Chapter 15. Probability, continued. The sample space of a random experiment is the set of all possible outcomes of the experiment.


 Janel Elaine Gregory
 2 years ago
 Views:
Transcription
1 Chapter 15. Probability, continued Review: The sample space of a random experiment is the set of all possible outcomes of the experiment. We have to count outcomes; a useful tool is the multiplication rule: count outcomes in two or more stages; suppose in each stage the number of choices doesn't depend on choices made in other stages; then the number of outcomes is the product of the number of choices in each stage. Example: five candidates in an election, the one with the most votes is President, 2 nd most is VP, 3 rd most is Secretary. Outcomes: Section 3. Permutations and combinations A standard deck of cards has four suits (Hearts, Diamonds, Spades and Clubs) and 13 values (Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2), making 4 * 13 = 52 cards. A poker hand consists of five cards. How many different poker hands are there? There are 52 * 51 * 50 * 49 * 48 = 311,875,200 hands, if we pay attention to the order in which the cards are dealt. 1
2 In general, if you have n items, and you choose r of them, and you care about the order in which you choose them, then the number of choices is called the number of permutations on n objects taken r at a time = np r = n * (n 1) * (n 2) *... * (n r + 1) (notice that there are r numbers being multiplied). So the number of 4 member committees (in which order matters) from 35 people = 35 P 4 = 35 * 34 * 33 * 32. The number of 5 card hands (in which order matters) from 52 cards = 52 P 5 = 52 * 51 * 50 * 49 * 48. In draw poker, the order we receive the cards doesn t matter. When we counted the ordered hands above, we counted each unordered hand many times. How many ordered hands does each unordered hand give rise to? There are 5 * 4 * 3 * 2 * 1 = 5! = 120 orderings of a five card hand. So the number of unordered hands is 52 * 51 * 50 * 49 * 48)/( 5 * 4 * 3 * 2 * 1) = 311,875,200/120 = 2,598,960 In general, if you have n items, and you choose r of them, and you don t care about the order in which you choose them, then the number of choices is called the number of combinations of n objects taken r at a time = nc r = n P r / r! = [n * (n 1) * (n 2) *... * (n r + 1)]/[r * (r 1) * (r 2) *...*1] 2
3 Suppose there are 10 horses entered in a race. In how many ways can one pick a) the top three finishers regardless of order? b) the first, second, and thirdplace finishers in the race? b) asks for the number of permutations = 10 P 3 = 10 * 9 * 8 = 720. a) asks for the number of combinations = 10 C 3 = 720/3! = 720/6 = 120. Look at book's problems and determine if you are asked for permutations or combinations 3
4 Sections 4. Probability spaces The probability of something happening is a number between 0 and 1; if a random experiment is carried out repeatedly, the probability of an outcome is supposed to represent the fraction of the experiments for which that outcome is expected to occur. The book gives the example of a basketball player who, over a long period of time, shoots 2135/2362 =.904 of his free throws. So one might conclude that if he has a free throw attempt (that s the random experiment in this case),.904 is the probability of success and =.096 is the probability of failure. In general, for some random experiment, you must first determine the sample space S = {o 1,...,o N }. Next you must assign a probability Pr (o i ) to each outcome o i. Each probability is between 0 and 1, and Pr (o 1 ) + + Pr (o N ) = 1. This is called a probability assignment. If we toss a fair coin two times, and note on each toss whether it is heads or tails, then the sample space S = {HH, HT, TH, TT}. Since each of these outcomes is equally likely, Pr (HH) = Pr (HT) = Pr (TH) = Pr (TT) = ¼ =.25. If we toss a fair coin two times, and note the number of heads, then the sample space S = {2, 1, 0}. But these outcomes are not equally likely, since HT and TH both result in an outcome of 1. So Pr (2) = Pr (0) =.25, and Pr (1) =.5. 4
5 p580 #38b. Consider the sample space S = {o 1, o 2, o 3, o 4 }. Suppose you are given Pr (o 1 ) + Pr (o 2 ) = Pr (o 3 ) + Pr (o 4 ). If Pr (o 1 ) =.15 and Pr (o 3 ) =.22, find the probability assignment. (So we need to find Pr (o 2 ) and Pr (o 4 )). Answer: Pr (o 1 ) + Pr (o 2 ) = Pr (o 3 ) + Pr (o 4 ) =.5 (since they are equal and add to 1), so Pr (o 2 ) = =.35 and Pr (o 4 ) = =.28. There are 8 players (call them P 1,, P 8 ) entered in a chess tournament. Our random experiment is to hold the tournament and note the winner. The sample space is S = { P 1,, P 8 }. We are told that P 1 has a 25% chance of winning, P 2 has a 15% chance, P 3 has a 5% chance, and all others have an equal chance. Find the probability assignment. Answer: We are given that Pr (P 1 ) =.25, Pr (P 2 ) =.15 and Pr (P 3 ) =.05. So the probability of one of these three players winning is Pr (P 1 ) + Pr (P 2 ) + Pr (P 3 ) = =.45. Thus the probability of one of the other players winning is Pr (P 4 ) + Pr (P 5 ) + Pr (P 6 ) + Pr (P 7 ) + Pr (P 8 ) = =.55. Also, Pr (P 4 ) = = Pr (P 8 ), so each is 1/5 of.55, that is Pr (P 4 ) = = Pr (P 8 ) =.55/5 =.11. 5
6 Events. An event is any subset of the sample space. The probability of an event is the sum of the probabilities of the outcomes in that event. If we toss a fair coin two times, and note on each toss whether it is heads or tails, then the sample space S = {HH, HT, TH, TT}. Here is a listing of some of the events: Description of event Set of outcomes Probability Toss two heads {HH}.25 Toss one head {HT, TH}.5 Toss at least one head {HH, HT, TH}.75 Toss two heads or two tails {HH, TT}.5 Toss at most two heads {HH, HT, TH, TT} 1 Toss three heads { } 0 The last two events listed are called the certain event and the impossible event. p.580 # 44. Consider the random experiment where a student takes a fourquestion truefalse quiz. Write out the event described by each of the following statements: a) Exactly two of the answers given are T s (T for true). b) At least two of the answers given are T s. c) At most two of the answers given are T s. d) The first two answers given are T s. Answers: a) {TTFF, TFTF, TFFT, FTTF, FTFT, FFTT} b) {TTFF, TFTF, TFFT, FTTF, FTFT, FFTT, TTTF, TTFT, TFTT, FTTT, TTTT} c) {FFFF, TFFF, FTFF, FFTF, FFFT, TTFF, TFTF, TFFT, FTTF, FTFT, FFTT} d) {TTFF, TTTF, TTFT, TTTT} 6
Remember to leave your answers as unreduced fractions.
Probability Worksheet 2 NAME: Remember to leave your answers as unreduced fractions. We will work with the example of picking poker cards out of a deck. A poker deck contains four suits: diamonds, hearts,
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 informationCombinations If 5 sprinters compete in a race, how many different ways can the medals for first, second and third place, be awarded
Combinations If 5 sprinters compete in a race, how many different ways can the medals for first, second and third place, be awarded If 5 sprinters compete in a race and the fastest 3 qualify for the relay
More informationLECTURE 3. Probability Computations
LECTURE 3 Probability Computations Pg. 67, #42 is one of the hardest problems in the course. The answer is a simple fraction there should be a simple way to do it. I don t know a simple way I used the
More informationPROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA
PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet
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 informationSECTION 105 Multiplication Principle, Permutations, and Combinations
105 Multiplication Principle, Permutations, and Combinations 761 54. Can you guess what the next two rows in Pascal s triangle, shown at right, are? Compare the numbers in the triangle with the binomial
More informationMath 2001 Homework #10 Solutions
Math 00 Homework #0 Solutions. Section.: ab. For each map below, determine the number of southerly paths from point to point. Solution: We just have to use the same process as we did in building Pascal
More informationMATH 2200 PROBABILITY AND STATISTICS M2200FL083.1
MATH 2200 PROBABILITY AND STATISTICS M2200FL083.1 In almost all problems, I have given the answers to four significant digits. If your answer is slightly different from one of mine, consider that to be
More informationnumber of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.
12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.
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 informationQuestion: What is the probability that a fivecard 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 14.3. section. The Probability of an Event
4.3 Probability (43) 727 4.3 PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques
More information3. Discrete Probability. CSE 312 Autumn 2011 W.L. Ruzzo
3. Discrete Probability CSE 312 Autumn 2011 W.L. Ruzzo sample spaces Sample space: S is the set of all possible outcomes of an experiment (Ω in your text book Greek uppercase omega) Coin flip: S = {Heads,
More information2.5 Conditional Probabilities and 2Way Tables
2.5 Conditional Probabilities and 2Way Tables Learning Objectives Understand how to calculate conditional probabilities Understand how to calculate probabilities using a contingency or 2way table It
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 informationThe study of probability has increased in popularity over the years because of its wide range of practical applications.
6.7. Probability. The study of probability has increased in popularity over the years because of its wide range of practical applications. In probability, each repetition of an experiment is called a trial,
More informationJan 17 Homework Solutions Math 151, Winter 2012. Chapter 2 Problems (pages 5054)
Jan 17 Homework Solutions Math 11, Winter 01 Chapter Problems (pages 0 Problem In an experiment, a die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample
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 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 informationLesson 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 informationDiscrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22
CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette
More informationUNDERGROUND TONK LEAGUE
UNDERGROUND TONK LEAGUE WWW.TONKOUT.COM RULES Players are dealt (5) five cards to start. Player to left of dealer has first play. Player must draw a card from the deck or Go For Low. If a player draws
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 informationPoker. 10,Jack,Queen,King,Ace. 10, Jack, Queen, King, Ace of the same suit Five consecutive ranks of the same suit that is not a 5,6,7,8,9
Poker Poker is an ideal setting to study probabilities. Computing the probabilities of different will require a variety of approaches. We will not concern ourselves with betting strategies, however. Our
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 10401 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 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 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 information8.3 Probability Applications of Counting Principles
8. Probability Applications of Counting Principles In this section, we will see how we can apply the counting principles from the previous two sections in solving probability problems. Many of the probability
More informationSection 64 Multiplication Principle, Permutations, and Combinations
78 SEQUENCES, SERIES, ND PROBBILITY (B) Find the cycles per second for C, three notes higher than. 91. Puzzle. If you place 1 on the first square of a chessboard, on the second square, on the third, and
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 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 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 informationI. WHAT IS PROBABILITY?
C HAPTER 3 PROBABILITY Random Experiments I. WHAT IS PROBABILITY? The weatherman on 0 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and
More information6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0.
Name: Date:. For each of the following scenarios, determine the appropriate distribution for the random variable X. A) A fair die is rolled seven times. Let X = the number of times we see an even number.
More informationMATH 105: Finite Mathematics 65: Combinations
MATH 105: Finite Mathematics 65: Combinations Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline 1 Developing Combinations 2 s of Combinations 3 Combinations vs. Permutations 4 Conclusion
More informationCombinatorics 3 poker hands and Some general probability
Combinatorics 3 poker hands and Some general probability Play cards 13 ranks Heart 4 Suits Spade Diamond Club Total: 4X13=52 cards You pick one card from a shuffled deck. What is the probability that it
More informationSection 65 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 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 informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6sided dice. What s the probability of rolling at least one 6? There is a 1
More informationPermutations & Combinations
Permutations & Combinations Extension 1 Mathematics HSC Revision Multiplication Rule If one event can occur in m ways, a second event in n ways and a third event in r, then the three events can occur in
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 informationThe Casino Lab STATION 1: CRAPS
The Casino Lab Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will
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 datagenerating
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 information9.2 The Multiplication Principle, Permutations, and Combinations
9.2 The Multiplication Principle, Permutations, and Combinations Counting plays a major role in probability. In this section we shall look at special types of counting problems and develop general formulas
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 informationCoin Flip Questions. Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT.
Coin Flip Questions Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. 1 How many ways can you get exactly 1 head? 2 How many ways can you get exactly 2 heads?
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 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 informationHomework 6 (due November 4, 2009)
Homework 6 (due November 4, 2009 Problem 1. On average, how many independent games of poker are required until a preassigned player is dealt a straight? Here we define a straight to be cards of consecutive
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Ch.  Problems to look at Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability
More informationPOKER LOTTO LOTTERY GAME CONDITIONS These Game Conditions apply, until amended or revised, to the POKER LOTTO lottery game.
POKER LOTTO LOTTERY GAME CONDITIONS These Game Conditions apply, until amended or revised, to the POKER LOTTO lottery game. 1.0 Rules 1.1 POKER LOTTO is governed by the Rules Respecting Lottery Games of
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 informationBasic concepts in probability. Sue Gordon
Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are
More informationBasic Probability Theory I
A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population
More informationTopic 1 Probability spaces
CSE 103: Probability and statistics Fall 2010 Topic 1 Probability spaces 1.1 Definition In order to properly understand a statement like the chance of getting a flush in fivecard poker is about 0.2%,
More informationStandard 12: The student will explain and evaluate the financial impact and consequences of gambling.
TEACHER GUIDE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Priority Academic Student Skills Personal Financial
More informationProbability and Hypothesis Testing
B. Weaver (3Oct25) Probability & Hypothesis Testing. PROBABILITY AND INFERENCE Probability and Hypothesis Testing The area of descriptive statistics is concerned with meaningful and efficient ways of
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 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 informationCombinatorial Proofs
Combinatorial Proofs Two Counting Principles Some proofs concerning finite sets involve counting the number of elements of the sets, so we will look at the basics of counting. Addition Principle: If A
More informationCombinations and Permutations
Combinations and Permutations What's the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
More informationMath 118 Study Guide. This study guide is for practice only. The actual question on the final exam may be different.
Math 118 Study Guide This study guide is for practice only. The actual question on the final exam may be different. Convert the symbolic compound statement into words. 1) p represents the statement "It's
More informationStandard 12: The student will explain and evaluate the financial impact and consequences of gambling.
STUDENT MODULE 12.1 GAMBLING PAGE 1 Standard 12: The student will explain and evaluate the financial impact and consequences of gambling. Risky Business Simone, Paula, and Randy meet in the library every
More informationThe temporary new rules and amendments authorize casino licensees to. offer a supplemental wager in the game of three card poker known as the three
Progressive Wager and Envy Bonus In Three Card Poker Accounting And Internal Controls Gaming Equipment Rules Of The Game Temporary Amendments: N.J.A.C. 19:451.20; 19:461.10A; and 19:4720.1, 20.6, 20.10
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 information111 Permutations and Combinations
Fundamental Counting Principal 111 Permutations and Combinations Using the Fundamental Counting Principle 1a. A makeyourownadventure story lets you choose 6 starting points, gives 4 plot choices, and
More informationHomework 2 Solutions
CSE 21  Winter 2012 Homework #2 Homework 2 Solutions 2.1 In this homework, we will consider ordinary decks of playing cards which have 52 cards, with 13 of each of the four suits (Hearts, Spades, Diamonds
More information4. Binomial Expansions
4. Binomial Expansions 4.. Pascal's Triangle The expansion of (a + x) 2 is (a + x) 2 = a 2 + 2ax + x 2 Hence, (a + x) 3 = (a + x)(a + x) 2 = (a + x)(a 2 + 2ax + x 2 ) = a 3 + ( + 2)a 2 x + (2 + )ax 2 +
More informationBasics of Counting. The product rule. Product rule example. 22C:19, Chapter 6 Hantao Zhang. Sample question. Total is 18 * 325 = 5850
Basics of Counting 22C:19, Chapter 6 Hantao Zhang 1 The product rule Also called the multiplication rule If there are n 1 ways to do task 1, and n 2 ways to do task 2 Then there are n 1 n 2 ways to do
More informationAcing Math (One Deck At A Time!): A Collection of Math Games. Table of Contents
Table of Contents Introduction to Acing Math page 5 Card Sort (Grades K  3) page 8 Greater or Less Than (Grades K  3) page 9 Number Battle (Grades K  3) page 10 Place Value Number Battle (Grades 16)
More informationComplement. If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A.
Complement If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A. For example, if A is the event UNC wins at least 5 football games, then A c is the
More informationBEGINNER S BRIDGE NOTES. Leigh Harding
BEGINNER S BRIDGE NOTES Leigh Harding PLAYING THE CARDS IN TRUMP CONTRACTS Don t play a single card until you have planned how you will make your contract! The plan will influence decisions you will have
More informationContemporary Mathematics Online Math 1030 Sample Exam I Chapters 1214 No Time Limit No Scratch Paper Calculator Allowed: Scientific
Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 1214 No Time Limit No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the lefthand margin. You
More informationProbabilities of Poker Hands with Variations
Probabilities of Poker Hands with Variations Jeff Duda Acknowledgements: Brian Alspach and Yiu Poon for providing a means to check my numbers Poker is one of the many games involving the use of a 52card
More information4.4 Conditional Probability
4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school.
More informationAlgebra 2 C Chapter 12 Probability and Statistics
Algebra 2 C Chapter 12 Probability and Statistics Section 3 Probability fraction Probability is the ratio that measures the chances of the event occurring For example a coin toss only has 2 equally likely
More informationTexas Hold em. From highest to lowest, the possible five card hands in poker are ranked as follows:
Texas Hold em Poker is one of the most popular card games, especially among betting games. While poker is played in a multitude of variations, Texas Hold em is the version played most often at casinos
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 7, due Wedneday, March 14 Happy Pi Day! (If any errors are spotted, please email them to morrison at math dot berkeley dot edu..5.10 A croissant
More informationMAT 1000. Mathematics in Today's World
MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities
More informationMath 210. 1. Compute C(1000,2) (a) 499500. (b) 1000000. (c) 2. (d) 999000. (e) None of the above.
Math 210 1. Compute C(1000,2) (a) 499500. (b) 1000000. (c) 2. (d) 999000. 2. Suppose that 80% of students taking calculus have previously had a trigonometry course. Of those that did, 75% pass their calculus
More informationCounting principle, permutations, combinations, probabilities
Counting Methods Counting principle, permutations, combinations, probabilities Part 1: The Fundamental Counting Principle The Fundamental Counting Principle is the idea that if we have a ways of doing
More informationSt. Hilary CARD PARTY WEDNESDAYS House Rules Version 3 Rules of 4 hand single deck Pinochle based on Hoyle
St. Hilary CARD PARTY WEDNESDAYS House Rules Version 3 Rules of 4 hand single deck Pinochle based on Hoyle Principal Rule of Play We are a congenial, respectful and attentive people of God. The primary
More informationLesson 8: The Difference Between Theoretical Probabilities and Estimated Probabilities
The Difference Between Theoretical Probabilities and Estimated Probabilities Student Outcomes Given theoretical probabilities based on a chance experiment, students describe what they expect to see when
More information6.3 Conditional Probability and Independence
222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted
More informationChapter 3: The basic concepts of probability
Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording
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 information1. The sample space S is the set of all possible outcomes. 2. An event is a set of one or more outcomes for an experiment. It is a sub set of S.
1 Probability Theory 1.1 Experiment, Outcomes, Sample Space Example 1 n psychologist examined the response of people standing in line at a copying machines. Student volunteers approached the person first
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 informationChapter 4 Probability
The Big Picture of Statistics Chapter 4 Probability Section 42: Fundamentals Section 43: Addition Rule Sections 44, 45: Multiplication Rule Section 47: Counting (next time) 2 What is probability?
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 informationExample Hand Say you are dealt the following cards: Suits Suits are ranked in the following order.
Chinese Poker or 13 Card Poker This game is played with 4 players. Each player is dealt 13 cards. The object is to arrange them into 3 groups: one consisting of 3 cards (front hand), and two consisting
More informationSTAT 201 INTRODUCTION TO BUSINESS STATISTICS PROBABILITY REVIEW QUESTIONS
STAT 201 INTRODUCTION TO BUSINESS STATISTICS PROBABILITY REVIEW QUESTIONS Question 1: Five standard sixsided dice are rolled. What is the probability of getting the same number on all five dice? The probability
More informationThe game also features three optional bonus bets.
Type of Game The game of Commission Free Fortune Gow Poker utilizes a playerdealer position and is a California game. The playerdealer shall collect all losing wagers, pay all winning wagers, and may
More informationQuiz CHAPTER 16 NAME: UNDERSTANDING PROBABILITY AND LONG TERM EXPECTATIONS
Quiz CHAPTER 16 NAME: UNDERSTANDING PROBABILITY AND LONG TERM EXPECTATIONS 1. Give two examples of ways that we speak about probability in our every day lives. NY REASONABLE ANSWER OK. EXAMPLES: 1) WHAT
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 information2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.
Math 142 September 27, 2011 1. 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
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 information