CSE21 - Math for Algorithm and Systems Analysis
|
|
|
- Stella Allen
- 5 years ago
- Views:
Transcription
1 CSE21 - Math for Algorithm and Systems Analysis Midterm Next week One 8.5 x 11 sheet of notes (both sides), no electronics #Partitions Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
2 Topics for today The Enigma revisited Study sessions, office hours. Practice Midterm questions Composing permutations, taking inverses. Expected Value, Variance (biased coin) Partitions & Sterling Numbers (of the second kind) Counting functions Derangements Rubalcaba 4/25/ / 18
3 Midterm / Homework Study Sessions / Culture Office Hours / Study Sessions Week 4: Friday (4/25) 2-4:30pm at The Loft (HW4, general questions) Sat-Sun (4/26-4/27) Adams Avenue Unplugged, Little Italy Art Walk Sunday (4/27) 11am-Noon EBU3B B260A (with Kacy & Tracy) Week 5: Wed. (4/30) 2-4:00pm at The Loft (Midterm Q&A w/ TAs, tutors) Week 5: Wed. (4/30) 6-7:00pm at EBU3B 4122 Week 5: Thurs. (5/1) 7-9pm CENTER 119 (Midterm Q&A) Week 5: Friday (5/2) 12-12:40 EBU3B 4122 Week 5: Friday (5/2) 1-1:50 Midterm Friday (5/2) 6-9pm Graffiti Beach, 2220 Fern Street (post midterm DJ set) Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
4 Partitions The Stirling numbers of the second kind, written S(n,k) or t n k u, count the number of ways to partition a set of n labeled objects into k nonempty unlabeled subsets. Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
5 Partitions The Stirling numbers of the second kind, written S(n,k) or t n k u, count the number of ways to partition a set of n labeled objects into k nonempty unlabeled subsets. First let s partition the set of three elements ta,b,cu. tta,b,cuu 3 3 tta,bu, tcuu tta,cu, tbuu ttau, tb,cuu ttau, tbu, tcuu 3 1 `2 3 1 `2 3 1 `2 3 1 `1`1 Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
6 Partitions The Stirling numbers of the second kind, written S(n,k) or t n k u, count the number of ways to partition a set of n labeled objects into k nonempty unlabeled subsets. First let s partition the set of three elements ta,b,cu. tta,b,cuu 3 3 tta,bu, tcuu tta,cu, tbuu ttau, tb,cuu ttau, tbu, tcuu 3 1 `2 3 1 `2 3 1 `2 3 1 `1`1 Sp3,2q 3 There are three ways of partitioning a set of three labeled objects into two unlabeled subsets. Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
7 Partitions The Stirling numbers of the second kind, written S(n,k) or t n k u, count the number of ways to partition a set of n labeled objects into k nonempty unlabeled subsets. First let s partition the set of three elements ta,b,cu. tta,b,cuu 3 3 tta,bu, tcuu tta,cu, tbuu ttau, tb,cuu ttau, tbu, tcuu 3 1 `2 3 1 `2 3 1 `2 3 1 `1`1 Sp3,2q 3 There are three ways of partitioning a set of three labeled objects into two unlabeled subsets. Sp3,1q 1 There is one way of partitioning a set of three labeled objects into one unlabeled subset. Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
8 James Stirling ( ) The Stirling numbers of the second kind, written S(n,k) or t n k u, count the number of ways to partition a set of n labeled objects into k nonempty unlabeled subsets. Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
9 James Stirling ( ) The Stirling numbers of the second kind, written S(n,k) or t n k u, count the number of ways to partition a set of n labeled objects into k nonempty unlabeled subsets. Here is a general formula. " * n 1 k k! kÿ ˆk p 1q k j j n. j j 0 Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
10 Partitions of a labeled set of four elements Sp4,1q 1 Sp4,2q 7 Sp4,3q 6 Sp4,4q 1 Note the four elements are labeled by position in the figure. Ignore the lines in the diagram, they represent partition refinement. Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
11 Partitions of a labeled set of four elements Sp4,1q Sp4,2q 4 ` `1, 4 2 `2 Sp4,3q `1`1 Sp4,4q `1`1`1 Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
12 Partitions of a labeled set of five elements Sp5,1q 1 Sp5,2q 15 Sp5,3q 25 Sp5,4q 10 Sp5,5q 1 Note the five elements are labeled by position in the figure. Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
13 Partitions of a labeled set of five elements Sp5,1q Sp5,2q 10 ` `2, 5 4 `1 Sp5,3q 15 ` `2`1, 5 3 `1`1 Sp5,4q `1`1`1 Sp5,5q `1`1`1`1 Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
14 Expected Value Let s look at tossing a biased coin three times, where the probability of heads is 0.3. Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
15 Expected Value Let s look at tossing a biased coin three times, where the probability of heads is 0.3. Let X be the number of heads tossed. Ppxq : t0,1,2,3u Ñ r0,1s X PpX q ˆ3 0 p0.3q 0 p0.7q ˆ3 1 p0.3q 1 p0.7q ˆ3 2 p0.3q 2 p0.7q ˆ3 3 p0.3q 3 p0.7q Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
16 Expected Value X PpX q EpX q 3ÿ X PpX q 0 Pp0q `1 Pp1q `2 Pp2q `3 Pp3q k 0 EpX q ` ` ` Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
17 Expected Value X PpX q EpX q 3ÿ X PpX q 0 Pp0q `1 Pp1q `2 Pp2q `3 Pp3q k 0 EpX q ` ` ` Note, this is an example of a Binomial distribution Bpn,pq with n 3 and p 0.3. EpX q n p Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
18 Two Enigma Rotors Rubalcaba 4/25/ / 18
19 Two Enigma Rotors (right rotor exploded view) Rubalcaba 4/25/ / 18
20 An Enigma Rotor (exploded view) You can see the 26 wires that make up the function map from the set of letters ta,b,...z u to ta,b,...z u. This function is a permutation (all Enigma rotors were permutations on 26 letters). Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
21 The Nine Permutations of the Enigma Enter the stecker (plugboard) on the right, follow a wire through the stecker, right rotor, middle rotor, left rotor, Umkehrwalze (reflector plate), then backwards through the left, middle, right rotors and finally backwards through the plugboard. Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
22 Wiring of the Enigma (simple 4 letter, 3 rotor model) A is pressed and D is illuminated (Note the plugboard wire swapping the roles of S and D) Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
23 Quiz (4/25/2014) π and ρ are permutations on t1,2,3,4,5u π p2,3,5q ρ p1,5q 1 Compose π ρ A. p1,5q B. p1,2,3,5q C. p1,2,5,3q D. p1,5,2,3q E. p2,5,3q 2 Compose ρ π A. p1,5q B. p1,2,3,5q C. p1,2,5,3q D. p1,5,2,3q E. p2,5,3q 3 Find the inverse of π A. p1,5q B. p1,2,3,5q C. p1,2,5,3q D. p1,5,2,3q E. p2,5,3q 4 π : t1,2,3,4,5u Ñ t1,2,3,4,5u A. π is one-to-one but not onto B. π is onto but not one-to-one C. π is neither one-to-one nor onto D. π is a bijection 5 What s the probability that you guessed all previous answers correct? `1 5 3 `1 4 Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
24 Sample Midterm Questions (draft) 1 List all derangements of the four element set t1,2,3,4u in cycle notation. 2 What is the probability that a random permutation on four letters is a derangement? 3 Find pp2,3,4,5q p1,3,5q p1,2qp3,5qq 1 4 Given an alphabet of just 6 letters, ta,b,c,d,e,f u how many plugboard wires would give the most possible arrangements? 5 How many ways are there to connect two plugboard wires with a restricted alphabet of five letters? How does this relate to Sterling Numbers of the second kind (explain the connection or lack of connection)? 6 How many ways are there to solve w+x+y+z=10, if each of w,x,y,z are positive integers? 7 What is the probability of getting a full house dealt from standard deck of 52 cards? 8 How many partitions of a set of five labeled elements ta,b,c,d,eu have two unlabeled subsets? 9 In a class with 22 students, what is the probability that at least two students share a birthday? 10 In a class with 22 students, what is the probability that no students share a birthday? 11 How many injective functions are there from D t1,2,3,4u to C ta,b,g,yu? 12 How many surjective functions are there from D t1,2,3,4u to C tred,blueu? 13 Encrypt this message with the Casear cipher: MIDTERM 14 What is the probability that a random graph Gp15,0.2q has 15 edges? What is the expected number of edges? 15 List all partitions of five labeled elements that have three unlabeled subsets of size 3,1, and 1. Is this Sp5,3q? 16 Calculate the number of ways to partition a set of five labeled objects into three labeled subsets of size 3,1,1 in two different ways, one with multinomial coefficients and one way with binomial coefficients Rubalcaba (rrrubalcaba@eng.ucsd.edu) 4/25/ / 18
Introductory 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
UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis
![UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis](/thumbs/82/84793934.jpg "UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis")
UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis Lecture 7 Class URL: http://vlsicad.ucsd.edu/courses/cse21-s14/ Lecture 7 Notes Goals for this week: Unit FN Functions
CMath 55 PROFESSOR KENNETH A. RIBET. Final Examination May 11, :30AM 2:30PM, 100 Lewis Hall
CMath 55 PROFESSOR KENNETH A. RIBET Final Examination May 11, 015 11:30AM :30PM, 100 Lewis Hall Please put away all books, calculators, cell phones and other devices. You may consult a single two-sided
MA 524 Midterm Solutions October 16, 2018
MA 524 Midterm Solutions October 16, 2018 1. (a) Let a n be the number of ordered tuples (a, b, c, d) of integers satisfying 0 a < b c < d n. Find a closed formula for a n, as well as its ordinary generating
CPSC 467: Cryptography and Computer Security
CPSC 467: Cryptography and Computer Security Michael J. Fischer Lecture 5b September 11, 2013 CPSC 467, Lecture 5b 1/11 Stream ciphers CPSC 467, Lecture 5b 2/11 Manual stream ciphers Classical stream ciphers
Finite and Infinite Sets
Finite and Infinite Sets MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Basic Definitions Definition The empty set has 0 elements. If n N, a set S is said to have
Discussion : Independence 1.6: Counting. Qingyang Xue based on slides from Zack While February 7, University of Massachusetts Amherst
Discussion 2 1.5: Independence 1.6: Counting Qingyang Xue based on slides from Zack While February 7, 2019 University of Massachusetts Amherst 1 Table of Contents 1. Preliminaries 2. Quiz 1 Review 3. Practice
Multiple Choice Questions for Review
Review Questions Multiple Choice Questions for Review 1. Suppose there are 12 students, among whom are three students, M, B, C (a Math Major, a Biology Major, a Computer Science Major. We want to send
Principle 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.
Reading 14 : Counting
CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Instructors: Beck Hasti, Gautam Prakriya Reading 14 : Counting In this reading we discuss counting. Often, we are interested in the cardinality
Permutation Groups. Every permutation can be written as a product of disjoint cycles. This factorization is unique up to the order of the factors.
Permutation Groups 5-9-2013 A permutation of a set X is a bijective function σ : X X The set of permutations S X of a set X forms a group under function composition The group of permutations of {1,2,,n}
Math236 Discrete Maths with Applications
Math236 Discrete Maths with Applications P. Ittmann UKZN, Pietermaritzburg Semester 1, 2012 Ittmann (UKZN PMB) Math236 2012 1 / 43 The Multiplication Principle Theorem Let S be a set of k-tuples (s 1,
Cryptography. Module in Autumn Term 2016 University of Birmingham. Lecturers: Mark D. Ryan and David Galindo
Lecturers: Mark D. Ryan and David Galindo. Cryptography 2017. Slide: 1 Cryptography Module in Autumn Term 2016 University of Birmingham Lecturers: Mark D. Ryan and David Galindo Slides originally written
Week 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
Section Summary. Permutations Combinations Combinatorial Proofs
Section 6.3 Section Summary Permutations Combinations Combinatorial Proofs Permutations Definition: A permutation of a set of distinct objects is an ordered arrangement of these objects. An ordered arrangement
CSE 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
Problem Set 2. Counting
Problem Set 2. Counting 1. (Blitzstein: 1, Q3 Fred is planning to go out to dinner each night of a certain week, Monday through Friday, with each dinner being at one of his favorite ten restaurants. i
STAT Statistics I Midterm Exam One. Good Luck!
STAT 515 - Statistics I Midterm Exam One Name: Instruction: You can use a calculator that has no connection to the Internet. Books, notes, cellphones, and computers are NOT allowed in the test. There are
Counting in Algorithms
Counting Counting in Algorithms How many comparisons are needed to sort n numbers? How many steps to compute the GCD of two numbers? How many steps to factor an integer? Counting in Games How many different
Week 1: Probability models and counting
Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model
Algebra. Recap: Elements of Set Theory.
January 14, 2018 Arrangements and Derangements. Algebra. Recap: Elements of Set Theory. Arrangements of a subset of distinct objects chosen from a set of distinct objects are permutations [order matters]
Due Friday February 17th before noon in the TA drop box, basement, AP&M. HOMEWORK 3 : HAND IN ONLY QUESTIONS: 2, 4, 8, 11, 13, 15, 21, 24, 27
Exercise Sheet 3 jacques@ucsd.edu Due Friday February 17th before noon in the TA drop box, basement, AP&M. HOMEWORK 3 : HAND IN ONLY QUESTIONS: 2, 4, 8, 11, 13, 15, 21, 24, 27 1. A six-sided die is tossed.
and problem sheet 7
1-18 and 15-151 problem sheet 7 Solutions to the following five exercises and optional bonus problem are to be submitted through gradescope by 11:30PM on Friday nd November 018. Problem 1 Let A N + and
Partitions and Permutations
Chapter 5 Partitions and Permutations 5.1 Stirling Subset Numbers 5.2 Stirling Cycle Numbers 5.3 Inversions and Ascents 5.4 Derangements 5.5 Exponential Generating Functions 5.6 Posets and Lattices 1 2
Discrete Structures Lecture Permutations and Combinations
Introduction Good morning. Many counting problems can be solved by finding the number of ways to arrange a specified number of distinct elements of a set of a particular size, where the order of these
Chapter 6.1. Cycles in Permutations
Chapter 6.1. Cycles in Permutations Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 6.1. Cycles in Permutations Math 184A / Fall 2017 1 / 27 Notations for permutations Consider a permutation in 1-line
Probability Theory. Mohamed I. Riffi. Islamic University of Gaza
Probability Theory Mohamed I. Riffi Islamic University of Gaza Table of contents 1. Chapter 1 Probability Properties of probability Counting techniques 1 Chapter 1 Probability Probability Theorem P(φ)
The Product Rule The Product Rule: A procedure can be broken down into a sequence of two tasks. There are n ways to do the first task and n
Chapter 5 Chapter Summary 5.1 The Basics of Counting 5.2 The Pigeonhole Principle 5.3 Permutations and Combinations 5.5 Generalized Permutations and Combinations Section 5.1 The Product Rule The Product
With Question/Answer Animations. Chapter 6
With Question/Answer Animations Chapter 6 Chapter Summary The Basics of Counting The Pigeonhole Principle Permutations and Combinations Binomial Coefficients and Identities Generalized Permutations and
Math 3012 Applied Combinatorics Lecture 2
August 20, 2015 Math 3012 Applied Combinatorics Lecture 2 William T. Trotter trotter@math.gatech.edu The Road Ahead Alert The next two to three lectures will be an integrated approach to material from
Counting. Chapter 6. With Question/Answer Animations
. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education. Counting Chapter
CS1802 Week 3: Counting Next Week : QUIZ 1 (30 min)
CS1802 Discrete Structures Recitation Fall 2018 September 25-26, 2018 CS1802 Week 3: Counting Next Week : QUIZ 1 (30 min) Permutations and Combinations i. Evaluate the following expressions. 1. P(10, 4)
2. 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
Lecture 1. Permutations and combinations, Pascal s triangle, learning to count
18.440: Lecture 1 Permutations and combinations, Pascal s triangle, learning to count Scott Sheffield MIT 1 Outline Remark, just for fun Permutations Counting tricks Binomial coefficients Problems 2 Outline
Math 1111 Math Exam Study Guide
Math 1111 Math Exam Study Guide The math exam will cover the mathematical concepts and techniques we ve explored this semester. The exam will not involve any codebreaking, although some questions on the
Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 22 Fall 2017 Homework 2 Drew Armstrong Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman: Section 1.2, Exercises 5, 7, 13, 16. Section 1.3, Exercises,
Lecture 1: Introduction
Lecture 1: Introduction Instructor: Omkant Pandey Spring 2018 (CSE390) Instructor: Omkant Pandey Lecture 1: Introduction Spring 2018 (CSE390) 1 / 13 Cryptography Most of us rely on cryptography everyday
CHAPTER 7 Probability
CHAPTER 7 Probability 7.1. Sets A set is a well-defined collection of distinct objects. Welldefined means that we can determine whether an object is an element of a set or not. Distinct means that we can
Sec$on Summary. Permutations Combinations Combinatorial Proofs
Section 6.3 Sec$on Summary Permutations Combinations Combinatorial Proofs 2 Coun$ng ordered arrangements Ex: How many ways can we select 3 students from a group of 5 students to stand in line for a picture?
CSE 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
Math 447 Test 1 February 25, Spring 2016
Math 447 Test 1 February 2, Spring 2016 No books, no notes, only scientific (non-graphic calculators. You must show work, unless the question is a true/false or fill-in-the-blank question. Name: Question
Combinatorics 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
Midterm practice super-problems
Midterm practice super-problems These problems are definitely harder than the midterm (even the ones without ), so if you solve them you should have no problem at all with the exam. However be aware that
HIGH SCHOOL MATHEMATICS CONTEST Sponsored by THE MATHEMATICS DEPARTMENT of WESTERN CAROLINA UNIVERSITY. LEVEL I TEST March 23, 2017
HIGH SCHOOL MATHEMATICS CONTEST Sponsored by THE MATHEMATICS DEPARTMENT of WESTERN CAROLINA UNIVERSITY LEVEL I TEST March 23, 2017 Prepared by: John Wagaman, Chairperson Nathan Borchelt DIRECTIONS: Do
Section The Multiplication Principle and Permutations
Section 2.1 - The Multiplication Principle and Permutations Example 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small, medium, and large). How many different
Cardinality revisited
Cardinality revisited A set is finite (has finite cardinality) if its cardinality is some (finite) integer n. Two sets A,B have the same cardinality iff there is a one-to-one correspondence from A to B
Math 1111 Math Exam Study Guide
Math 1111 Math Exam Study Guide The math exam will cover the mathematical concepts and techniques we ve explored this semester. The exam will not involve any codebreaking, although some questions on the
CSE 312 Midterm Exam May 7, 2014
Name: CSE 312 Midterm Exam May 7, 2014 Instructions: You have 50 minutes to complete the exam. Feel free to ask for clarification if something is unclear. Please do not turn the page until you are instructed
Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
Section 6.1 #16 What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1
50 Counting Questions
50 Counting Questions Prob-Stats (Math 3350) Fall 2012 Formulas and Notation Permutations: P (n, k) = n!, the number of ordered ways to permute n objects into (n k)! k bins. Combinations: ( ) n k = n!,
a) 2, 4, 8, 14, 22, b) 1, 5, 6, 10, 11, c) 3, 9, 21, 39, 63, d) 3, 0, 6, 15, 27, e) 3, 8, 13, 18, 23,
Pre-alculus Midterm Exam Review Name:. Which of the following is an arithmetic sequence?,, 8,,, b),, 6, 0,, c), 9,, 9, 6, d), 0, 6,, 7, e), 8,, 8,,. What is a rule for the nth term of the arithmetic sequence
LAMC Junior Circle February 3, Oleg Gleizer. Warm-up
LAMC Junior Circle February 3, 2013 Oleg Gleizer oleg1140@gmail.com Warm-up Problem 1 Compute the following. 2 3 ( 4) + 6 2 Problem 2 Can the value of a fraction increase, if we add one to the numerator
November 6, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern
Distribution of Primes
Distribution of Primes Definition. For positive real numbers x, let π(x) be the number of prime numbers less than or equal to x. For example, π(1) = 0, π(10) = 4 and π(100) = 25. To use some ciphers, we
MA/CSSE 473 Day 13. Student Questions. Permutation Generation. HW 6 due Monday, HW 7 next Thursday, Tuesday s exam. Permutation generation
MA/CSSE 473 Day 13 Permutation Generation MA/CSSE 473 Day 13 HW 6 due Monday, HW 7 next Thursday, Student Questions Tuesday s exam Permutation generation 1 Exam 1 If you want additional practice problems
CIS 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,
CS1800 Discrete Structures Fall 2016 Profs. Aslam, Gold, Ossowski, Pavlu, & Sprague 7 November, CS1800 Discrete Structures Midterm Version C
CS1800 Discrete Structures Fall 2016 Profs. Aslam, Gold, Ossowski, Pavlu, & Sprague 7 November, 2016 CS1800 Discrete Structures Midterm Version C Instructions: 1. The exam is closed book and closed notes.
ECS 20 (Spring 2013) Phillip Rogaway Lecture 1
ECS 20 (Spring 2013) Phillip Rogaway Lecture 1 Today: Introductory comments Some example problems Announcements course information sheet online (from my personal homepage: Rogaway ) first HW due Wednesday
Topics 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
Probability. Engr. Jeffrey T. Dellosa.
Probability Engr. Jeffrey T. Dellosa Email: jtdellosa@gmail.com Outline Probability 2.1 Sample Space 2.2 Events 2.3 Counting Sample Points 2.4 Probability of an Event 2.5 Additive Rules 2.6 Conditional
Combinatorics. Chapter Permutations. Counting Problems
Chapter 3 Combinatorics 3.1 Permutations Many problems in probability theory require that we count the number of ways that a particular event can occur. For this, we study the topics of permutations and
UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis
![UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis](/thumbs/87/96626175.jpg "UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis")
UCSD CSE 21, Spring 2014 [Section B00] Mathematics for Algorithm and System Analysis Lecture 3 Class URL: http://vlsicad.ucsd.edu/courses/cse21-s14/ Lecture 3 Notes Goal for today: CL Section 3 Subsets,
Midterm 2 6:00-8:00pm, 16 April
CS70 2 Discrete Mathematics and Probability Theory, Spring 2009 Midterm 2 6:00-8:00pm, 16 April Notes: There are five questions on this midterm. Answer each question part in the space below it, using the
6.041/6.431 Spring 2009 Quiz 1 Wednesday, March 11, 7:30-9:30 PM.
6.04/6.43 Spring 09 Quiz Wednesday, March, 7:30-9:30 PM. Name: Recitation Instructor: TA: Question Part Score Out of 0 3 all 40 2 a 5 b 5 c 6 d 6 3 a 5 b 6 c 6 d 6 e 6 f 6 g 0 6.04 Total 00 6.43 Total
JIGSAW ACTIVITY, TASK # Make sure your answer in written in the correct order. Highest powers of x should come first, down to the lowest powers.
JIGSAW ACTIVITY, TASK #1 Your job is to multiply and find all the terms in ( 1) Recall that this means ( + 1)( + 1)( + 1)( + 1) Start by multiplying: ( + 1)( + 1) x x x x. x. + 4 x x. Write your answer
HOMEWORK ASSIGNMENT 5
HOMEWORK ASSIGNMENT 5 MATH 251, WILLIAMS COLLEGE, FALL 2006 Abstract. These are the instructor s solutions. 1. Big Brother The social security number of a person is a sequence of nine digits that are not
Important Distributions 7/17/2006
Important Distributions 7/17/2006 Discrete Uniform Distribution All outcomes of an experiment are equally likely. If X is a random variable which represents the outcome of an experiment of this type, then
MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability. Preliminary Concepts, Formulas, and Terminology
MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability Preliminary Concepts, Formulas, and Terminology Meanings of Basic Arithmetic Operations in Mathematics Addition: Generally
Discrete Mathematics with Applications MATH236
Discrete Mathematics with Applications MATH236 Dr. Hung P. Tong-Viet School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Pietermaritzburg Campus Semester 1, 2013 Tong-Viet
MATH 1115, Mathematics for Commerce WINTER 2011 Toby Kenney Homework Sheet 6 Model Solutions
MATH, Mathematics for Commerce WINTER 0 Toby Kenney Homework Sheet Model Solutions. A company has two machines for producing a product. The first machine produces defective products % of the time. The
Math 365 Wednesday 2/20/19 Section 6.1: Basic counting
Math 365 Wednesday 2/20/19 Section 6.1: Basic counting Exercise 19. For each of the following, use some combination of the sum and product rules to find your answer. Give an un-simplified numerical answer
2. How many bit strings of length 10 begin with 1101? a b. 2 6 c. 2 4 d. None of the above.
This test consists of 25 equally weighted questions. 1. Given a two-step procedure where there are n1 ways to do Task 1, and n2 ways to do Task 2 after completing Task 1, then there are ways to do the
A Cryptosystem Based on the Composition of Reversible Cellular Automata
A Cryptosystem Based on the Composition of Reversible Cellular Automata Adam Clarridge and Kai Salomaa Technical Report No. 2008-549 Queen s University, Kingston, Canada {adam, ksalomaa}@cs.queensu.ca
n(s)=the number of ways an event can occur, assuming all ways are equally likely to occur. p(e) = n(e) n(s)
The following story, taken from the book by Polya, Patterns of Plausible Inference, Vol. II, Princeton Univ. Press, 1954, p.101, is also quoted in the book by Szekely, Classical paradoxes of probability
In how many ways can the letters of SEA be arranged? In how many ways can the letters of SEE be arranged?
-Pick up Quiz Review Handout by door -Turn to Packet p. 5-6 In how many ways can the letters of SEA be arranged? In how many ways can the letters of SEE be arranged? - Take Out Yesterday s Notes we ll
CHAPTER 8 Additional Probability Topics
CHAPTER 8 Additional Probability Topics 8.1. Conditional Probability Conditional probability arises in probability experiments when the person performing the experiment is given some extra information
Simple Counting Problems
Appendix F Counting Principles F1 Appendix F Counting Principles What You Should Learn 1 Count the number of ways an event can occur. 2 Determine the number of ways two or three events can occur using
DVA325 Formal Languages, Automata and Models of Computation (FABER)
DVA325 Formal Languages, Automata and Models of Computation (FABER) Lecture 1 - Introduction School of Innovation, Design and Engineering Mälardalen University 11 November 2014 Abu Naser Masud FABER November
Math 14 Lecture Notes Ch. 3.3
3.3 Two Basic Rules of Probability If we want to know the probability of drawing a 2 on the first card and a 3 on the 2 nd card from a standard 52-card deck, the diagram would be very large and tedious
Combinatorics problems
Combinatorics problems Sections 6.1-6.4 Math 245, Spring 2011 1 How to solve it There are four main strategies for solving counting problems that we will look at: Multiplication principle: A man s wardrobe
Chapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
Section : Combinations and Permutations
Section 11.1-11.2: Combinations and Permutations Diana Pell A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words
Unit Nine Precalculus Practice Test Probability & Statistics. Name: Period: Date: NON-CALCULATOR SECTION
Name: Period: Date: NON-CALCULATOR SECTION Vocabulary: Define each word and give an example. 1. discrete mathematics 2. dependent outcomes 3. series Short Answer: 4. Describe when to use a combination.
Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations
Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations Objective(s): Vocabulary: I. Fundamental Counting Principle: Two Events: Three or more Events: II. Permutation: (top of p. 684)
10-1. 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
On Hultman Numbers. 1 Introduction
47 6 Journal of Integer Sequences, Vol 0 (007, Article 076 On Hultman Numbers Jean-Paul Doignon and Anthony Labarre Université Libre de Bruxelles Département de Mathématique, cp 6 Bd du Triomphe B-050
Fundamental Counting Principle
Lesson 88 Probability with Combinatorics HL2 Math - Santowski Fundamental Counting Principle Fundamental Counting Principle can be used determine the number of possible outcomes when there are two or more
CSE 1400 Applied Discrete Mathematics Permutations
CSE 1400 Applied Discrete Mathematics Department of Computer Sciences College of Engineering Florida Tech Fall 2011 1 Cyclic Notation 2 Re-Order a Sequence 2 Stirling Numbers of the First Kind 2 Problems
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even
Math 4610, Problems to be Worked in Class
Math 4610, Problems to be Worked in Class Bring this handout to class always! You will need it. If you wish to use an expanded version of this handout with space to write solutions, you can download one
CS 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
Solutions to Problem Set 7
Massachusetts Institute of Technology 6.4J/8.6J, Fall 5: Mathematics for Computer Science November 9 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised November 3, 5, 3 minutes Solutions to Problem
aabb abab abba baab baba bbaa permutations of these. But, there is a lot of duplicity in this list, each distinct word (such as 6! 3!2!1!
Introduction to COMBINATORICS In how many ways (permutations) can we arrange n distinct objects in a row?answer: n (n ) (n )... def. = n! EXAMPLE (permuting objects): What is the number of different permutations
Permutations 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;
CSE 21 Practice Final Exam Winter 2016
CSE 21 Practice Final Exam Winter 2016 1. Sorting and Searching. Give the number of comparisons that will be performed by each sorting algorithm if the input list of length n happens to be of the form
A Probability Work Sheet
A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair six-sided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we
Lecture 3 Presentations and more Great Groups
Lecture Presentations and more Great Groups From last time: A subset of elements S G with the property that every element of G can be written as a finite product of elements of S and their inverses is
Purple. Used by Japanese government. Not used for tactical military info. Used to send infamous 14-part message
Purple Purple 1 Purple Used by Japanese government o Diplomatic communications o Named for color of binder cryptanalysts used o Other Japanese ciphers: Red, Coral, Jade, etc. Not used for tactical military
Theory of Probability - Brett Bernstein
Theory of Probability - Brett Bernstein Lecture 3 Finishing Basic Probability Review Exercises 1. Model flipping two fair coins using a sample space and a probability measure. Compute the probability of
The Problem. Tom Davis December 19, 2016
The 1 2 3 4 Problem Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 19, 2016 Abstract The first paragraph in the main part of this article poses a problem that can be approached