Exam 2 Review (Sections Covered: 3.1, 3.3, , 7.1) Solutro. 1. Write a system of linear inequalities that describes the shaded region.
|
|
- Nelson McCoy
- 5 years ago
- Views:
Transcription
1 Exam 2 Review (Sections Covered: ) Solutro 1 Write a system of linear inequalities that describes the shaded region :) 5x +2y 30 x +2y " 2) 12 x : 0 y Z 0 : Line as 0 TO ± 30 True Line (2) 0+0 ± 12 Tr 2 Write a system of linear inequalities that describes the solution set D 7x +6y 86 =f (2) x + y < 10 x 2 (3) 0 y 2 1 4) 5 (2) Line 1 i ) False Line 12 ) O +0<10 True Line (3) 0>0 True Line 14 ) 025 False (2) 17
2 3 Solve the linear programming problem Maximize P =3x +5y Ll ) subject to 2x + y apple 16 ( 801 ( 016 ) 14 2x +3y apple ) x 0 y 0 (a) Find the corner points of the solution set theline Ll ) 0+0[6 Linlk True # i±% E*s (b) Find the maximum "±M t# K Fall 2016 Maya Johnson
3 4 Solve the linear programming problem Minimize C =2x +4y li ) subject to 4x +2y ) 121 2x +3y 30 ( 1507 x 0 y 0 (O: (a) Find the corner points of the solution set Lionel o+oz4o o±i# E #EF Points : = Line : ONFalse (b) Find the Min maximum Effete rsoiyot8ime# 3 Fall 2016 Maya Johnson
4 5 Let the universal set U = {u v w x y z} with sets A = {uvyz} B = {xyz} andc = {w x} Determine whether the following statements are true or false (a) x y 2 B (b) {x y z} B (c) {u w} 2 A (d) {u y} A True False True % 6 Let the universal set U = { } with sets A = { 202} B = { 3113} and C = { 2 1 3} Determinewhetherthefollowingstatementsaretrueorfalse (a) A has 20 subsets (b) B c =? Fat (c) (A \ B) c = U : :alee : (d) (A [ C) ={ } rue (e)? 2 C False (f) A and B are disjoint sets True 7 Let U be a universal set with sets A and B Determine whether the following statements are true or false (a) (U) c =? True (b) (?) c = U True (c) (A \ B) c = A c \ B c False (d) B \? = B (e)? B (f) A [? =? True False False 4 Fall 2016 Maya Johnson
5 8 Write venn diagrams to represent each of the following sets (a) A \ B \ C c (b) A c [ B [ C 0$08 (c) (A \ B) [ C *eemaa% ao (d) (A [ B c ) \ C IAUBYNC = Angolan c ) 5 Fall 2016 Maya Johnson
6 9 Let U = { } A = { } B = { } andc = { } Find each set using roster notation (a) (A \ B) [ C (b) (A [ B [ C) c 0VCaCaI9i6i2s13@lAuBUCKAcnBcnctecDloctt05IYaTiniaA (c) (A \ B \ C) c 10 Let U = { } A = { } B = { } and C = { } Listtheelementsofeachset (a) A c \ (B \ C c ) 'M 618 }={6@ (b) (A [ B c ) [ (B \ C c ) laub9u{ 6183={ }UBCU 6 18 } (c) (A [ B) c \ C c ltttnpscncy µ 6 Fall 2016 Maya Johnson
7 11 If n(b) =13n(A [ B) =24andn(A \ B) =6findn(A) NAUB )=nla)thlb ) NCANB 24 = MA ) ) NCA) = ' 12 In a survey of 400 people a pet food manufacturer found that 250 owned a bird 150 owned a snake and 75 owned neither a bird or a snake (a) How many owned a bird or a snake? (b) How many owned both a bird and a snake? nl Bnp (a) n( Bus )a=4g 5 Lb ) 325= nl BAS ) = = In a survey of 300 members of a local sports club 180 members indicated that they plan to attend the next Summer or Winter Olympic Games 150 members indicated that they plan to attend the next Summer Olympic Games and 90 indicated that they plan to attend the next Winter Olympic Games How many members of the club plan to attend (a) Both of the games? (b) Exactly one of the games? (c) The Summer Olympic Games only? ;Ii : :QQD (d) None of the games? Ld 7 Fall 2016 Maya Johnson
8 14 Let A and B be subsets of a universal set U and suppose n(u) =48n(A) =13n(B) =23and n(a \ B) = 8 Compute: (a) n(a c \ B) (b) n(b c ) (c) n(a c [ B c ) (d) How many subsets does B have? (e) How many proper subsets does B have? ' 00 lab ( c 20 +5=400 " (d) 223= = ( e) 15 Let A B andc be sets in a universal set U We are given n(u) =66n(A) =32n(B) =33 n(c) =33n(A \ B) =16n(A \ C) =10n(B \ C) =18n(A \ B \ C c ) = 9 Find the following values (a) n((a [ B [ C) c ) (b) n((a c \ B c ) [ C) (c) n((a c [ B c ) \ C) semi ee#nojx@qo0qy regions =@ (c)add t Ll =# with two as 8 Fall 2016 Maya Johnson
9 16 Use the following information to determine the number of people in each region of the Venn Diagram r Agroupof295studentswereaskedwhichofthesesportstheyparticipatedinduringhighschool 44 students participated in all of these sports 87 students participated in basketball and track 39 students participated in basketball and tennis but not track r 79 students participated in track but not tennis 155 students participated in basketball 73*43= students did not participate in tennis 103 students participated in exactly one sport r 3629=380 a = b = Tennis a b Track c c = d e g f d = e = f = 4 44=430 h b = 295 Basketball g = h = µ ' =290 : 32 }9# 9 Fall 2016 Maya Johnson
10 Use the following information to determine the number of people in each region of the Venn Diagram 251 people were asked which of these instruments that they could play: Piano Drums or Guitar 20 people could play none of these instruments 34 people could play all three of these instruments 79 people could play drums or guitar but could not play piano 115 people could play guitar : 130 people could play at least two of these instruments 3%34= people could play piano and guitar but could not play drums 78 people could play piano and drums a = b = Guitar Piano b c = a c lag283424=290 d = e =240 d f e = h g Drums f = g = 24=260 h = C = = Fall 2016 Maya Johnson
11 18 Agroupofstudentswereaskedwhichofthesesportstheyplay Theinformationwasrecorded in the Venn Diagram Use the the Venn Diagram to answer these questions Let T =TennisF = Football and B =Basketball a =43 Football Tennis b =38 a b e c c =45 d =8 d f e =22 g f =16 h Basketball g =12 h =35 (a) How many students play Football or Basketball but not Tennis? a + d tg a =630 (b) How many students do not play Football? Ctftg th = =@ (c) How many students play tennis or do not play basketball? b + C tetf tat h = =+360 n(t c \ (B [ F )) at d tg =630 (d) n(t [ (B \ F c )) btctetf tg = t 12= Fall 2016 Maya Johnson
12 +0 19 In recent years a state has issued license plates using a combination of three digits followed by three letters of the alphabet followed by another three digits How many di erent license plates can be issued using this configuration? et 26*61*10=263 ; loti D 20 Complete the following (a) How many sevendigit telephone numbers are possible if the first digit must be nonzero? b a 0% it = 900 (b) How many international directdialing numbers for calls within the United States and Canada are possible if each number consists of a 1 plus a threedigit area code (the first digit of which must be nonzero) and a number of the type described in part (a)? I = Astatemakeslicenseplateswiththreelettersfollowedbyfourdigits (a) How many license plates are possible? =175760%21 (b) If no repetition of the letters is permitted how many di erent license plates are possible? a 2 21 e I IO :D (c) If no repetition of letters or digits is permitted how many di erent license plates are possible? 26 E 7=78624%5 (d) How many license plates have no repetition of letters or digits and begin with a vowel? D 9 & I = Fall 2016 Maya Johnson
13 22 A company car that has a seating capacity of eight is to be used by eight employees who have formed a car pool If only three of these employees can drive how many possible seating arrangements are there for the group? 31 III I6 = There are four families attending a concert together Each family consists of 1 male and 2 females In how many ways can they be seated in a row of twelve seats if (a) There are no restrictions? 12! = (b) Each family is seated together? 1# ' =3! (c) The members of each gender are seated together? 12 total people =3h@ 3131 :3! 4! si#e=ifnstt!g Alex Mark Sue Bill and Maggie attend the movie theater Assume that Sue and Maggie are female and that Alex Mark and Bill are male How many ways can they be seated if (a) There are no restrictions? 5! = # 5 total people (b) The females sit together and the males sit together? 3 = 2! 3! 4=240 (c) Mark and Sue want to sit together? t=2!c4 #yt I! = Fall 2016 Maya Johnson
14 25 At a college library exhibition of faculty publications two mathematics books four social science books and three biology books will be displayed on a shelf (Assume that none of the books are alike) (a) In how many ways can the nine books be arranged on the shelf? 9! (b) In how many ways can the nine books be arranged on the shelf if books on the same subject matter are placed together? 1 # =2! 4! 3! 3! Find the number of distinguishable arrangements of each of the following words (a) acdbens 7! (b) baaaben # (c) aaabbba HE =D 27 In how many ways can a subcommittee of six be chosen from a Senate committee of six Democrats and five Republicans if (a) All members are eligible? C ( U 61=4620 Total of 11 choose 6 (b) The subcommittee must consist of three Republicans and three Democrats? Clb } ) CK 31= Fall 2016 Maya Johnson
15 19 28 In how many di erent ways can a panel of 12 jurors and 2 alternates be chosen from a group of 16 prospective jurors? C ) Cl 412 ) zw@ 29 From a shipment of 25 transistors 6 of which are defective a sample of 9 transistors is selected at random Total of 25 6 defective nondef choose 9 (a) In how many di erent ways can the sample be selected? C 12591=20*29754 (b) How many samples contain exactly 3 defective transistors? C 1631 Cll9 67= (c) How many samples contain no defective transistors? C 11991= (d) How many samples contain at least 5 defective transistors? C ( 65 ) C 119 4) tcl 66 ) CI 19 3) = Fall 2016 Maya Johnson
16 30 A box contains 8 red marbles 8 green marbles and 10 black marbles A sample of 12 marbles is to be picked from the box (a) How many samples contain at least 1 red marble? Total ways Cl ) C Total 26 choose 12 nd on 't wait ( ) 9639 a (b) How many samples contain exactly 4 red marbles and exactly 3 black marbles? C ( 84 ) CC 10 3) C ( 851= (c) How many samples contain exactly 7 red marbles or exactly 6 green marbles? CH ) tcl 86 ) CH 6) I C (d) How many samples contain exactly 5 green marbles or exactly 3 black marbles? 1187 ) + Cl 103 cl 169) CHE ) Choi 3) CH 41= Suppose we have 20 people on a committee How many subcommittees contain one president one vice president and six cabinet members? 2011*161= Ten runners are competing in a halfmarathon How many ways can we award one 1st place prize one 2nd place prize one 3rd place prize and four 4th place prizes? k 4 CHIT =25# 16 Fall 2016 Maya Johnson
17 33 Consider the sample space S = {s t n} How many total events are there for this sample space? # of events = # of subsets = 23=80 34 Let S = {5 9 12} be a sample space associated with an experiment (a) List all events of this experiment { 5499 } if 12 } I { } (b) How many events of S contain the number 5? 4 (c) How many events of S contain the number 12 or the number 5? 35 An experiment consists of tossing a coin and observing the side that lands up and then rolling a fair 4sided die and observing the number rolled Let H and T represent heads and tails respectively (a) Describe the sample space S corresponding to this experiment :HI H2 H 3 H 4 Tl TZ T 3 T 4 } (b) What is the event E 1 that an even number is rolled? { H 2 H 4 T 2 T 4 } (c) What is the event E 2 that a head is tossed or a 3 is rolled? { Hi He H 344 T 3 } (d) What is the event E 3 that a tail is tossed and an odd number is rolled? { T ' T3 } 17 Fall 2016 Maya Johnson
18 5 36 The numbers and 7 are written on separate pieces of paper and put into a hat Two pieces of paper are drawn at the same time and the product of the numbers is recorded Find the sample space S a { A jar contains 8 marbles numbered 1 through 8 An experiment consists of randomly selecting a marble from the jar observing the number drawn and then randomly selecting a card from a standard deck and observing the suit of the card (hearts diamonds clubs or spades) (a) How many outcomes are in the sample space for this experiment? 8 4 = 32 (b) How many outcomes are in the event an even number is drawn? (c) How many outcomes are in the event a number more than 1 is drawn and a red card is drawn? * two of the suits are red 7 a 2 =@ (d) How many outcomes are in the event a number less than 2 is drawn or a club is not drawn? I = = Fall 2016 Maya Johnson
Section 7.1 Experiments, Sample Spaces, and Events
Section 7.1 Experiments, Sample Spaces, and Events Experiments An experiment is an activity with observable results. 1. Which of the follow are experiments? (a) Going into a room and turning on a light.
More informationDRAFT. M118 Exam Jam Concise. Contents. Chapter 2: Set Theory 2. Chapter 3: Combinatorics 3. Chapter 4: Probability 4. Chapter 5: Statistics 6
Contents Chapter 2: Set Theory 2 Chapter 3: Combinatorics 3 Chapter 4: Probability 4 Chapter 5: Statistics 6 Chapter 6: Linear Equations and Matrix Algebra 8 Chapter 7: Linear Programming: Graphical Solutions
More informationM118 Exam Jam. Contents. Chapter 2: Set Theory 2. Chapter 3: Combinatorics 5. Chapter 4: Probability 7. Chapter 5: Statistics 12
Contents Chapter 2: Set Theory 2 Chapter 3: Combinatorics 5 Chapter 4: Probability 7 Chapter 5: Statistics 12 Chapter 6: Linear Equations and Matrix Algebra 17 Chapter 7: Linear Programming: Graphical
More informationChapter 2: Set Theory 2. Chapter 3: Combinatorics 3. Chapter 4: Probability 4. Chapter 5: Statistics 5
M118 Exam Jam Concise s Contents Chapter 2: Set Theory 2 Chapter 3: Combinatorics 3 Chapter 4: Probability 4 Chapter 5: Statistics 5 Chapter 6: Linear Equations and Matrix Algebra 7 Chapter 7: Linear Programming:
More informationExam III Review Math-132 (Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1, 8.2, 8.3)
1 Exam III Review Math-132 (Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.6, 8.1, 8.2, 8.3) On this exam, questions may come from any of the following topic areas: - Union and intersection of sets - Complement of
More informationIf S = {O 1, O 2,, O n }, where O i is the i th elementary outcome, and p i is the probability of the i th elementary outcome, then
1.1 Probabilities Def n: A random experiment is a process that, when performed, results in one and only one of many observations (or outcomes). The sample space S is the set of all elementary outcomes
More information3.2 Probability Rules
3.2 Probability Rules The idea of probability rests on the fact that chance behavior is predictable in the long run. In the last section, we used simulation to imitate chance behavior. Do we always need
More informationTopic 5: Probability. 5.4 Combined Events and Conditional Probability Paper 1
Topic 5: Probability Standard Level 5.4 Combined Events and Conditional Probability Paper 1 1. In a group of 16 students, 12 take art and 8 take music. One student takes neither art nor music. The Venn
More information(a) Fill in the missing probabilities in the table. (b) Calculate P(F G). (c) Calculate P(E c ). (d) Is this a uniform sample space?
Math 166 Exam 1 Review Sections L.1-L.2, 1.1-1.7 Note: This review is more heavily weighted on the new material this week: Sections 1.5-1.7. For more practice problems on previous material, take a look
More informationProblem # Number of points 1 /20 2 /20 3 /20 4 /20 5 /20 6 /20 7 /20 8 /20 Total /150
Name Student ID # Instructor: SOLUTION Sergey Kirshner STAT 516 Fall 09 Practice Midterm #1 January 31, 2010 You are not allowed to use books or notes. Non-programmable non-graphic calculators are permitted.
More informationPROBABILITY.
PROBABILITY PROBABILITY(Basic Terminology) Random Experiment: If in each trial of an experiment conducted under identical conditions, the outcome is not unique, but may be any one of the possible outcomes,
More informationMath 3201 Unit 1 Set Theory
Math 3201 Unit 1 Set Theory Overview In this unit, we will organize information into. We will use diagrams to illustrate between sets and subsets and use to describe sets. We will determine the in each
More informationBasic Set Concepts (2.1)
1 Basic Set Concepts (2.1) I. Set A collection of objects whose contents can be clearly determined. Capitol letters usually name a set. Elements are the contents in a set. Sets can be described using words,
More informationUNIT 5 ~ Probability: What Are the Chances? 1
UNIT 5 ~ Probability: What Are the Chances? 1 6.1: Simulation Simulation: The of chance behavior, based on a that accurately reflects the phenomenon under consideration. (ex 1) Suppose we are interested
More informationMidterm Review Honors ICM Name: Per: Remember to show work to receive credit! Circle your answers! Sets and Probability
Midterm Review Honors ICM Name: Per: Remember to show work to receive credit! Circle your answers! Unit 1 Sets and Probability 1. Let U denote the set of all the students at Green Hope High. Let D { x
More informationSection 2.4 Bernoulli Trials
Section 2.4 Bernoulli Trials A bernoulli trial is a repeated experiment with the following properties: 1. There are two outcomes of each trial: success and failure. 2. The probability of success in each
More informationChapter 5 : Probability. Exercise Sheet. SHilal. 1 P a g e
1 P a g e experiment ( observing / measuring ) outcomes = results sample space = set of all outcomes events = subset of outcomes If we collect all outcomes we are forming a sample space If we collect some
More informationCombinatorial Analysis
Chapter 1 Combinatorial Analysis STAT 302, Department of Statistics, UBC 1 A starting example: coin tossing Consider the following random experiment: tossing a fair coin twice There are four possible outcomes,
More informationDO NOT WRITE BELOW THIS LINE Total page 1: / 20 points. Total page 2: / 20 points
MATH M118: Finite Mathematics Sample Department Final Examination (The actual final examination will be identical to this sample in length, format, and difficulty.) Directions: Place your name and student
More information2. Linda paid $38 for a jacket that was on sale for 25% of the original price. What was the original price of the jacket?
KCATM 011 Word Problems: Team 1. A restaurant s fixed price dinner includes an appetizer, an entrée, and dessert. If the restaurant offers 4 different types of appetizers, 5 different types of entrees,
More informationChapter 2 PROBABILITY SAMPLE SPACE
Chapter 2 PROBABILITY Key words: Sample space, sample point, tree diagram, events, complement, union and intersection of an event, mutually exclusive events; Counting techniques: multiplication rule, permutation,
More informationMTH 201 Applied Mathematics Sample Final Exam Questions. 1. The augmented matrix of a system of equations (in two variables) is:
MTH 201 Applied Mathematics Sample Final Exam Questions 1. The augmented matrix of a system of equations (in two variables) is: 2 1 6 4 2 12 Which of the following is true about the system of equations?
More informationThe probability of an event is viewed as a numerical measure of the chance that the event will occur.
Chapter 5 This chapter introduces probability to quantify randomness. Section 5.1: How Can Probability Quantify Randomness? The probability of an event is viewed as a numerical measure of the chance that
More informationProbability and Statistics Notes
Probability and Statistics Notes Chapter One Jesse Crawford Department of Mathematics Tarleton State University (Tarleton State University) Chapter One Notes 1 / 71 Outline 1 A Sketch of Probability and
More information{ 10,12,17} { 4, 7,13}
Math 13 L Shipley Exam #3 Review Spring 2011 Exam #3 will cover chapter 6, and the first two sections of chapter 7. This review will give you an idea about what types of concepts will be covered on the
More informationELEG 3143 Probability & Stochastic Process Ch. 1 Probability
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 1 Probability Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Applications Elementary Set Theory Random
More informationIntermediate Math Circles November 8, 2017 Probability II
Intersection of Events and Independence Consider two groups of pairs of events Intermediate Math Circles November 8, 017 Probability II Group 1 (Dependent Events) A = {a sales associate has training} B
More informationUnit 1 Day 1. Set Operations & Venn Diagrams
Unit 1 Day 1 Set Operations & Venn Diagrams Honors ICM Get out your signed syllabus form Get out paper and a pencil for notes! Has everyone accessed the website? Math Riddles Mr. Smith has 4 daughters.
More informationProbability 5-4 The Multiplication Rules and Conditional Probability
Outline Lecture 8 5-1 Introduction 5-2 Sample Spaces and 5-3 The Addition Rules for 5-4 The Multiplication Rules and Conditional 5-11 Introduction 5-11 Introduction as a general concept can be defined
More informationChapter. Probability
Chapter 3 Probability Section 3.1 Basic Concepts of Probability Section 3.1 Objectives Identify the sample space of a probability experiment Identify simple events Use the Fundamental Counting Principle
More informationTopic -2. Probability. Larson & Farber, Elementary Statistics: Picturing the World, 3e 1
Topic -2 Probability Larson & Farber, Elementary Statistics: Picturing the World, 3e 1 Probability Experiments Experiment : An experiment is an act that can be repeated under given condition. Rolling a
More informationMath P (A 1 ) =.5, P (A 2 ) =.6, P (A 1 A 2 ) =.9r
Math 3070 1. Treibergs σιι First Midterm Exam Name: SAMPLE January 31, 2000 (1. Compute the sample mean x and sample standard deviation s for the January mean temperatures (in F for Seattle from 1900 to
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Express the set using the roster method. 1) {x x N and x is greater than 7} 1) A) {8,9,10,...}
More information4/17/2012. NE ( ) # of ways an event can happen NS ( ) # of events in the sample space
I. Vocabulary: A. Outcomes: the things that can happen in a probability experiment B. Sample Space (S): all possible outcomes C. Event (E): one outcome D. Probability of an Event (P(E)): the likelihood
More informationChapter 6. Probability
Chapter 6 robability Suppose two six-sided die is rolled and they both land on sixes. Or a coin is flipped and it lands on heads. Or record the color of the next 20 cars to pass an intersection. These
More informationSenior Math Circles November 19, 2008 Probability II
University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles November 9, 2008 Probability II Probability Counting There are many situations where
More informationChapter 8 Sequences, Series, and Probability
Chapter 8 Sequences, Series, and Probability Overview 8.1 Sequences and Series 8.2 Arithmetic Sequences and Partial Sums 8.3 Geometric Sequences and Partial Sums 8.5 The Binomial Theorem 8.6 Counting Principles
More informationIntroduction to Probability Theory
Introduction to Probability Theory Overview The concept of probability is commonly used in everyday life, and can be expressed in many ways. For example, there is a 50:50 chance of a head when a fair coin
More informationNuevo examen - 02 de Febrero de 2017 [280 marks]
Nuevo examen - 0 de Febrero de 0 [0 marks] Jar A contains three red marbles and five green marbles. Two marbles are drawn from the jar, one after the other, without replacement. a. Find the probability
More information1 The Basic Counting Principles
1 The Basic Counting Principles The Multiplication Rule If an operation consists of k steps and the first step can be performed in n 1 ways, the second step can be performed in n ways [regardless of how
More informationLesson One Hundred and Sixty-One Normal Distribution for some Resolution
STUDENT MANUAL ALGEBRA II / LESSON 161 Lesson One Hundred and Sixty-One Normal Distribution for some Resolution Today we re going to continue looking at data sets and how they can be represented in different
More informationIM3 DEC EXAM PREP MATERIAL DEC 2016
1. Given the line x 3 y 5 =1; Paper 1 - CALCULATOR INACTIVE a. Determine the slope of this line. b. Write the equation of this line in function form. c. Evaluate f ( 12). d. Solve for x if 95 = f (x).
More information1. Use the Fundamental Counting Principle. , that n events, can occur is a 1. a 2. a 3. a n
A Permutations A Permutations and Combinations (pp 96 99) Making an organized list or using a tree diagram are just two of the methods that can help count the number of ways to perform a task Other methods
More information= 2 5 Note how we need to be somewhat careful with how we define the total number of outcomes in b) and d). We will return to this later.
PROBABILITY MATH CIRCLE (ADVANCED /27/203 The likelyhood of something (usually called an event happening is called the probability. Probability (informal: We can calculate probability using a ratio: want
More informationMath II Final Exam Question Bank Fall 2016
Math II Final Exam Question Bank Fall 2016 Name: Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which figure shows the flag on the left after it has been
More informationFall Math 140 Week-in-Review #5 courtesy: Kendra Kilmer (covering Sections 3.4 and 4.1) Section 3.4
Section 3.4 A Standard Maximization Problem has the following properties: The objective function is to be maximized. All variables are non-negative. Fall 2017 Math 140 Week-in-Review #5 courtesy: Kendra
More informationP (A B) P ((B C) A) P (B A) = P (B A) + P (C A) P (A) = P (B A) + P (C A) = Q(A) + Q(B).
Lectures 7-8 jacques@ucsdedu 41 Conditional Probability Let (Ω, F, P ) be a probability space Suppose that we have prior information which leads us to conclude that an event A F occurs Based on this information,
More informationMATH 1310 (College Mathematics for Liberal Arts) - Final Exam Review (Revised: Fall 2016)
MATH 30 (College Mathematics for Liberal Arts) - Final Exam Review (Revised: Fall 206) This Review is comprehensive but should not be the only material used to study for the Final Exam. It should not be
More informationName: Teacher: DO NOT OPEN THE EXAMINATION PAPER UNTIL YOU ARE TOLD BY THE SUPERVISOR TO BEGIN. Mathematics 3201 PRE-PUBLIC EXAMINATION JUNE 2014
Name: Teacher: DO NOT OPEN THE EXAMINATION PAPER UNTIL YOU ARE TOLD BY THE SUPERVISOR TO BEGIN Mathematics 301 PRE-PUBLIC EXAMINATION JUNE 014 Value: 100 Marks Duration: 3 Hours General Instructions This
More informationMath st Homework. First part of Chapter 2. Due Friday, September 17, 1999.
Math 447. 1st Homework. First part of Chapter 2. Due Friday, September 17, 1999. 1. How many different seven place license plates are possible if the first 3 places are to be occupied by letters and the
More informationMath 493 Final Exam December 01
Math 493 Final Exam December 01 NAME: ID NUMBER: Return your blue book to my office or the Math Department office by Noon on Tuesday 11 th. On all parts after the first show enough work in your exam booklet
More information, x {1, 2, k}, where k > 0. Find E(X). (2) (Total 7 marks)
1.) The probability distribution of a discrete random variable X is given by 2 x P(X = x) = 14, x {1, 2, k}, where k > 0. Write down P(X = 2). Show that k = 3. (1) Find E(X). (Total 7 marks) 2.) In a group
More informationI - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability
What is Probability? the chance of an event occuring eg 1classical probability 2empirical probability 3subjective probability Section 2 - Probability (1) Probability - Terminology random (probability)
More informationBasic Concepts of Probability. Section 3.1 Basic Concepts of Probability. Probability Experiments. Chapter 3 Probability
Chapter 3 Probability 3.1 Basic Concepts of Probability 3.2 Conditional Probability and the Multiplication Rule 3.3 The Addition Rule 3.4 Additional Topics in Probability and Counting Section 3.1 Basic
More informationMath Exam 1 Review. NOTE: For reviews of the other sections on Exam 1, refer to the first page of WIR #1 and #2.
Math 166 Fall 2008 c Heather Ramsey Page 1 Math 166 - Exam 1 Review NOTE: For reviews of the other sections on Exam 1, refer to the first page of WIR #1 and #2. Section 1.5 - Rules for Probability Elementary
More informationConditional Probability
Conditional Probability Sometimes our computation of the probability of an event is changed by the knowledge that a related event has occurred (or is guaranteed to occur) or by some additional conditions
More informationMATH STUDENT BOOK. 12th Grade Unit 9
MATH STUDENT BOOK 12th Grade Unit 9 Unit 9 COUNTING PRINCIPLES MATH 1209 COUNTING PRINCIPLES INTRODUCTION 1. PROBABILITY DEFINITIONS, SAMPLE SPACES, AND PROBABILITY ADDITION OF PROBABILITIES 11 MULTIPLICATION
More informationStat 225 Week 1, 8/20/12-8/24/12, Notes: Set Theory
Stat 225 Week 1, 8/20/12-8/24/12, Notes: Set Theory The Fall 2012 Stat 225 T.A.s September 7, 2012 The material in this handout is intended to cover general set theory topics. Information includes (but
More informationSection F Ratio and proportion
Section F Ratio and proportion Ratio is a way of comparing two or more groups. For example, if something is split in a ratio 3 : 5 there are three parts of the first thing to every five parts of the second
More informationLECTURE NOTES by DR. J.S.V.R. KRISHNA PRASAD
.0 Introduction: The theory of probability has its origin in the games of chance related to gambling such as tossing of a coin, throwing of a die, drawing cards from a pack of cards etc. Jerame Cardon,
More informationName: Practice Final Exam May 8, 2012
Math 00 Finite Math Practice Final Exam May 8, 0 Name: Be sure that you have all 7 pages of the test. The exam lasts for hours. The Honor Code is in effect for this examination, including keeping your
More informationName: Teacher: DO NOT OPEN THE EXAMINATION PAPER UNTIL YOU ARE TOLD BY THE SUPERVISOR TO BEGIN. Mathematics 3201 PRE-PUBLIC EXAMINATION JUNE 2014
Name: Teacher: DO NOT OPEN THE EXAMINATION PAPER UNTIL YOU ARE TOLD BY THE SUPERVISOR TO BEGIN Mathematics 3201 PRE-PUBLIC EXAMINATION JUNE 2014 Value: 100 Marks Duration: 3 Hours General Instructions
More informationSTA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6. With SOLUTIONS
STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6 With SOLUTIONS Mudunuru Venkateswara Rao, Ph.D. STA 2023 Fall 2016 Venkat Mu ALL THE CONTENT IN THESE SOLUTIONS PRESENTED IN BLUE AND BLACK
More informationMAT2377. Ali Karimnezhad. Version September 9, Ali Karimnezhad
MAT2377 Ali Karimnezhad Version September 9, 2015 Ali Karimnezhad Comments These slides cover material from Chapter 1. In class, I may use a blackboard. I recommend reading these slides before you come
More information324 Stat Lecture Notes (1) Probability
324 Stat Lecture Notes 1 robability Chapter 2 of the book pg 35-71 1 Definitions: Sample Space: Is the set of all possible outcomes of a statistical experiment, which is denoted by the symbol S Notes:
More informationThe enumeration of all possible outcomes of an experiment is called the sample space, denoted S. E.g.: S={head, tail}
Random Experiment In random experiments, the result is unpredictable, unknown prior to its conduct, and can be one of several choices. Examples: The Experiment of tossing a coin (head, tail) The Experiment
More informationSTAT 201 Chapter 5. Probability
STAT 201 Chapter 5 Probability 1 2 Introduction to Probability Probability The way we quantify uncertainty. Subjective Probability A probability derived from an individual's personal judgment about whether
More information1. Solve the following system of equations. 5x + y = 2z 13 3x + 3z = y 4y 2z = 2x + 12
Math 166 Final Exam Review Note: This review does not cover every concept that could be tested on a final. Please also take a look at previous Week in Reviews for more practice problems. Every instructor
More informationSTA 2023 EXAM-2 Practice Problems. Ven Mudunuru. From Chapters 4, 5, & Partly 6. With SOLUTIONS
STA 2023 EXAM-2 Practice Problems From Chapters 4, 5, & Partly 6 With SOLUTIONS Mudunuru, Venkateswara Rao STA 2023 Spring 2016 1 1. A committee of 5 persons is to be formed from 6 men and 4 women. What
More informationAre Spinners Really Random?
Are Spinners Really Random? 2 2 3 3 2 1 1 1 3 4 4 6 4 5 5 Classroom Strategies Blackline Master IV - 13 Page 193 Spin to Win! 2 5 10 Number of Coins Type of Coin Page 194 Classroom Strategies Blackline
More informationSome Basic Concepts of Probability and Information Theory: Pt. 1
Some Basic Concepts of Probability and Information Theory: Pt. 1 PHYS 476Q - Southern Illinois University January 18, 2018 PHYS 476Q - Southern Illinois University Some Basic Concepts of Probability and
More informationStatistics 100 Exam 2 March 8, 2017
STAT 100 EXAM 2 Spring 2017 (This page is worth 1 point. Graded on writing your name and net id clearly and circling section.) PRINT NAME (Last name) (First name) net ID CIRCLE SECTION please! L1 (MWF
More information$ and det A = 14, find the possible values of p. 1. If A =! # Use your graph to answer parts (i) (iii) below, Working:
& 2 p 3 1. If A =! # $ and det A = 14, find the possible values of p. % 4 p p" Use your graph to answer parts (i) (iii) below, (i) Find an estimate for the median score. (ii) Candidates who scored less
More informationNumber Theory and Counting Method. Divisors -Least common divisor -Greatest common multiple
Number Theory and Counting Method Divisors -Least common divisor -Greatest common multiple Divisors Definition n and d are integers d 0 d divides n if there exists q satisfying n = dq q the quotient, d
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MGF 1106 Exam #2 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Six students, A, B, C, D, E, F, are to give speeches to
More information4. Probability of an event A for equally likely outcomes:
University of California, Los Angeles Department of Statistics Statistics 110A Instructor: Nicolas Christou Probability Probability: A measure of the chance that something will occur. 1. Random experiment:
More informationFinal Practice 1 Date: Section: Name: . Find: n(a) 2) na= ( ) 25
Final Practice 1 Date: Section: Name: 1. Is the set A={ x x is a rational nmber less than 6 } finite or infinite 1) infinite 2. Let A = { 32, 33, 34, 35,...,55,56}. Find: n(a) 2) na= ( ) 25 3. Fill in
More informationTOPIC 12 PROBABILITY SCHEMATIC DIAGRAM
TOPIC 12 PROBABILITY SCHEMATIC DIAGRAM Topic Concepts Degree of Importance References NCERT Book Vol. II Probability (i) Conditional Probability *** Article 1.2 and 1.2.1 Solved Examples 1 to 6 Q. Nos
More informationSTAT 516 Answers Homework 2 January 23, 2008 Solutions by Mark Daniel Ward PROBLEMS. = {(a 1, a 2,...) : a i < 6 for all i}
STAT 56 Answers Homework 2 January 23, 2008 Solutions by Mark Daniel Ward PROBLEMS 2. We note that E n consists of rolls that end in 6, namely, experiments of the form (a, a 2,...,a n, 6 for n and a i
More informationSection 4.2 Basic Concepts of Probability
Section 4.2 Basic Concepts of Probability 2012 Pearson Education, Inc. All rights reserved. 1 of 88 Section 4.2 Objectives Identify the sample space of a probability experiment Identify simple events Use
More informationSouth Pacific Form Seven Certificate
141/1 South Pacific Form Seven Certificate INSTRUCTIONS MATHEMATICS WITH STATISTICS 2015 QUESTION and ANSWER BOOKLET Time allowed: Two and a half hours Write your Student Personal Identification Number
More informationPrinciples of Mathematics 12
Principles of Mathematics 12 Examination Booklet Sample 2007/08 Form A DO NOT OPEN ANY EXAMINATION MATERIALS UNTIL INSTRUCTED TO DO SO. FOR FURTHER INSTRUCTIONS REFER TO THE RESPONSE BOOKLET. Contents:
More informationDiscussion 01. b) What is the probability that the letter selected is a vowel?
STAT 400 Discussion 01 Spring 2018 1. Consider the following experiment: A letter is chosen at random from the word STATISTICS. a) List all possible outcomes and their probabilities. b) What is the probability
More informationProbability the chance that an uncertain event will occur (always between 0 and 1)
Quantitative Methods 2013 1 Probability as a Numerical Measure of the Likelihood of Occurrence Probability the chance that an uncertain event will occur (always between 0 and 1) Increasing Likelihood of
More informationMath 1313 Experiments, Events and Sample Spaces
Math 1313 Experiments, Events and Sample Spaces At the end of this recording, you should be able to define and use the basic terminology used in defining experiments. Terminology The next main topic in
More informationMAT Mathematics in Today's World
MAT 1000 Mathematics in Today's World Last Time We discussed the four rules that govern probabilities: 1. Probabilities are numbers between 0 and 1 2. The probability an event does not occur is 1 minus
More informationHW2 Solutions, for MATH441, STAT461, STAT561, due September 9th
HW2 Solutions, for MATH44, STAT46, STAT56, due September 9th. You flip a coin until you get tails. Describe the sample space. How many points are in the sample space? The sample space consists of sequences
More informationCHAPTER 3 PROBABILITY TOPICS
CHAPTER 3 PROBABILITY TOPICS 1. Terminology In this chapter, we are interested in the probability of a particular event occurring when we conduct an experiment. The sample space of an experiment is the
More informationChapter 3: Probability 3.1: Basic Concepts of Probability
Chapter 3: Probability 3.1: Basic Concepts of Probability Objectives Identify the sample space of a probability experiment and a simple event Use the Fundamental Counting Principle Distinguish classical
More informationCh 14 Randomness and Probability
Ch 14 Randomness and Probability We ll begin a new part: randomness and probability. This part contain 4 chapters: 14-17. Why we need to learn this part? Probability is not a portion of statistics. Instead
More informationChapter 7: Section 7-1 Probability Theory and Counting Principles
Chapter 7: Section 7-1 Probability Theory and Counting Principles D. S. Malik Creighton University, Omaha, NE D. S. Malik Creighton University, Omaha, NE Chapter () 7: Section 7-1 Probability Theory and
More information1 Combinatorial Analysis
ECE316 Notes-Winter 217: A. K. Khandani 1 1 Combinatorial Analysis 1.1 Introduction This chapter deals with finding effective methods for counting the number of ways that things can occur. In fact, many
More informationMath 3201 Sample Exam. PART I Total Value: 50% 1. Given the Venn diagram below, what is the number of elements in both A and B, n(aub)?
Math 0 Sample Eam PART I Total : 50%. Given the Venn diagram below, what is the number of elements in both A and B, n(aub)? 6 8 A green white black blue red ellow B purple orange. Given the Venn diagram
More informationLecture Lecture 5
Lecture 4 --- Lecture 5 A. Basic Concepts (4.1-4.2) 1. Experiment: A process of observing a phenomenon that has variation in its outcome. Examples: (E1). Rolling a die, (E2). Drawing a card form a shuffled
More informationSection 7.2 Definition of Probability
Section 7.2 Definition of Probability Question: Suppose we have an experiment that consists of flipping a fair 2-sided coin and observing if the coin lands on heads or tails? From section 7.1 we should
More informationSolutionbank S1 Edexcel AS and A Level Modular Mathematics
file://c:\users\buba\kaz\ouba\s1_5_a_1.html Exercise A, Question 1 For each of the following experiments, identify the sample space and find the probability of the event specified. Throwing a six sided
More informationMGF 1106: Exam 1 Solutions
MGF 1106: Exam 1 Solutions 1. (15 points total) True or false? Explain your answer. a) A A B Solution: Drawn as a Venn diagram, the statement says: This is TRUE. The union of A with any set necessarily
More informationProbability, Conditional Probability and Bayes Rule IE231 - Lecture Notes 3 Mar 6, 2018
Probability, Conditional Probability and Bayes Rule IE31 - Lecture Notes 3 Mar 6, 018 #Introduction Let s recall some probability concepts. Probability is the quantification of uncertainty. For instance
More information1. Determine whether each of the following stochastic matrices is a transition matrix for a Markov process that is regular, absorbing, or neither.
Math 166 Final Exam Review Note: This review does not cover every concept that could be tested on a final. Please also take a look at previous Week in Reviews for more practice problems. Every instructor
More informationSenior Math Circles March 3, 2010 Counting Techniques and Probability II
1 University of Waterloo Faculty of Mathematics Senior Math Circles March 3, 2010 Counting Techniques and Probability II Centre for Education in Mathematics and Computing Counting Rules Multiplication
More information