have to find the permutations of 8 distinct objects taken 4 at a time So far, we have considered permutations of n distinct objects taken r at a time, but what if we have inside of the set, some subgroups of identical elements? In that case we have to take out all the permutaions of the identical subgroups. Example Find all thedistinct permutations of the word ATLANTA. The word consists of 7 letters, and the number of permutations is given by 7!, but we have to consider that there are two subgroups of identical elements. The first one associated to the letter A with three members and the second one associated to the letter T with two members, that is, we need to take the permutations of each subgroup out of the final result. Therefore, the total number of permutations is have P (4,2 ) = 7! 3!2! = 420

6 In general we obtain the folowing result Permutations of n Objects, Not all Distincts Given a set of n objects in which n 1 are alike and of one kind, n 2 are alike and of another kind,, and n m are alike and of yet another kind, so that n 1 + n n m = n Then the number of permutations of these objects taken n at a time are given by Example n! n 1!n 2!...n m! If nine students are taking a test and they will be graded by three instructors, taking each one three exams. In how many ways can the tests be directed to any of the three instructors? We can think in the nine tests as an arrange of nine places, and each one is going to be associated to one of the instructors, but each instructor receives three tests, then the problem is equivalent to find the permutions of nine elements with three subgroups of identical elements.thus, the total number of ways the nine test can be distributed to the instructors is given by 9! 3!3!3! = 1680 Now, when the order is not important in the arrangement, that is, when we want to select r objects from a set of n objects without any regard to the order in which these objects are selected, then we are talking about Combinations. In this way to find the number of combinations de n elements taken r at a time, from the number of permutations P (n,r ) = n! (n r )!

7 We have to take out the permutations of each arrangement of r elements, that is, Combinations of n objects The number of combinations of n distinct objects taken r at a time is given by OBS C(n,r) gives us the number of subsets with r ( n) elements from a set with n elements. Examples C (n,r ) = P (n,r ) r! 1) How many poker hands of 5 cards can be dealt from a standerd deck of 52 cards? = n! r! (n r )! n! r! (n r )! ( where r n ) Since the order does not matter, we just want to find the combinations of 52 distinct elements taken 5 at a time. 2) A Senate investigation subcommittee of four members is to be selected from a Senate committee of ten members. Determine the number of ways in which this can be done. C (52,5 ) = 52! 5! (52 5 )! = 52! 5! 47! = 2,598,960 Since in the selection of the members the order does not matter, we need to find the combinations of 10 members taken 4 at a time. C (10,4 ) = 10! 4! (10 4 )! = 10! 4!6! = 210

8 3) The members of a string quartet consisting of two violinists, a violist, and a cellist are to be selected from a group of six violinists, three violists, and two cellists. a. In how many ways can the string quartet be formed? b. In how many ways can the string quartet be formed if one of the violinists is to be designed as the first violinist and the other is to be designated as the second violinist? b. In this case the order is important for the violinists. The violinists can be selected in P(6,2) ways The violists can be selected in C(3,1) ways The cellists can be selected in C(2,1) ways and by the multiplication principle we have P(6,2) C(3,2) C(2,1) = (30)(3)(2) = 180 a. The violinists can be selected in C(6,2) ways The violists can be selected in C(3,1) ways The cellists can be selected in C(2,1) ways 4) The Psicos, a pop group, are planning a concert tour with performances to be given in five cities: San Francisco, Los Angeles, San Diego, Denver, and Las Vegas. In how many ways can they arrange their itinerary if and by the multiplication principle we have C(6,2) C(3,1) C(2,1) = (15)(3)(2) = 90 a. There are no restricions. b. The three performances in California must be given consecutively?

9 In both cases the order is important a. P(5,5) = 5! = 120 b.- Now the three cities of california can be thought as a block and the other cities as two more blocks. Thus, the number of ways of performing in California and the other cities is P(3,3). On the other hand inside of the California block, we have P(3,3) ways to arrange the itinerary. Therefore, using the multiplication principle we have. (3!)(3!) = 36 There are : C(5,2) ways the investor can select the aerospace companies C(3,2) ways the investor can select the energy companies C(4,2) ways the investor can select the electronics companies Therefore by the multiplication principle, there are C(5,2)C(3,2)C(4,2) = 180 5) Suppose that an investor has decided to purchase shares in the stocks of two aerospace companies, two energy development companies,and two electronic companies. In how many ways can the investor select the group of six companies from the investment from the recommended list of five aerospace companies, three energy development companies, and four electronics companies?

10 Probability. Experiments, Sample Space, and Events Sample Point, Smaple Space, and Event There are many terms used in the probability theory and we will start giving some basic definitions. Definition An experiment is an activity with observable results. Example 1)Tossing a coin and observing whether it falls heads or tails 2) Rolling a die and looking the number falling uppermost 3) Testing a light bulb from a batch and observing whether is deffective or not In the discussion of expermients, we use the following terms: Sample Point: An outcome of an experiment. Sample Space: The set consisting of all possible sample points of an experiment. Event: A subset of a sample space of an experiment. From the above set of definitions we can see that the sample space plays the role of a universal set, that is, it will be our reference set in the study of probability. In this sense the sample space represents the certain event, that is, the event that must occurs because it contains all the possible outacomes of the experiment. On the other hand, the empty set Ø is called the impossible event, since the empty set contains no elements.

11 Probability. Experiments, Sample Space, and Events Example Uniond of Two events The union of two events E and F is the event E F Describe the sample space associated with the experiment of tossing a coin and observing whether it falls heads or tails. What are the events of this experiment? The sample space is given by S = { H, T } And all the possible subsets ( events ) are Ø, {H}, {T}, S Now, since every event is a subset of the sample space, then we can form new events using the set operations defined before. Thus, if E and F are events associated to an experiment with sample space S then we define the : Intersection of Two Events The instersection of two events is the event E F Complement of an Event The complement of the event E is the evemt E c Thus, Ven diagrams will be useful in solving problems dealing with events associated to some given experiment. Definition The events E and F are called mutually exclusive if E F = Ø

12 Probability. Experiments, Sample Space, and Events Examples 1) An experiment consists of tossing a coin three times and observing the resulting sequence of heads and tails a. Describe the sample space of the experiment. b. Determine the event E that exactly two heads appear. c. Determine the event F that at least one head appears. a. Since every time the coin has two different ways of landing, Head or Tail, then we have: S= { HHH,HHT,HTH,THH,HTT,THT,TTH, TTT} b. E ={HHT, HTH, THH} c. F = { HHH,HHT,HTH,THH,HTT,THT,TTH } 2) An experiment consist of rolling a pair of dice and observing the number that falls uppermost on each die. a. Describe an appropriate sample space for this experiment. b. Determine the events that E 1,E 2,, E 12, that the sum of the numbers falling uppermost is 1, 2,, 12. a. Since the first die has 6 possible otpions and the second die has also 6 possibilities, then by the multiplication principle we have 36 possible outcomes: S = { (1,1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2,1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3,1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4,1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5,1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6,1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) }

13 Probability. Experiments, Sample Space, and Events Examples b. E 1 = Ø E 2 = {(1, 1)} E 3 = {(1, 2), (2,1)} E 4 = {(1, 3), (2, 2), (3, 1)} E 5 = {(1, 4), (2, 3), (3, 2), (4, 1)} E 6 = {(1, 5),(2, 4). (3, 3), (4, 2), (5, 1)} E 7 = {(1, 6), (2,5), (3, 4), (4, 3), (5, 2), ( 6, 1)} E 8 = {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2) } E 9 = {(3, 6), (4, 5), (5, 4), (6, 3) } 3) An experiment consist of recording, in order of their births, the sex composition of a three-child family in which the children were born at different times. a.- Describe an appropriate sample space S for this experiment. b. Describe the event E that there are two girls and a boy in the family c. Describe the event F That the oldest child is a girl d. Describe the event G that the oldest child is a girl and the yougest child is a boy. a. Using a tree diagram E 10 = {(4, 6), (5, 5), (6, 4) } E 11 = {5, 6), (6,5)} E 12 = {(6, 6)}

14 Probability. Experiments, Sample Space, and Events Examples The sample space is given by S = { bbb, bbg, bgb, gbb, bgg,gbg,ggb, ggg } b. E = {bgg, gbg, ggb} c. F = { gbb, gbg, ggb, ggg} d. G = { gbb, ggb}

15

### RANDOM EXPERIMENTS AND EVENTS

Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In day-to-day 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

### 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

### STAT 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

### In how many ways can the letters of SEA be arranged? In how many ways can the letters of SEE be arranged?

-Pick up Quiz Review Handout by door -Turn to Packet p. 5-6 In how many ways can the letters of SEA be arranged? In how many ways can the letters of SEE be arranged? - Take Out Yesterday s Notes we ll

### Probability and Counting Rules. Chapter 3

Probability and Counting Rules Chapter 3 Probability as a general concept can be defined as the chance of an event occurring. Many people are familiar with probability from observing or playing games of

### MATH 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

### 8.2 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

### Section : Combinations and Permutations

Section 11.1-11.2: 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

### 4.1 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

### 7.1 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

### Probability. 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

### Chapter 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.

### Chapter 4. Probability and Counting Rules. McGraw-Hill, Bluman, 7 th ed, Chapter 4

Chapter 4 Probability and Counting Rules McGraw-Hill, Bluman, 7 th ed, Chapter 4 Chapter 4 Overview Introduction 4-1 Sample Spaces and Probability 4-2 Addition Rules for Probability 4-3 Multiplication

### Test 2 Review Solutions

Test Review Solutions. A family has three children. Using b to stand for and g to stand for, and using ordered triples such as bbg, find the following. a. draw a tree diagram to determine the sample space

### Chapter 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.

### Lesson 10: Using Simulation to Estimate a Probability

Lesson 10: Using Simulation to Estimate a Probability Classwork In previous lessons, you estimated probabilities of events by collecting data empirically or by establishing a theoretical probability model.

### CS 361: Probability & Statistics

January 31, 2018 CS 361: Probability & Statistics Probability Probability theory Probability Reasoning about uncertain situations with formal models Allows us to compute probabilities Experiments will

### Section 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

### 1. 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,

### Grade 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.

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Chapter 3: Practice SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) A study of 000 randomly selected flights of a major

### Name: 1. Match the word with the definition (1 point each - no partial credit!)

Chapter 12 Exam Name: Answer the questions in the spaces provided. If you run out of room, show your work on a separate paper clearly numbered and attached to this exam. SHOW ALL YOUR WORK!!! Remember

### Counting 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

### PROBABILITY. 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

### Discrete 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

### Chapter 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

### CHAPTER 7 Probability

CHAPTER 7 Probability 7.1. Sets A set is a well-defined 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

### Chapter 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,

### EECS 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

### If 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

### Math 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

### Define 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

### STOR 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

### 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

### Day 7. At least one and combining events

Day 7 At least one and combining events Day 7 Warm-up 1. You are on your way to Hawaii and of 15 possible books, you can only take 10. How many different collections of 10 books can you take? 2. Domino

### November 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

### Section 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

### STAT 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

### I. 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

### Probability Concepts and Counting Rules

Probability Concepts and Counting Rules Chapter 4 McGraw-Hill/Irwin Dr. Ateq Ahmed Al-Ghamedi Department of Statistics P O Box 80203 King Abdulaziz University Jeddah 21589, Saudi Arabia ateq@kau.edu.sa

### Probability. 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!

### Probability - 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

### Probability (Devore Chapter Two)

Probability (Devore Chapter Two) 1016-351-01 Probability Winter 2011-2012 Contents 1 Axiomatic Probability 2 1.1 Outcomes and Events............................... 2 1.2 Rules of Probability................................

### Chapter 1: Sets and Probability

Chapter 1: Sets and Probability Section 1.3-1.5 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

### CHAPTER 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

### Contents 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

### Exam Time. Final Exam Review. TR class Monday December 9 12:30 2:30. These review slides and earlier ones found linked to on BlackBoard

Final Exam Review These review slides and earlier ones found linked to on BlackBoard Bring a photo ID card: Rocket Card, Driver's License Exam Time TR class Monday December 9 12:30 2:30 Held in the regular

### Block 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.

### Probability 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

### Elementary Combinatorics

184 DISCRETE MATHEMATICAL STRUCTURES 7 Elementary Combinatorics 7.1 INTRODUCTION Combinatorics deals with counting and enumeration of specified objects, patterns or designs. Techniques of counting are

### Probability 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

### Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results:

Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results: Outcome Frequency 1 8 2 8 3 12 4 7 5 15 8 7 8 8 13 9 9 10 12 (a) What is the experimental probability

### A 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

### Name: Class: Date: Probability/Counting Multiple Choice Pre-Test

Name: _ lass: _ ate: Probability/ounting Multiple hoice Pre-Test Multiple hoice Identify the choice that best completes the statement or answers the question. 1 The dartboard has 8 sections of equal area.

### Name: 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

### CSC/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

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Ch. 3 Probability 3.1 Basic Concepts of Probability and Counting 1 Find Probabilities 1) A coin is tossed. Find the probability that the result is heads. A) 0. B) 0.1 C) 0.9 D) 1 2) A single six-sided

### Mathematical 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:

### Exercise Class XI Chapter 16 Probability Maths

Exercise 16.1 Question 1: Describe the sample space for the indicated experiment: A coin is tossed three times. A coin has two faces: head (H) and tail (T). When a coin is tossed three times, the total

### MULTIPLE 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

### Chapter 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

### Probability: Terminology and Examples Spring January 1, / 22

Probability: Terminology and Examples 18.05 Spring 2014 January 1, 2017 1 / 22 Board Question Deck of 52 cards 13 ranks: 2, 3,..., 9, 10, J, Q, K, A 4 suits:,,,, Poker hands Consists of 5 cards A one-pair

### November 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

### heads 1/2 1/6 roll a die sum on 2 dice 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 1, 2, 3, 4, 5, 6 heads tails 3/36 = 1/12 toss a coin trial: an occurrence

trial: an occurrence roll a die toss a coin sum on 2 dice sample space: all the things that could happen in each trial 1, 2, 3, 4, 5, 6 heads tails 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 example of an outcome:

### Section 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

### Chapter 4: Introduction to Probability

MTH 243 Chapter 4: Introduction to Probability Suppose that we found that one of our pieces of data was unusual. For example suppose our pack of M&M s only had 30 and that was 3.1 standard deviations below

### Probability as a general concept can be defined as the chance of an event occurring.

3. Probability 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. Probability as a general

### The Product Rule The Product Rule: A procedure can be broken down into a sequence of two tasks. There are n ways to do the first task and n

Chapter 5 Chapter Summary 5.1 The Basics of Counting 5.2 The Pigeonhole Principle 5.3 Permutations and Combinations 5.5 Generalized Permutations and Combinations Section 5.1 The Product Rule The Product

### Diamond ( ) (Black coloured) (Black coloured) (Red coloured) ILLUSTRATIVE EXAMPLES

CHAPTER 15 PROBABILITY Points to Remember : 1. In the experimental approach to probability, we find the probability of the occurence of an event by actually performing the experiment a number of times

### Class 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:

### 4.3 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:

### MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #1 - SPRING DR. DAVID BRIDGE

MATH 205 - CALCULUS & STATISTICS/BUSN - PRACTICE EXAM # - SPRING 2006 - DR. DAVID BRIDGE TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. Tell whether the statement is

### The next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following:

CS 70 Discrete Mathematics for CS Fall 2004 Rao Lecture 14 Introduction to Probability The next several lectures will be concerned with probability theory. We will aim to make sense of statements such

### Probability 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,

### Finite 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

### CS 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

### Section 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

### Chapter 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

### CSC/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

### Theory 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

### Combinatorics: 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

### Probability 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

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

More 9.-9.3 Practice Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. ) In how many ways can you answer the questions on

### University of Connecticut Department of Mathematics

University of Connecticut Department of Mathematics Math 070Q Exam A Fall 07 Name: TA Name: Discussion: Read This First! This is a closed notes, closed book exam. You cannot receive aid on this exam from

### MAT104: 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

### Fundamentals 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

### Elementary 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

### Lecture 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

### CISC 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

### CS100: DISCRETE STRUCTURES. Lecture 8 Counting - CH6

CS100: DISCRETE STRUCTURES Lecture 8 Counting - CH6 Lecture Overview 2 6.1 The Basics of Counting: THE PRODUCT RULE THE SUM RULE THE SUBTRACTION RULE THE DIVISION RULE 6.2 The Pigeonhole Principle. 6.3

### 7.4 Permutations and Combinations

7.4 Permutations and Combinations The multiplication principle discussed in the preceding section can be used to develop two additional counting devices that are extremely useful in more complicated counting

### Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples

Rosen, Discrete Mathematics and Its Applications, 6th edition Extra Examples Section 6.1 An Introduction to Discrete Probability Page references correspond to locations of Extra Examples icons in the textbook.

### STAT 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

### STAT 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

### 1. 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

### Math 146 Statistics for the Health Sciences Additional Exercises on Chapter 3

Math 46 Statistics for the Health Sciences Additional Exercises on Chapter 3 Student Name: Find the indicated probability. ) If you flip a coin three times, the possible outcomes are HHH HHT HTH HTT THH

### Permutations and Combinations

Motivating question Permutations and Combinations A) Rosen, Chapter 5.3 B) C) D) Permutations A permutation of a set of distinct objects is an ordered arrangement of these objects. : (1, 3, 2, 4) is a