2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.
|
|
- Gwendoline Manning
- 7 years ago
- Views:
Transcription
1 Math 142 September 27, How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are re-arranged. How many of them start with the letter H? The first letter is fixed, leaving three other letters to re-arrange. Hence, 3! = 6 ways. 4. An password contains 4 letters (all lower case, all 26 are allowed) followed by 4 numbers (digits 0-9). How many passwords could be created? (26)(26)(26)(26)(10)(10)(10)(10) = 4,569,760,000. This is the multiplication principle. 5. A race consists of 10 runners. a) How many ways can 1 st -2 nd -3 rd place be filled? (10)(9)(8) = 720. This is also P(10,3). b) Of the remaining seven runners, the next 4 get a ribbon. How many ways can the ribbons be handed out? Since order does not matter, use combination: C(7,4) = 35 ways. 6. A computer byte is a string of 8 digits consisting of 0s and 1s only, for example, is a byte. How many possible bytes are there? 2 =256. Here, use the multiplication principle. 7. From a group of 8 people, a committee of 5 is to be formed. How many committees are possible? Since a committee does not imply order matters, use combination: C(8,5) = From a group of 10 men and 12 women, a committee of 3 men and 4 women is to be formed. How many committees are possible? Since order does not matter, use combinations: Men: C(10,3) = 120, women: C(12,4) = 495. There will be (120)(495) = 59,400 possible committees. 9. Evaluate!. Hint: your calculator may freeze. Try simplifying first.!! = =10,100. Don t depend on your calculator to do big factorials like this. It will! freeze. Instead, reduce as shown, and cancel. 10. How many ways can three six-sided dice be rolled? (6)(6)(6) = 216 ways. (Mult. Princ.) 11. If two dice are rolled, how many of the rolls show a sum of 9? 4 of them do. Hint: write out all 36 outcomes and count off the ones that sum to 9. Writing out the entire sample space is a very big hint, if you catch my drift. 12. If repeats are allowed, you use the multiplication principle. If repeats are not allowed and order matters, you use permutation. If repeats are not allowed and order does not matter, you use combination.
2 MAT 142 Intro to Probability Worksheet (Oct. 4, 2011) 1. A jar of M&M candies contains 12 brown, 4 yellow, 2 blue, 5 red, 3 green and 4 orange. You select one at random. Find the probability that you select one that is: (Leave answers as fractions) a. Brown: 12/30 b. Green or orange: 7/30 c. Not red: 25/30 d. Yellow and blue: 0. It s impossible to select one candy that is both colors simultaneously. Read the questions carefully. And indicates simultaneously, and or indicated union. 2. Shoppers at a local department store were asked to complete a survey of their shopping experience. The results are shown in the table below: Satisfied Not satisfied Total Made a purchase Did not make a purchase Totals a. What is the probability that a shopper selected at random made a purchase? 624/1522. Reduction is not necessary. b. What are the odds that a shopper selected at random was satisfied with the service? 845:677 Must be in odds form! 3. In a study of fatal car accidents, some accidents were attributed to high speed or drunk driving. The probability that the fatality was attributed to high speeds was 0.57 The probability that the fatality was attributed to drunk driving was 0.62 The probability that high speed and drunk driving caused the accident was 0.41 Hint: Make a Venn Diagram a. What is the probability that a fatal accident will be attributed to high speeds or drunk driving? This is the union. b. What is the probability that a fatal accident will be attributed to neither cause? 0.22 c. What is the probability that the fatality will be attributed to drunk driving, but not high speeds? If a single card is selected from a standard 52 card deck, what is the probability that a red face card is selected? 26/52 5. If a single card is drawn from a standard 52 card deck, what is the probability that we obtain a face card or a red card? P(Red) = 26/52, P(Face) = 12/52. P(Both) = 6/52. Therefore, P(Red or Face) = P(Red) + P(Face) P(Both) =26/ /52 6/52 = 32/52.
3 6. If 5 cards are drawn from a standard 52 card deck, what is the probability that the result is 3 red cards and 2 black cards? (, ) (, ) (, ) =0.325, about a 32.5% probability. 7. If the odds for E occurring is 2:5, what is the probability of E? 2/7. Remember, Odds for is written success:failure. Probability is success/total. There are 2 ways to succeed, 5 ways to fail, 7 total. Also, use correct notation. Fractions in place of the colon for odds will be marked wrong! As will the other way around. 8. If a fair coin is tossed three times, write down the sample space S. Then find the probability that exactly two heads occur. S = {ttt, tth, tht, thh, htt, hth, hht, hhh}, P(2 heads) = 3/8 9. If the probability of E occurring is 0.26, what is the probability E does not occur? = 0.74 Look for key words that indicate an ordering or not. Look for phrases like with repetition, which means items can be re-used, or without repetition, which means items cannot be re-used. The formulas for permutation and combination are given but know how to find them on your calculator too. Know how to reduce a large factorial expression. Some calculators will freeze. Always be on the lookout for the words or indicating union, and and indicating intersection (both). A lot of you are still making rookie mistakes by misreading the problems and not picking up on these fine details. If a problem asks for probability, leave your answer in fraction or decimal form. If a problem asks for odds, leave the answer in the colon format. Read the question carefully: odds for is success:failure, odds against is failure:success. Probability is always success/total and they are related by failure + success = total. Draw Venns, Tables or Trees as needed. They help a lot. In conditional probabilities, remember to reduce the sample space accordingly. Read the questions carefully. Look for words like given to indicate a conditional probability. READ ALL QUESTIONS SLOWLY AND CAREFULLY! This is not a race.
4 Math 142 October 11, Two candidates, Smith and Wilson, are running for mayor. The voting breakdown is shown below in the table: Rep (R) Dem (D) Indep (I) Total Smith (S) Wilson (W) Total A voter is selected at random. Determine these probabilities: a) The probability the voter voted for Smith, given the voter was Republican. ( ):72/118 b) The probability the voter was Independent, given the voter voted for Wilson. ( ) 22/129 c) Determine ( ): 68/129. d) Determine ( ): 110/ A jar has 15 red, 20 orange and 22 blue candies. Two candies are drawn without replacement. Find these probabilities: a) The second candy is blue given the first was red. 22/56 b) The second is orange given the first was blue. 20/56 c) Both candies were blue. 22/57 times 21/56 = or 11/76 (both are the same). d) (2 pts extra credit) Both candies are of different color. It s easier to figure the probability of getting the same color first: Two blues is (from part c). Two reds is 0.066, and two oranges is The probability all are the same color is the sum: = Therefore, the probability the two candies are different color is = Tourists to Las Vegas are surveyed. 52% visit Hoover Dam, 31% visit the Strip, and 14% visit both the Strip and Hoover Dam. Determine the following probabilities. You may leave your answer in decimal format. Hint: draw a Venn. a) The probability a tourist visited the Hoover Dam given the tourist visited the Strip. 0.14/0.31 = 0.45 b) The probability a tourist visited Strip given the tourist visited Hoover Dam. 0.14/0.52 = 0.27 c) The probability a tourist did not visit Hoover Dam given the tourist did not visit the Strip. The probability a tourist did not visit the strip is The probability the tourist visited neither place is Thus, the probability is 0.31/0.69 = 0.45
5 4. You roll a single die once. If it lands a 6, you get $10. Otherwise, you get nothing. The cost to play is free. What is the expected value of one roll of this die? There is a 1/6 chance of winning $10, 5/6 chance of winning nothing. The EV is (1/6)(10) + (5/6)(0) = 10/6 = $ A bag has 20 tokens in it. They all feel the same. One is gold colored and worth $20. Two are silver colored and worth $5 each. The other 17 are worth nothing. For $3, you can reach in and randomly grab one token. What is the expected value of this game? Subtract out the $3 cost when figuring the winnings: You have a 1/20 chance of netting $17, a 2/20 chance of netting $2, and a 17/20 chance of losing your $3. The EV is (1/20)(17) + (2/20)(2) + (17/20)(-3) = -30/20 = This means on average, you ll lose $1.50 per game. In the long term, it s a bad game. 6. A lottery sells 100 tickets for $1 each. One ticket is the winner, with a jackpot of $75. The rest are worthless, and you lose your $1. Your friend s bright idea is to buy all the tickets. Use Expected Value to explain why this is a lousy idea. Show your calculation and give a one sentence explanation. The EV is (1/100)(74) + (99/100)(-1) = -25/100 = -$0.25. The EV is negative so you ll lose in the long term. If you spent $100 to get $75, you have lost $ A roulette wheel has 38 slots. The cost to play is $1. If the ball lands in a slot you picked, you win $36. Otherwise, you lose the $1. a) Find the Expected Value of one play. (1/38)(35) + (37/38)(-1) = -2/38 = -$ Is this game in your favor? (Y/N) No. You ll lose on average a little over five cents per game in the long term. b) If you played 100 games, how much up or down can you expect to be? 100 times the EV; (100)(- $0.053) = -$5.30. You ll be down about $5.30. c) What is the fair price to play this game? Fair price = cost + EV = $1 + (-$0.053) = $0.947, or about 95 cents.
Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314
Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space
More informationChapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.
Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive
More informationProbability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1 www.math12.com
Probability --QUESTIONS-- Principles of Math - Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..
More informationChapter 4 - Practice Problems 1
Chapter 4 - Practice Problems SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) Compare the relative frequency formula
More informationChapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.
Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More informationV. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE
V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay $ to play. A penny and a nickel are flipped. You win $ if either
More informationFind the indicated probability. 1) If a single fair die is rolled, find the probability of a 4 given that the number rolled is odd.
Math 0 Practice Test 3 Fall 2009 Covers 7.5, 8.-8.3 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. ) If a single
More informationContemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4?
Contemporary Mathematics- MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than
More information36 Odds, Expected Value, and Conditional Probability
36 Odds, Expected Value, and Conditional Probability What s the difference between probabilities and odds? To answer this question, let s consider a game that involves rolling a die. If one gets the face
More informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6-sided dice. What s the probability of rolling at least one 6? There is a 1
More informationProbability. Section 9. Probability. Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space)
Probability Section 9 Probability Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space) In this section we summarise the key issues in the basic probability
More informationMA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.
MA 5 Lecture 4 - Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More informationSection 7C: The Law of Large Numbers
Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half
More informationAP Stats - Probability Review
AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose
More informationChapter 4 - Practice Problems 2
Chapter - Practice Problems 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. 1) If you flip a coin three times, the
More informationExam. Name. How many distinguishable permutations of letters are possible in the word? 1) CRITICS
Exam Name How many distinguishable permutations of letters are possible in the word? 1) CRITICS 2) GIGGLE An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm,
More informationElementary Statistics and Inference. Elementary Statistics and Inference. 16 The Law of Averages (cont.) 22S:025 or 7P:025.
Elementary Statistics and Inference 22S:025 or 7P:025 Lecture 20 1 Elementary Statistics and Inference 22S:025 or 7P:025 Chapter 16 (cont.) 2 D. Making a Box Model Key Questions regarding box What numbers
More informationProbability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes
Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Practice Test Chapter 9 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the odds. ) Two dice are rolled. What are the odds against a sum
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 informationEXAM. Exam #3. Math 1430, Spring 2002. April 21, 2001 ANSWERS
EXAM Exam #3 Math 1430, Spring 2002 April 21, 2001 ANSWERS i 60 pts. Problem 1. A city has two newspapers, the Gazette and the Journal. In a survey of 1, 200 residents, 500 read the Journal, 700 read the
More informationCh. 13.2: Mathematical Expectation
Ch. 13.2: Mathematical Expectation Random Variables Very often, we are interested in sample spaces in which the outcomes are distinct real numbers. For example, in the experiment of rolling two dice, we
More informationProbability and Expected Value
Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are
More information1 Combinations, Permutations, and Elementary Probability
1 Combinations, Permutations, and Elementary Probability Roughly speaking, Permutations are ways of grouping things where the order is important. Combinations are ways of grouping things where the order
More information2. Three dice are tossed. Find the probability of a) a sum of 4; or b) a sum greater than 4 (may use complement)
Probability Homework Section P4 1. A two-person committee is chosen at random from a group of four men and three women. Find the probability that the committee contains at least one man. 2. Three dice
More informationHoover High School Math League. Counting and Probability
Hoover High School Math League Counting and Probability Problems. At a sandwich shop there are 2 kinds of bread, 5 kinds of cold cuts, 3 kinds of cheese, and 2 kinds of dressing. How many different sandwiches
More informationA probability experiment is a chance process that leads to well-defined outcomes. 3) What is the difference between an outcome and an event?
Ch 4.2 pg.191~(1-10 all), 12 (a, c, e, g), 13, 14, (a, b, c, d, e, h, i, j), 17, 21, 25, 31, 32. 1) What is a probability experiment? A probability experiment is a chance process that leads to well-defined
More informationPERMUTATIONS AND COMBINATIONS
PERMUTATIONS AND COMBINATIONS Mathematics for Elementary Teachers: A Conceptual Approach New Material for the Eighth Edition Albert B. Bennett, Jr., Laurie J. Burton and L. Ted Nelson Math 212 Extra Credit
More informationProbability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to
More informationReview for Test 2. Chapters 4, 5 and 6
Review for Test 2 Chapters 4, 5 and 6 1. You roll a fair six-sided die. Find the probability of each event: a. Event A: rolling a 3 1/6 b. Event B: rolling a 7 0 c. Event C: rolling a number less than
More informationLecture 13. Understanding Probability and Long-Term Expectations
Lecture 13 Understanding Probability and Long-Term Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).
More informationIntroduction to Discrete Probability. Terminology. Probability definition. 22c:19, section 6.x Hantao Zhang
Introduction to Discrete Probability 22c:19, section 6.x Hantao Zhang 1 Terminology Experiment A repeatable procedure that yields one of a given set of outcomes Rolling a die, for example Sample space
More informationChapter 16: law of averages
Chapter 16: law of averages Context................................................................... 2 Law of averages 3 Coin tossing experiment......................................................
More informationProbability and Venn diagrams UNCORRECTED PAGE PROOFS
Probability and Venn diagrams 12 This chapter deals with further ideas in chance. At the end of this chapter you should be able to: identify complementary events and use the sum of probabilities to solve
More informationSession 8 Probability
Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome
More informationActivities/ Resources for Unit V: Proportions, Ratios, Probability, Mean and Median
Activities/ Resources for Unit V: Proportions, Ratios, Probability, Mean and Median 58 What is a Ratio? A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a
More informationCurriculum Design for Mathematic Lesson Probability
Curriculum Design for Mathematic Lesson Probability This curriculum design is for the 8th grade students who are going to learn Probability and trying to show the easiest way for them to go into this class.
More informationThe overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES
INTRODUCTION TO CHANCE VARIABILITY WHAT DOES THE LAW OF AVERAGES SAY? 4 coins were tossed 1600 times each, and the chance error number of heads half the number of tosses was plotted against the number
More informationPattern matching probabilities and paradoxes A new variation on Penney s coin game
Osaka Keidai Ronshu, Vol. 63 No. 4 November 2012 Pattern matching probabilities and paradoxes A new variation on Penney s coin game Yutaka Nishiyama Abstract This paper gives an outline of an interesting
More informationExpected Value. 24 February 2014. Expected Value 24 February 2014 1/19
Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery
More informationSection 6.2 Definition of Probability
Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will
More informationBasic Probability Theory II
RECAP Basic Probability heory II Dr. om Ilvento FREC 408 We said the approach to establishing probabilities for events is to Define the experiment List the sample points Assign probabilities to the sample
More informationStat 20: Intro to Probability and Statistics
Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. Childers-Day UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
STATISTICS/GRACEY PRACTICE TEST/EXAM 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the given random variable as being discrete or continuous.
More informationMathematics Higher Level
Mathematics Higher Level for the IB Diploma Exam Preparation Guide Paul Fannon, Vesna Kadelburg, Ben Woolley, Stephen Ward INTRODUCTION ABOUT THIS BOOK If you are using this book, you re probably getting
More informationMath 3C Homework 3 Solutions
Math 3C Homework 3 s Ilhwan Jo and Akemi Kashiwada ilhwanjo@math.ucla.edu, akashiwada@ucla.edu Assignment: Section 2.3 Problems 2, 7, 8, 9,, 3, 5, 8, 2, 22, 29, 3, 32 2. You draw three cards from a standard
More informationSection 6.1 Discrete Random variables Probability Distribution
Section 6.1 Discrete Random variables Probability Distribution Definitions a) Random variable is a variable whose values are determined by chance. b) Discrete Probability distribution consists of the values
More informationQuestion of the Day. Key Concepts. Vocabulary. Mathematical Ideas. QuestionofDay
QuestionofDay Question of the Day What is the probability that in a family with two children, both are boys? What is the probability that in a family with two children, both are boys, if we already know
More informationCh. 13.3: More about Probability
Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the
More informationChapter 5 A Survey of Probability Concepts
Chapter 5 A Survey of Probability Concepts True/False 1. Based on a classical approach, the probability of an event is defined as the number of favorable outcomes divided by the total number of possible
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,
More informationStatistics 100A Homework 2 Solutions
Statistics Homework Solutions Ryan Rosario Chapter 9. retail establishment accepts either the merican Express or the VIS credit card. total of percent of its customers carry an merican Express card, 6
More informationExam Style Questions. Revision for this topic. Name: Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser
Name: Exam Style Questions Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser You may use tracing paper if needed Guidance 1. Read each question carefully before you begin answering
More informationMath 202-0 Quizzes Winter 2009
Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile
More informationBayesian Tutorial (Sheet Updated 20 March)
Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that
More informationEDEXCEL FUNCTIONAL SKILLS PILOT
EEXEL FUNTIONAL SKILLS PILOT Maths Level hapter 7 Working with probability SETION K Measuring probability 9 Remember what you have learned 3 raft for Pilot Functional Maths Level hapter 7 Pearson Education
More informationHow To Know When A Roulette Wheel Is Random
226 Part IV Randomness and Probability Chapter 14 From Randomness to Probability 1. Roulette. If a roulette wheel is to be considered truly random, then each outcome is equally likely to occur, and knowing
More informationA Few Basics of Probability
A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study
More informationFor two disjoint subsets A and B of Ω, say that A and B are disjoint events. For disjoint events A and B we take an axiom P(A B) = P(A) + P(B)
Basic probability A probability space or event space is a set Ω together with a probability measure P on it. This means that to each subset A Ω we associate the probability P(A) = probability of A with
More informationWe rst consider the game from the player's point of view: Suppose you have picked a number and placed your bet. The probability of winning is
Roulette: On an American roulette wheel here are 38 compartments where the ball can land. They are numbered 1-36, and there are two compartments labeled 0 and 00. Half of the compartments numbered 1-36
More informationMethods Used for Counting
COUNTING METHODS From our preliminary work in probability, we often found ourselves wondering how many different scenarios there were in a given situation. In the beginning of that chapter, we merely tried
More informationWeek 5: Expected value and Betting systems
Week 5: Expected value and Betting systems Random variable A random variable represents a measurement in a random experiment. We usually denote random variable with capital letter X, Y,. If S is the sample
More informationCh5: Discrete Probability Distributions Section 5-1: Probability Distribution
Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.
More informationSection 6-5 Sample Spaces and Probability
492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)
More informationQuestion: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationPROBABILITY. Chapter. 0009T_c04_133-192.qxd 06/03/03 19:53 Page 133
0009T_c04_133-192.qxd 06/03/03 19:53 Page 133 Chapter 4 PROBABILITY Please stand up in front of the class and give your oral report on describing data using statistical methods. Does this request to speak
More informationMath Games For Skills and Concepts
Math Games p.1 Math Games For Skills and Concepts Original material 2001-2006, John Golden, GVSU permission granted for educational use Other material copyright: Investigations in Number, Data and Space,
More information14.4. Expected Value Objectives. Expected Value
. Expected Value Objectives. Understand the meaning of expected value. 2. Calculate the expected value of lotteries and games of chance.. Use expected value to solve applied problems. Life and Health Insurers
More informationHomework Assignment #2: Answer Key
Homework Assignment #2: Answer Key Chapter 4: #3 Assuming that the current interest rate is 3 percent, compute the value of a five-year, 5 percent coupon bond with a face value of $,000. What happens if
More informationThat s Not Fair! ASSESSMENT #HSMA20. Benchmark Grades: 9-12
That s Not Fair! ASSESSMENT # Benchmark Grades: 9-12 Summary: Students consider the difference between fair and unfair games, using probability to analyze games. The probability will be used to find ways
More information7.S.8 Interpret data to provide the basis for predictions and to establish
7 th Grade Probability Unit 7.S.8 Interpret data to provide the basis for predictions and to establish experimental probabilities. 7.S.10 Predict the outcome of experiment 7.S.11 Design and conduct an
More informationPROBABILITY. The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE
PROBABILITY 53 Chapter 3 PROBABILITY The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE 3. Introduction In earlier Classes, we have studied the probability as
More informationDecision Making Under Uncertainty. Professor Peter Cramton Economics 300
Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate
More informationReady, Set, Go! Math Games for Serious Minds
Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 -
More informationChapter 13 & 14 - Probability PART
Chapter 13 & 14 - Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14 - Probability 1 / 91 Why Should We Learn Probability Theory? Dr. Joseph
More informationChapter 3. Probability
Chapter 3 Probability Every Day, each us makes decisions based on uncertainty. Should you buy an extended warranty for your new DVD player? It depends on the likelihood that it will fail during the warranty.
More informationDecimals and Percentages
Decimals and Percentages Specimen Worksheets for Selected Aspects Paul Harling b recognise the number relationship between coordinates in the first quadrant of related points Key Stage 2 (AT2) on a line
More informationMathematical goals. Starting points. Materials required. Time needed
Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about
More information(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball.
Examples for Chapter 3 Probability Math 1040-1 Section 3.1 1. Draw a tree diagram for each of the following situations. State the size of the sample space. (a) You flip a coin three times. (b) You draw
More informationChapter 16. Law of averages. Chance. Example 1: rolling two dice Sum of draws. Setting up a. Example 2: American roulette. Summary.
Overview Box Part V Variability The Averages Box We will look at various chance : Tossing coins, rolling, playing Sampling voters We will use something called s to analyze these. Box s help to translate
More informationConditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of conditional probability and independence
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More informationIntroduction to Probability
3 Introduction to Probability Given a fair coin, what can we expect to be the frequency of tails in a sequence of 10 coin tosses? Tossing a coin is an example of a chance experiment, namely a process which
More informationPROBABILITY. SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and 1.
PROBABILITY SIMPLE PROBABILITY SIMPLE PROBABILITY is the likelihood that a specific event will occur, represented by a number between 0 and. There are two categories of simple probabilities. THEORETICAL
More informationBasic Probability. Probability: The part of Mathematics devoted to quantify uncertainty
AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.
More information2.5 Conditional Probabilities and 2-Way Tables
2.5 Conditional Probabilities and 2-Way Tables Learning Objectives Understand how to calculate conditional probabilities Understand how to calculate probabilities using a contingency or 2-way table It
More informationPERMUTATIONS and COMBINATIONS. If the order doesn't matter, it is a Combination. If the order does matter it is a Permutation.
Page 1 PERMUTATIONS and COMBINATIONS If the order doesn't matter, it is a Combination. If the order does matter it is a Permutation. PRACTICE! Determine whether each of the following situations is a Combination
More informationChapter 7 Probability and Statistics
Chapter 7 Probability and Statistics In this chapter, students develop an understanding of data sampling and making inferences from representations of the sample data, with attention to both measures of
More informationUsing Permutations and Combinations to Compute Probabilities
Using Permutations and Combinations to Compute Probabilities Student Outcomes Students distinguish between situations involving combinations and situations involving permutations. Students use permutations
More informationDefinition and Calculus of Probability
In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the
More informationDetermine the empirical probability that a person selected at random from the 1000 surveyed uses Mastercard.
Math 120 Practice Exam II Name You must show work for credit. 1) A pair of fair dice is rolled 50 times and the sum of the dots on the faces is noted. Outcome 2 4 5 6 7 8 9 10 11 12 Frequency 6 8 8 1 5
More informationAMS 5 CHANCE VARIABILITY
AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and
More informationProbability and Compound Events Examples
Probability and Compound Events Examples 1. A compound event consists of two or more simple events. ossing a die is a simple event. ossing two dice is a compound event. he probability of a compound event
More informationHigh School Statistics and Probability Common Core Sample Test Version 2
High School Statistics and Probability Common Core Sample Test Version 2 Our High School Statistics and Probability sample test covers the twenty most common questions that we see targeted for this level.
More informationProbability definitions
Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a data-generating
More informationUNIT 7A 118 CHAPTER 7: PROBABILITY: LIVING WITH THE ODDS
11 CHAPTER 7: PROBABILITY: LIVING WITH THE ODDS UNIT 7A TIME OUT TO THINK Pg. 17. Birth orders of BBG, BGB, and GBB are the outcomes that produce the event of two boys in a family of. We can represent
More informationMath 408, Actuarial Statistics I, Spring 2008. Solutions to combinatorial problems
, Spring 2008 Word counting problems 1. Find the number of possible character passwords under the following restrictions: Note there are 26 letters in the alphabet. a All characters must be lower case
More informationAll the examples in this worksheet and all the answers to questions are available as answer sheets or videos.
BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Numbers 3 In this section we will look at - improper fractions and mixed fractions - multiplying and dividing fractions - what decimals mean and exponents
More informationEveryday Math Online Games (Grades 1 to 3)
Everyday Math Online Games (Grades 1 to 3) FOR ALL GAMES At any time, click the Hint button to find out what to do next. Click the Skip Directions button to skip the directions and begin playing the game.
More information13.0 Central Limit Theorem
13.0 Central Limit Theorem Discuss Midterm/Answer Questions Box Models Expected Value and Standard Error Central Limit Theorem 1 13.1 Box Models A Box Model describes a process in terms of making repeated
More information