teacher s guide getting Started
|
|
- Oswald Booker
- 5 years ago
- Views:
Transcription
1 teacher s guide getting Started Benjamin Dickman Brookline, MA Purpose in this two-day lesson, students are asked to choose the best possible painting from a group provided to them. certain restrictions prevent students from going back to previously viewed paintings, so choosing the best is not as straightforward as just looking at all of them and deciding. the objective of this lesson is to use ordering and logical thinking to create probabilistic strategies that have greater chances of success than just random selection. conditional probability is also explored as a way to evaluate the strategies further. Prerequisites knowledge of factorials is helpful but not necessary. Materials Required: none. Suggested: none. Optional: Playing cards to represent paintings of greater and lesser value. Worksheet 1 guide the first three pages of the lesson constitute the first day s work. Students are introduced to the problem of choosing a painting for an art gallery. Shrinking the problem down to a situation in which there are only two or three paintings helps students to create a strategy for picking the best painting possible. the idea of conditional probability is introduced near the end of the first day. Worksheet 2 guide the fourth page of the lesson constitutes the second day s work. Students are urged to create a general formula for conditional probability and then expand their process of picking a painting to larger selections of paintings. ccssm Addressed S-cP.1: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). S-cP.3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 225
2 Picking A PAinting Student name: Date: An anonymous donor has decided to give her art collection to various museums. Each museum is allowed to choose one painting, and, because you have a discerning eye for brushwork, the national gallery of Art has requested that you choose on their behalf. Furthermore, because the paintings all are different, you are confident that no two of them are equally good. Adolph von Menzel [Public domain], via Wikimedia commons time constraints and other museums vying for the paintings force you to follow a few rules: You cannot view any painting before it is shown officially; Paintings will be shown one at a time, in a random order; For each painting, you must either choose it or reject it; if you choose a painting, you must leave with it; if you reject a painting, you cannot return to it later; the total number of paintings is known ahead of time; and You know the relative rankings of paintings that were shown and have no external knowledge. Leading Question How will you decide which painting to choose if your goal is to pick the best painting possible? 226
3 Student name: Date: 1. is it a good idea always to pick the first painting shown? What about the last one? What other strategies could you use? 2. Suppose there are only two paintings. What is the chance that the first painting shown is the best one? What is the chance that the last painting shown is the best? 3. What if there are three paintings? What is the chance that the first painting shown is the best? What is the chance the second one shown is best? What is the chance the third one shown is best? 4. For three paintings, there will be the best painting (A), the second best painting (B), and the worst painting (c). What are the different orderings in which the three paintings could be shown? How many of these orderings are there in all? the set of all possible outcomes is known as the sample space. 227
4 Student name: Date: 5. A friend has a suggestion. Whatever painting is shown first, reject it! then, as soon as you see a painting better than the first one, select it! When will this friend s suggested strategy be successful in obtaining the best painting? When will it fail? What is the probability that the best painting out of the entire set will be selected if this strategy is followed? 6. in the cases where c is shown first, what is the probability of choosing the best painting out of the entire set using the strategy from question 5? What about the cases where B and A are shown first? 7. What if there are four paintings? in how many orders can they be arranged? create your own strategy to pick a painting. What is the probability that your strategy will be successful in selecting the best painting? 8. in the case where the worst painting is shown first (out of four), what is the probability of choosing the best painting out of the entire set using the strategy from question 5? What about the cases when other paintings are shown first? 228
5 Student name: Date: 9. How many orderings are possible for five paintings? Six? create and evaluate strategies for when there are many paintings. What difficulties might emerge? 10. create a general formula for calculating the probability if you know the quality of the first painting shown. 11. What if there were 100 paintings? create a strategy that will help you pick the best painting at least 1/4 of the time try first dividing the paintings into two equal sets, one of the first 50 paintings shown, and the other containing the last 50 paintings shown. 12. How might you generalize the question of choosing the best painting? What are some related questions you can ask? 229
6 teacher s guide Possible Solutions the solutions shown represent only some possible solution methods. Please evaluate students solution methods on the basis of mathematical validity. 1. Always picking the first or last painting will result in the same probability of picking the best painting. With n paintings, there will be probability 1/n of picking the best. 2. Using the same idea as question 1 where n = 2, P(first shown is the best) = P(last shown is the best) = 1/2. 3. Similar to the last two, P(first shown is the best) = P(second shown is the best) = P(last shown is the best) = 1/3. 4. Six orderings create the sample space: (A, B, c), (A, c, B), (B, A, c), (B, c, A), (c, A, B), and (c, B, A). 5. the strategy suggested by a friend is successful for the subset of the sample space (c, A, B), (B, A, c), and (B, c, A), and fails for the rest. Since it is successful for 3 of the 6 orderings, the probability of the strategy being successful is P(strategy is successful) = 3/6 = 1/2. 6. When B is shown first, the best, A, is chosen 2/2 times. When A is shown first, the best is chosen 0/2 times, and when c is shown first the best is chosen 1/2 times. 7. For four paintings, there are 4! = 24 orderings possible. the same strategy as before (reject the first painting, and pick the next one shown that is better than the first) will be successful with probability 11/24. Another option is to choose to reject the first two paintings, then choose the next one shown that is better than both of the first two. this will be successful with probability 10/24 = 5/ if the paintings are ordered A, B, c, and D as in question 4, then if D is shown first, the best, A, is chosen 2/6 times. When c is shown first the best is chosen 3/6 times. When B is shown first the best is chosen 6/6 times and when A is shown first the best is chosen 0/6 times. this total aligns with the answer to question 7, 11/ For n paintings, there are n! possible orderings. thus, for 5 paintings, there are 5! = 120 orderings and for 6 there are 6! = 720 orderings. Using similar strategies as with the previous problems, students may choose to view some number, k, of paintings before deciding when to stop viewing and choose a painting that is better than any of the ones already viewed. As n grows, computations may become very tedious very quickly. 10. the formula should bear some resemblance to conditional probability (i.e., given two events, X and Y, then the probability of X occurring given that Y has occurred is P(X given Y) = P(X and Y)/P(Y). 11. Reject any painting shown in the first half, and choose the next painting shown that is better than any of those shown in first half. this strategy will succeed at least when the second best painting is in the first half, and the best painting is in the second half. the probability of this is (50/100)(50/99) > 1/4. (in fact, there are other cases for which this strategy will work that will only increase the probability that it is successful.) 12. choosing the best painting is not always possible no matter what strategy is used. the best thing do is to increase the probability of choosing one of the best paintings (if not the best). Some possible questions are For a total of n paintings, how many should you pass on?, Are there other kinds of strategies one could use?, and What are the advantages and disadvantages of using a computer program to evaluate probabilities of success for different strategies? 230
7 teacher s guide Extending the Model the first extension is to define the optimal strategy for n paintings, and to do most of the proof that it is correct. First, a definition we will need: A candidate is a painting which you rank higher than any you have seen previously. We begin from the fact that the best painting among the n paintings is somewhere in the order in which the collection is shown. the strategy we will consider, which generalizes one in the lesson, is to examine and to rank relatively the first p 1 of the paintings shown. What is p? We will show how to find the best p as a function of n. then the strategy is to accept the first painting which is a candidate shown after the initial p 1 paintings. this means the first painting you see starting at p that is better than all of the first p 1 paintings is the one you choose. You are in a sense using the first p 1 paintings to get the lay of the land. What will this strategy do? if the very best of all the paintings happens to be among the first p 1 that you looked at, you have missed it and there is nothing you can do about it. in this case, your probability of getting the best painting is 0. this consideration will keep p from getting too large. if the best painting is later than p 1 in the order of presentation, that is, in the interval (p,, n), you have a chance. if the painting presented as p happens to be the overall best, which happens with probability 1/n, you will get it. Of course, 1/n is the probability of the best painting being at any particular position in the order. Suppose the overall best painting is at position k, where k p. Will you get that painting? if and only if the best painting among the first k 1 paintings is in fact among the first p 1 paintings! What s the probability of that? the probability is just (p 1)/(k 1). thus (p 1)/(k 1) is the probability that you will get the best painting given that it is at position k in the order. But the probability of that condition, as we have seen, is just 1/n. Hence the probability that you will get the best painting when it is at position k is (p 1)/(n(k 1)). Hence the probability that you get the best overall painting is the sum of these expressions from k = p to k = n. We write it out: the probability of success for this strategy is it remains to find the best p as a function of n. For n = 3 and 4, this was done in the lesson, and it is worth checking that the above formula gives the best answer. When you vary p with fixed n, the expression begins small when p = 2, is again small when p = n, and has a maximum in between. You look for the value of p where it stops increasing and starts decreasing. Let s also look at this for large n. About how big is that sum we just defined? For large n and p, it is approximately (p/n)(lnn lnp). if we set x = n/p, this is (lnx)/x. this has a maximum at x = e. So the best strategy, if n is at all large, is to pick p/n as close as we can to 1/e. We said at the beginning that we will do most of the proof that this is correct. We have omitted the argument that the optimal strategy is, in fact, to look at some number p 1 of paintings and then pick the best thereafter. this is eminently reasonable, but that s not a proof. the full story can be found, for example, in Fred Mosteller s Fifty challenging Problems in Probability with Solutions. And now, a second extension: An interesting modeling problem in a very similar spirit is what is sometimes called the theater problem. it concerns finding a parking space when you want to go to the theater. imagine an infinite road with parking spaces at the integers, most of which are filled as you approach the theater, which is at a known integer location. the model is actually most workable if you assume an infinite road. 231
8 teacher s guide Extending the Model You want a space as close to the theater as possible. When you consider a candidate, that is, a space that is available, you cannot tell what other closer spaces might be available. if you don t take this candidate, the space will no longer be there if you later decide you should have taken it. if you don t take a space by the time you pass the theater, you will have to take one a long way beyond the theater, and you will be unhappy. Assuming you know the location of the theater, and the probability that any space will be available, what is your best strategy? Once you understand this one, you can consider including in the model a (possibly high) cost of making an illegal U-turn and trying again! Reference Mosteller, F. (1987). Fifty challenging problems in probability with solutions. new York: Dover. 232
Section Summary. Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning
Section 7.1 Section Summary Finite Probability Probabilities of Complements and Unions of Events Probabilistic Reasoning Probability of an Event Pierre-Simon Laplace (1749-1827) We first study Pierre-Simon
More information8.2 Union, Intersection, and Complement of Events; Odds
8.2 Union, Intersection, and Complement of Events; Odds Since we defined an event as a subset of a sample space it is natural to consider set operations like union, intersection or complement in the context
More information12. 6 jokes are minimal.
Pigeonhole Principle Pigeonhole Principle: When you organize n things into k categories, one of the categories has at least n/k things in it. Proof: If each category had fewer than n/k things in it then
More informationNovember 6, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern
More informationProbability and Statistics. Copyright Cengage Learning. All rights reserved.
Probability and Statistics Copyright Cengage Learning. All rights reserved. 14.2 Probability Copyright Cengage Learning. All rights reserved. Objectives What Is Probability? Calculating Probability by
More informationPrinciple of Inclusion-Exclusion Notes
Principle of Inclusion-Exclusion Notes The Principle of Inclusion-Exclusion (often abbreviated PIE is the following general formula used for finding the cardinality of a union of finite sets. Theorem 0.1.
More informationMutually Exclusive Events
Mutually Exclusive Events Suppose you are rolling a six-sided die. What is the probability that you roll an odd number and you roll a 2? Can these both occur at the same time? Why or why not? Mutually
More informationBasic Probability Ideas. Experiment - a situation involving chance or probability that leads to results called outcomes.
Basic Probability Ideas Experiment - a situation involving chance or probability that leads to results called outcomes. Random Experiment the process of observing the outcome of a chance event Simulation
More informationFOURTH LECTURE : SEPTEMBER 18, 2014
FOURTH LECTURE : SEPTEMBER 18, 01 MIKE ZABROCKI I started off by listing the building block numbers that we have already seen and their combinatorial interpretations. S(n, k = the number of set partitions
More informationSOLUTIONS FOR PROBLEM SET 4
SOLUTIONS FOR PROBLEM SET 4 A. A certain integer a gives a remainder of 1 when divided by 2. What can you say about the remainder that a gives when divided by 8? SOLUTION. Let r be the remainder that a
More informationObjectives: - You are given a circuit with 2-4 resistors and a battery. The circuits are either series or parallel.
I: Solve Simple Circuits with Nontraditional Information Level 5 Prerequisite: Solve Complete Circuits Points To: Solve Circuits with Symbolic Algebra; Solve Combined Circuits One-Step Objectives: - You
More information7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count
7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count Probability deals with predicting the outcome of future experiments in a quantitative way. The experiments
More informationThe probability set-up
CHAPTER 2 The probability set-up 2.1. Introduction and basic theory We will have a sample space, denoted S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample
More informationTopics to be covered
Basic Counting 1 Topics to be covered Sum rule, product rule, generalized product rule Permutations, combinations Binomial coefficients, combinatorial proof Inclusion-exclusion principle Pigeon Hole Principle
More information[Independent Probability, Conditional Probability, Tree Diagrams]
Name: Year 1 Review 11-9 Topic: Probability Day 2 Use your formula booklet! Page 5 Lesson 11-8: Probability Day 1 [Independent Probability, Conditional Probability, Tree Diagrams] Read and Highlight Station
More informationCS100: DISCRETE STRUCTURES. Lecture 8 Counting - CH6
CS100: DISCRETE STRUCTURES Lecture 8 Counting - CH6 Lecture Overview 2 6.1 The Basics of Counting: THE PRODUCT RULE THE SUM RULE THE SUBTRACTION RULE THE DIVISION RULE 6.2 The Pigeonhole Principle. 6.3
More informationIn how many ways can we paint 6 rooms, choosing from 15 available colors? What if we want all rooms painted with different colors?
What can we count? In how many ways can we paint 6 rooms, choosing from 15 available colors? What if we want all rooms painted with different colors? In how many different ways 10 books can be arranged
More informationProbability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37
Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete
More informationPermutations and Combinations
Permutations and Combinations Introduction Permutations and combinations refer to number of ways of selecting a number of distinct objects from a set of distinct objects. Permutations are ordered selections;
More informationChapter 5: Probability: What are the Chances? Section 5.2 Probability Rules
+ Chapter 5: Probability: What are the Chances? Section 5.2 + Two-Way Tables and Probability When finding probabilities involving two events, a two-way table can display the sample space in a way that
More informationProbability Rules. 2) The probability, P, of any event ranges from which of the following?
Name: WORKSHEET : Date: Answer the following questions. 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if. 2) The probability,
More informationThe Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.)
The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you $ that if you give me $, I ll give you $2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If
More informationIdeas beyond Number. Teacher s guide to Activity worksheets
Ideas beyond Number Teacher s guide to Activity worksheets Learning objectives To explore reasoning, logic and proof through practical, experimental, structured and formalised methods of communication
More informationThe probability set-up
CHAPTER The probability set-up.1. Introduction and basic theory We will have a sample space, denoted S sometimes Ω that consists of all possible outcomes. For example, if we roll two dice, the sample space
More informationUnit 1 Day 1: Sample Spaces and Subsets. Define: Sample Space. Define: Intersection of two sets (A B) Define: Union of two sets (A B)
Unit 1 Day 1: Sample Spaces and Subsets Students will be able to (SWBAT) describe events as subsets of sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions,
More informationCIS 2033 Lecture 6, Spring 2017
CIS 2033 Lecture 6, Spring 2017 Instructor: David Dobor February 2, 2017 In this lecture, we introduce the basic principle of counting, use it to count subsets, permutations, combinations, and partitions,
More informationProbabilities and Probability Distributions
Probabilities and Probability Distributions George H Olson, PhD Doctoral Program in Educational Leadership Appalachian State University May 2012 Contents Basic Probability Theory Independent vs. Dependent
More informationNON-OVERLAPPING PERMUTATION PATTERNS. To Doron Zeilberger, for his Sixtieth Birthday
NON-OVERLAPPING PERMUTATION PATTERNS MIKLÓS BÓNA Abstract. We show a way to compute, to a high level of precision, the probability that a randomly selected permutation of length n is nonoverlapping. As
More informationn! = n(n 1)(n 2) 3 2 1
A Counting A.1 First principles If the sample space Ω is finite and the outomes are equally likely, then the probability measure is given by P(E) = E / Ω where E denotes the number of outcomes in the event
More informationSets. Definition A set is an unordered collection of objects called elements or members of the set.
Sets Definition A set is an unordered collection of objects called elements or members of the set. Sets Definition A set is an unordered collection of objects called elements or members of the set. Examples:
More informationGeorgia Department of Education Georgia Standards of Excellence Framework GSE Geometry Unit 6
How Odd? Standards Addressed in this Task MGSE9-12.S.CP.1 Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (or, and, not). MGSE9-12.S.CP.7
More informationStatistics Intermediate Probability
Session 6 oscardavid.barrerarodriguez@sciencespo.fr April 3, 2018 and Sampling from a Population Outline 1 The Monty Hall Paradox Some Concepts: Event Algebra Axioms and Things About that are True Counting
More informationDiscrete Mathematics: Logic. Discrete Mathematics: Lecture 15: Counting
Discrete Mathematics: Logic Discrete Mathematics: Lecture 15: Counting counting combinatorics: the study of the number of ways to put things together into various combinations basic counting principles
More informationBlock 1 - Sets and Basic Combinatorics. Main Topics in Block 1:
Block 1 - Sets and Basic Combinatorics Main Topics in Block 1: A short revision of some set theory Sets and subsets. Venn diagrams to represent sets. Describing sets using rules of inclusion. Set operations.
More informationThere are three types of mathematicians. Those who can count and those who can t.
1 Counting There are three types of mathematicians. Those who can count and those who can t. 1.1 Orderings The details of the question always matter. So always take a second look at what is being asked
More informationKenKen Strategies. Solution: To answer this, build the 6 6 table of values of the form ab 2 with a {1, 2, 3, 4, 5, 6}
KenKen is a puzzle whose solution requires a combination of logic and simple arithmetic and combinatorial skills. The puzzles range in difficulty from very simple to incredibly difficult. Students who
More informationNon-overlapping permutation patterns
PU. M. A. Vol. 22 (2011), No.2, pp. 99 105 Non-overlapping permutation patterns Miklós Bóna Department of Mathematics University of Florida 358 Little Hall, PO Box 118105 Gainesville, FL 326118105 (USA)
More informationDef: The intersection of A and B is the set of all elements common to both set A and set B
Def: Sample Space the set of all possible outcomes Def: Element an item in the set Ex: The number "3" is an element of the "rolling a die" sample space Main concept write in Interactive Notebook Intersection:
More informationCS 237 Fall 2018, Homework SOLUTION
0//08 hw03.solution.lenka CS 37 Fall 08, Homework 03 -- SOLUTION Due date: PDF file due Thursday September 7th @ :59PM (0% off if up to 4 hours late) in GradeScope General Instructions Please complete
More informationIntroductory Probability
Introductory Probability Combinations Nicholas Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK Agenda Assigning Objects to Identical Positions Denitions Committee Card Hands Coin Toss Counts
More informationThe study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability
The study of probability is concerned with the likelihood of events occurring Like combinatorics, the origins of probability theory can be traced back to the study of gambling games Still a popular branch
More informationProbability and Randomness. Day 1
Probability and Randomness Day 1 Randomness and Probability The mathematics of chance is called. The probability of any outcome of a chance process is a number between that describes the proportion of
More information18 Completeness and Compactness of First-Order Tableaux
CS 486: Applied Logic Lecture 18, March 27, 2003 18 Completeness and Compactness of First-Order Tableaux 18.1 Completeness Proving the completeness of a first-order calculus gives us Gödel s famous completeness
More informationSect Linear Equations in Two Variables
99 Concept # Sect. - Linear Equations in Two Variables Solutions to Linear Equations in Two Variables In this chapter, we will examine linear equations involving two variables. Such equations have an infinite
More informationKenken For Teachers. Tom Davis January 8, Abstract
Kenken For Teachers Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles January 8, 00 Abstract Kenken is a puzzle whose solution requires a combination of logic and simple arithmetic
More informationCompound Probability. Set Theory. Basic Definitions
Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 13
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 13 Introduction to Discrete Probability In the last note we considered the probabilistic experiment where we flipped a
More informationChapter 3: Elements of Chance: Probability Methods
Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 3-4 2014-2015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,
More informationSET THEORY AND VENN DIAGRAMS
Mathematics Revision Guides Set Theory and Venn Diagrams Page 1 of 26 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier SET THEORY AND VENN DIAGRAMS Version: 2.1 Date: 15-10-2015 Mathematics
More informationSample Spaces, Events, Probability
Sample Spaces, Events, Probability CS 3130/ECE 3530: Probability and Statistics for Engineers August 28, 2014 Sets A set is a collection of unique objects. Sets A set is a collection of unique objects.
More informationCSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability (solutions)
CSE 31: Foundations of Computing II Quiz Section #: Inclusion-Exclusion, Pigeonhole, Introduction to Probability (solutions) Review: Main Theorems and Concepts Binomial Theorem: x, y R, n N: (x + y) n
More informationTheory of Probability - Brett Bernstein
Theory of Probability - Brett Bernstein Lecture 3 Finishing Basic Probability Review Exercises 1. Model flipping two fair coins using a sample space and a probability measure. Compute the probability of
More informationTOPOLOGY, LIMITS OF COMPLEX NUMBERS. Contents 1. Topology and limits of complex numbers 1
TOPOLOGY, LIMITS OF COMPLEX NUMBERS Contents 1. Topology and limits of complex numbers 1 1. Topology and limits of complex numbers Since we will be doing calculus on complex numbers, not only do we need
More informationTHE ASSOCIATION OF MATHEMATICS TEACHERS OF NEW JERSEY 2018 ANNUAL WINTER CONFERENCE FOSTERING GROWTH MINDSETS IN EVERY MATH CLASSROOM
THE ASSOCIATION OF MATHEMATICS TEACHERS OF NEW JERSEY 2018 ANNUAL WINTER CONFERENCE FOSTERING GROWTH MINDSETS IN EVERY MATH CLASSROOM CREATING PRODUCTIVE LEARNING ENVIRONMENTS WEDNESDAY, FEBRUARY 7, 2018
More informationSALES AND MARKETING Department MATHEMATICS. Combinatorics and probabilities. Tutorials and exercises
SALES AND MARKETING Department MATHEMATICS 2 nd Semester Combinatorics and probabilities Tutorials and exercises Online document : http://jff-dut-tc.weebly.com section DUT Maths S2 IUT de Saint-Etienne
More informationStudents use absolute value to determine distance between integers on the coordinate plane in order to find side lengths of polygons.
Student Outcomes Students use absolute value to determine distance between integers on the coordinate plane in order to find side lengths of polygons. Lesson Notes Students build on their work in Module
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 4 Probability Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education School of Continuing
More informationThe Product Rule The Product Rule: A procedure can be broken down into a sequence of two tasks. There are n ways to do the first task and n
Chapter 5 Chapter Summary 5.1 The Basics of Counting 5.2 The Pigeonhole Principle 5.3 Permutations and Combinations 5.5 Generalized Permutations and Combinations Section 5.1 The Product Rule The Product
More informationelements in S. It can tricky counting up the numbers of
STAT-UB.003 Notes for Wednesday, 0.FEB.0. For many problems, we need to do a little counting. We try to construct a sample space S for which the elements are equally likely. Then for any event E, we will
More informationSuch a description is the basis for a probability model. Here is the basic vocabulary we use.
5.2.1 Probability Models When we toss a coin, we can t know the outcome in advance. What do we know? We are willing to say that the outcome will be either heads or tails. We believe that each of these
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even
More informationCSE 21 Mathematics for Algorithm and System Analysis
CSE 21 Mathematics for Algorithm and System Analysis Unit 1: Basic Count and List Section 3: Set CSE21: Lecture 3 1 Reminder Piazza forum address: http://piazza.com/ucsd/summer2013/cse21/hom e Notes on
More informationCC-13. Start with a plan. How many songs. are there MATHEMATICAL PRACTICES
CC- Interactive Learning Solve It! PURPOSE To determine the probability of a compound event using simple probability PROCESS Students may use simple probability by determining the number of favorable outcomes
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 informationThe next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Fall 2004 Rao Lecture 14 Introduction to Probability The next several lectures will be concerned with probability theory. We will aim to make sense of statements such
More informationAnalyzing Games: Solutions
Writing Proofs Misha Lavrov Analyzing Games: olutions Western PA ARML Practice March 13, 2016 Here are some key ideas that show up in these problems. You may gain some understanding of them by reading
More information10-1. Combinations. Vocabulary. Lesson. Mental Math. able to compute the number of subsets of size r.
Chapter 10 Lesson 10-1 Combinations BIG IDEA With a set of n elements, it is often useful to be able to compute the number of subsets of size r Vocabulary combination number of combinations of n things
More informationThe topic for the third and final major portion of the course is Probability. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 17 Introduction to Probability The topic for the third and final major portion of the course is Probability. We will aim to make sense of
More informationWeek 6: Advance applications of the PIE. 17 and 19 of October, 2018
(1/22) MA284 : Discrete Mathematics Week 6: Advance applications of the PIE http://www.maths.nuigalway.ie/ niall/ma284 17 and 19 of October, 2018 1 Stars and bars 2 Non-negative integer inequalities 3
More informationCSCI 2200 Foundations of Computer Science (FoCS) Solutions for Homework 7
CSCI 00 Foundations of Computer Science (FoCS) Solutions for Homework 7 Homework Problems. [0 POINTS] Problem.4(e)-(f) [or F7 Problem.7(e)-(f)]: In each case, count. (e) The number of orders in which a
More informationA Probability Work Sheet
A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair six-sided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we
More informationPoker: Probabilities of the Various Hands
Poker: Probabilities of the Various Hands 19 February 2014 Poker II 19 February 2014 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13
More informationCS1802 Week 6: Sets Operations, Product Sum Rule Pigeon Hole Principle (Ch )
CS1802 Discrete Structures Recitation Fall 2017 October 9-12, 2017 CS1802 Week 6: Sets Operations, Product Sum Rule Pigeon Hole Principle (Ch 8.5-9.3) Sets i. Set Notation: Draw an arrow from the box on
More information5 Elementary Probability Theory
5 Elementary Probability Theory 5.1 What is Probability? The Basics We begin by defining some terms. Random Experiment: any activity with a random (unpredictable) result that can be measured. Trial: one
More informationLESSON 2: THE INCLUSION-EXCLUSION PRINCIPLE
LESSON 2: THE INCLUSION-EXCLUSION PRINCIPLE The inclusion-exclusion principle (also known as the sieve principle) is an extended version of the rule of the sum. It states that, for two (finite) sets, A
More informationPoker: Probabilities of the Various Hands
Poker: Probabilities of the Various Hands 22 February 2012 Poker II 22 February 2012 1/27 Some Review from Monday There are 4 suits and 13 values. The suits are Spades Hearts Diamonds Clubs There are 13
More informationDay 5: Mutually Exclusive and Inclusive Events. Honors Math 2 Unit 6: Probability
Day 5: Mutually Exclusive and Inclusive Events Honors Math 2 Unit 6: Probability Warm-up on Notebook paper (NOT in notes) 1. A local restaurant is offering taco specials. You can choose 1, 2 or 3 tacos
More informationCSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability
CSE 312: Foundations of Computing II Quiz Section #2: Inclusion-Exclusion, Pigeonhole, Introduction to Probability Review: Main Theorems and Concepts Binomial Theorem: Principle of Inclusion-Exclusion
More informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More information4.4: The Counting Rules
4.4: The Counting Rules The counting rules can be used to discover the number of possible for a sequence of events. Fundamental Counting Rule In a sequence of n events in which the first one has k 1 possibilities
More informationMATHEMATICS E-102, FALL 2005 SETS, COUNTING, AND PROBABILITY Outline #1 (Probability, Intuition, and Axioms)
MATHEMATICS E-102, FALL 2005 SETS, COUNTING, AND PROBABILITY Outline #1 (Probability, Intuition, and Axioms) Last modified: September 19, 2005 Reference: EP(Elementary Probability, by Stirzaker), Chapter
More informationCombinatorics and Intuitive Probability
Chapter Combinatorics and Intuitive Probability The simplest probabilistic scenario is perhaps one where the set of possible outcomes is finite and these outcomes are all equally likely. A subset of the
More informationProblem Set 10 2 E = 3 F
Problem Set 10 1. A and B start with p = 1. Then they alternately multiply p by one of the numbers 2 to 9. The winner is the one who first reaches (a) p 1000, (b) p 10 6. Who wins, A or B? (Derek) 2. (Putnam
More informationLesson 16: The Computation of the Slope of a Non Vertical Line
++ Lesson 16: The Computation of the Slope of a Non Vertical Line Student Outcomes Students use similar triangles to explain why the slope is the same between any two distinct points on a non vertical
More informationAmazing Birthday Cards. Digital Lesson.com
1 3 5 7 9 1 7 1 1 1 9 1 3 1 5 2 1 2 3 2 5 2 7 2 9 3 1 Amazing Birthday Cards 1 6 1 7 1 8 1 9 2 0 21 22 23 2 4 2 5 2 6 2 7 28 29 30 31 Amazing Birthday Cards Amazing Birthday Cards Birthday Cards Number
More informationNOTES ON SEPT 13-18, 2012
NOTES ON SEPT 13-18, 01 MIKE ZABROCKI Last time I gave a name to S(n, k := number of set partitions of [n] into k parts. This only makes sense for n 1 and 1 k n. For other values we need to choose a convention
More informationProbability with Set Operations. MATH 107: Finite Mathematics University of Louisville. March 17, Complicated Probability, 17th century style
Probability with Set Operations MATH 107: Finite Mathematics University of Louisville March 17, 2014 Complicated Probability, 17th century style 2 / 14 Antoine Gombaud, Chevalier de Méré, was fond of gambling
More informationTeaching the TERNARY BASE
Features Teaching the TERNARY BASE Using a Card Trick SUHAS SAHA Any sufficiently advanced technology is indistinguishable from magic. Arthur C. Clarke, Profiles of the Future: An Inquiry Into the Limits
More informationStatistics Laboratory 7
Pass the Pigs TM Statistics 104 - Laboratory 7 On last weeks lab we looked at probabilities associated with outcomes of the game Pass the Pigs TM. This week we will look at random variables associated
More informationAlgorithmique appliquée Projet UNO
Algorithmique appliquée Projet UNO Paul Dorbec, Cyril Gavoille The aim of this project is to encode a program as efficient as possible to find the best sequence of cards that can be played by a single
More informationDefine and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)
12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the
More informationMathematics Success Grade 8
Mathematics Success Grade 8 T429 [OBJECTIVE] The student will solve systems of equations by graphing. [PREREQUISITE SKILLS] solving equations [MATERIALS] Student pages S207 S220 Rulers [ESSENTIAL QUESTIONS]
More information2. Combinatorics: the systematic study of counting. The Basic Principle of Counting (BPC)
2. Combinatorics: the systematic study of counting The Basic Principle of Counting (BPC) Suppose r experiments will be performed. The 1st has n 1 possible outcomes, for each of these outcomes there are
More informationIdentify Non-linear Functions from Data
Identify Non-linear Functions from Data Student Probe Identify which data sets display linear, exponential, or quadratic behavior. x -1 0 1 2 3 y -3-4 -3 0 5 x -2 0 2 4 6 y 9 4-1 -6-11 x -1 0 1 2 3 y ¼
More informationHypergeometric Probability Distribution
Hypergeometric Probability Distribution Example problem: Suppose 30 people have been summoned for jury selection, and that 12 people will be chosen entirely at random (not how the real process works!).
More informationChapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
More informationThese Are a Few of My Favorite Things
Lesson.1 Assignment Name Date These Are a Few of My Favorite Things Modeling Probability 1. A board game includes the spinner shown in the figure that players must use to advance a game piece around the
More informationMathematics Behind Game Shows The Best Way to Play
Mathematics Behind Game Shows The Best Way to Play John A. Rock May 3rd, 2008 Central California Mathematics Project Saturday Professional Development Workshops How much was this laptop worth when it was
More informationINDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2
INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2 WARM UP Students in a mathematics class pick a card from a standard deck of 52 cards, record the suit, and return the card to the deck. The results
More informationCHAPTER 7 Probability
CHAPTER 7 Probability 7.1. Sets A set is a well-defined collection of distinct objects. Welldefined means that we can determine whether an object is an element of a set or not. Distinct means that we can
More information