This is the set of outcomes which is not in A. From the axioms, we obtain: P (A) = 1 P (A).
|
|
- Leo White
- 7 years ago
- Views:
Transcription
1 Lectures 3 and 4 jacques@ucsd.edu 2.1 Two consequences of the axioms One of the main reasons for introducing sets at the beginning was to determine formulas for the probability of unions, intersections, and complements of sets, using the axioms of probability. These general rules apply in any probability space. Throughout this section, (Ω, F, P ) is a probability space. The first consequence of the axioms is the rule of complements. Recall the complement of a set A, which is denoted A 1 is Ω\A = {ω Ω : ω A}. This is the set of outcomes which is not in A. From the axioms, we obtain: Proposition 1 Let A be an event. Then P (A) = 1 P (A). Proof The events A and A partition Ω, by definition, so from Axioms 1 and 3, 1 = P (Ω) = P (A) + P (A). This gives the formula. The above formula is useful, as it is sometimes much easier to compute P (A) than P (A), or vice versa. The inclusion-exclusion formula is a general rule which helps to compute combinatorial probabilities, and follows as a useful generalization of Axiom 3 of probability. Proposition 2 (Inclusion-Exclusion Formula) Let A and B be events in a probability space. Then P (A B) = P (A) + P (B) P (A B). 1 Some books write A c instead 1
2 Proof We may write A B as a disjoint union of the following three sets: A B = (A\B) (B\A) (A B) So by Axiom 3 of probability, P (A B) = P (A\B) + P (B\A) + P (A B). Now A\B and A B are two disjoint sets whose union is A. So using Axiom 3, P (A) = P (A\B) + P (A B). This gives P (A\B) = P (A) P (A B). Similarly, P (B\A) = P (B) P (A B). Putting these two equations into the equation for P (A B), we get P (A B) = P (A) P (A B) + P (B) P (A B) + P (A B) = P (A) + P (B) P (A B). This completes the proof. We could actually take unions of even more sets, for example, for three sets one obtains an inclusion-exclusion formula for three sets: P (A B C)=P (A)+P (B)+P (C) P (A B) P (A C) P (B C)+P (A B C). 2.2 Computing probabilities. Let s do examples of how to use the formulas for sets and probabilities to compute a probability. First Example: Fair dice. Two dice are tossed, and the sum of the two numbers is recorded. The sample space Ω is the set of all ordered pairs of numbers up to six, i.e. Ω = {(1, 1), (1, 2), (2, 1),..., (6, 6)}. The events are all the subsets of Ω, and the probability measure we will use is P (A) = A 36. 2
3 Note that we could imagine this as the probability measure which comes from tossing two fair dice independently of one another. It is also the uniform measure on Ω, since P ({(i, j)}) = 1/36 for every pair (i, j) Ω this means that i was tossed on the first die, and j was tossed on the second. Consider now the event A that the sum of the two numbers on the dice is a single digit number, namely 1, 2, 3, 4, 5, 6, 7, 8 or 9. We could directly write down all the outcomes in A, there are 30 of them, and then we could compute P (A) = A /36 = 30/36 = 5/6. But it is much easier to use the complement rule: A is the event that the sum is 10, 11 or 12, so A = {(4, 6), (6, 4), (5, 5), (5, 6), (6, 5), (6, 6)} and then By the complement rule, P (A) = A 36 = 6 36 = 1 6. which agrees with the preceding answer. P (A) = 1 P (A) = 5 6 Second Example: Two fair dice. If two fair dice are thrown, what is the chance that at least one of the dice shows an odd number? Let A be the event that the first die shows an odd number, and let B be the event that the second shows an odd number. We are looking for P (A B) in the probability space (Ω, F, P ) given in the first example. Now we know by the inclusion-exclusion formula that We carefully write down A: P (A B) = P (A) + P (B) P (A B). A = {(1, 1), (1, 2),..., (1, 6), (3, 1), (3, 2),..., (3, 6), (5, 1), (5, 2),..., (5, 6)}. There are 18 outcomes in A, so P (A) = A 36 = 1 2. We could have argued this in a simpler way: half the numbers on the first die are odd, so there is a fifty percent chance that the first die shows an odd number, regardless of what the second die shows. Similarly, we get P (B) = 1/2. Now A B is the event that both numbers are odd, which is 1/4, since A B = {(1, 1), (3, 3), (5, 5), (1, 3), (3, 1), (1, 5), (5, 1), (3, 5), (5, 3)} 3
4 and so A B P (A B) = = = 1 4. Again we could have argued more simply: there are three odd numbers possible on the first die, and three on the second, so there are nine combinations which give two odd numbers. Finally, by inclusion-exclusion, P (A B) = = 3 4. We remark that whenever one sees in a probability problem the phrase at least one, this indicates the union of some events. The purpose of the next example is to illustrate how one has sometimes to use careful logic to compute the required probability in an efficient way. Third Example: Logic and sets. Two fair six-sided dice are thrown, with probability measure P (A) = A /36 in the same probability space as the last two examples. Let A be the event that the sum of the dice is odd, or at least one of the numbers is odd and both dice show different numbers. To compute P (A), we could list all outcomes (as an exercise) in A and we would find that there are 24 pairs of numbers in A, so P (A) = 24/36 = 2/3. However, A = B (C D) where B is the event that the sum of the dice is odd, C is the event that at least one of the numbers is odd, and D is the event that both dice show different numbers. Therefore P (A) = P (B (C D)) = P (B) + P (C D) P (B C D) using inclusion-exclusion. Observe that B C D = B, for if B occurs (sum is odd), then C must occur, and D must also occur. Therefore P (A) = P (C D). Now C D is the event that one of the dice shows an odd number and the other shows a different number. It is easier to deal with C D = C D by demorgan s law this is the event that both dice show an even number or both dice show the same number. Then by inclusion-exclusion P (C D) = P (C D) = P (C) + P (D) P (C D). 4
5 Now P (C) = 1/4 (probability both dice are even) and P (D) = 1/6 (probability both dice show the same number) and C D = {(2, 2), (4, 4), (6, 6)} (both dice are even and show the same number). Therefore By the complement rule, P (C D) = = 1 3. P (A) = P (C D) = 1 P (C D) = = 2 3. This answers the problem, but it is not the only way to do this problem. It is important to become familiar with manipulating sets to compute probabilities. Fourth Example: Probability in the integers. Let Ω = Z and let F consist of all subsets of Ω. Suppose we define P ({ω}) = 1 2. ω Then P defines a probability measure by letting for any A Ω P (A) = ω A 2 ω. To see this, certainly P (A) 0 for every A and P (Ω) = ω Z 2 ω = = 1 since the sum is a geometric series. In general, if P is a probability measure defined on the integers, then the requirement implies from elementary calculus that P ({ω}) = 1 ω Z lim P ({ω}) = 0 ω in other words, larger and larger integers must have smaller and smaller probabilities. Fifth Example: Combinatorial probability. We are going to spend some time later on looking at combinatorial probability, here we do a first example. Consider a standard deck of 52 cards (thirteen cards in each of four suits, diamonds, clubs, spades and hearts, and thirteen different card values 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A, four of each value 5
6 for example the diamonds and spades are shown below). Suppose we consider drawing five cards from the deck with the uniform probability measure on all five card hands. Thus the sample space Ω is the set of all five card hands, F is the set of all subsets of Ω and, for any event A, P (A) = A / Ω. Here Ω is very large: Ω = is the total number of five card hands. Let s compute the probability that at least two of the cards in a five card hand are of the same value (in poker, this is called a pair). This is a counting problem: how many ways can we fill five slots with five cards such that two of the cards are of the same suit (this is A ), and then we divide by Ω = to get the probability. It turns out to be easier to compute P (A) this is the probability that all cards have different values. There are 52 choices to fill the first slot, but now the second slot can t have a card of the same value, so there are only 48 choices for the second slot. Similarly there are 44 choices for the third, 40 for the fourth, and 36 for the last. So Therefore P (A) = = P (A) = So it is in fact more likely than not that all the cards will be different (but perhaps surprisingly close to 1/2). Figure 1 : Diamonds and spades 6
Statistics 100A Homework 2 Solutions
Statistics Homework Solutions Ryan Rosario Chapter 9. retail establishment accepts either the merican Express or the VIS credit card. total of percent of its customers carry an merican Express card, 6
More informationArkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers
More informationMath/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability
Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock
More 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 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 informationLecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University
Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments
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 informationMATHEMATICS 154, SPRING 2010 PROBABILITY THEORY Outline #3 (Combinatorics, bridge, poker)
Last modified: February, 00 References: MATHEMATICS 5, SPRING 00 PROBABILITY THEORY Outline # (Combinatorics, bridge, poker) PRP(Probability and Random Processes, by Grimmett and Stirzaker), Section.7.
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
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. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
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 informationBasic Probability Concepts
page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes
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 informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
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 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 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 informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More informationMath/Stat 394 Homework 2
Math/Stat 39 Homework Due Wednesday Jan 18 1. Six awards are to be given out among 0 students. How many ways can the awards be given out if (a one student will win exactly five awards. (b one student will
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationStatistics 100A Homework 1 Solutions
Chapter 1 tatistics 100A Homework 1 olutions Ryan Rosario 1. (a) How many different 7-place license plates are possible if the first 2 places are for letters and the other 5 for numbers? The first two
More informationSTAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia
STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that
More informationProbability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to
More informationMath 115 Spring 2011 Written Homework 5 Solutions
. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence
More informationPoker Probability from Wikipedia. Frequency of 5-card poker hands 36 0.00139% 0.00154% 72,192.33 : 1
Poker Probability from Wikipedia Frequency of 5-card poker hands The following enumerates the frequency of each hand, given all combinations of 5 cards randomly drawn from a full deck of 52 without replacement.
More informationINCIDENCE-BETWEENNESS GEOMETRY
INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationST 371 (IV): Discrete Random Variables
ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible
More informationRandom variables, probability distributions, binomial random variable
Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that
More informationConditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Conditional Probability, Independence and Bayes Theorem Class 3, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of conditional probability and independence
More informationRegions in a circle. 7 points 57 regions
Regions in a circle 1 point 1 region points regions 3 points 4 regions 4 points 8 regions 5 points 16 regions The question is, what is the next picture? How many regions will 6 points give? There's an
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationA Few Basics of Probability
A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study
More informationLecture 17 : Equivalence and Order Relations DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion
More informationCh. 13.3: More about Probability
Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationFibonacci Numbers and Greatest Common Divisors. The Finonacci numbers are the numbers in the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...
Fibonacci Numbers and Greatest Common Divisors The Finonacci numbers are the numbers in the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,.... After starting with two 1s, we get each Fibonacci number
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 informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationLecture 16 : Relations and Functions DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence
More informationWRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
More informationSYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me
SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBIN-CAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines
More informationSECTION 10-5 Multiplication Principle, Permutations, and Combinations
10-5 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 information3 Some Integer Functions
3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple
More informationSection 6.4: Counting Subsets of a Set: Combinations
Section 6.4: Counting Subsets of a Set: Combinations In section 6.2, we learnt how to count the number of r-permutations from an n-element set (recall that an r-permutation is an ordered selection of r
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation
More informationWeek 2: Conditional Probability and Bayes formula
Week 2: Conditional Probability and Bayes formula We ask the following question: suppose we know that a certain event B has occurred. How does this impact the probability of some other A. This question
More informationProbability definitions
Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a data-generating
More informationA Little Set Theory (Never Hurt Anybody)
A Little Set Theory (Never Hurt Anybody) Matthew Saltzman Department of Mathematical Sciences Clemson University Draft: August 21, 2013 1 Introduction The fundamental ideas of set theory and the algebra
More informationHomework 3 (due Tuesday, October 13)
Homework (due Tuesday, October 1 Problem 1. Consider an experiment that consists of determining the type of job either blue-collar or white-collar and the political affiliation Republican, Democratic,
More informationHow To Find The Sample Space Of A Random Experiment In R (Programming)
Probability 4.1 Sample Spaces For a random experiment E, the set of all possible outcomes of E is called the sample space and is denoted by the letter S. For the coin-toss experiment, S would be the results
More informationSample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
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 52-card
More informationProbability Models.S1 Introduction to Probability
Probability Models.S1 Introduction to Probability Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard The stochastic chapters of this book involve random variability. Decisions are
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationChapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way
Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)
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 informationFor a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i )
Probability Review 15.075 Cynthia Rudin A probability space, defined by Kolmogorov (1903-1987) consists of: A set of outcomes S, e.g., for the roll of a die, S = {1, 2, 3, 4, 5, 6}, 1 1 2 1 6 for the roll
More informationIra Fine and Thomas J. Osler Department of Mathematics Rowan University Glassboro, NJ 08028. osler@rowan.edu. 1. Introduction
1 08/0/00 THE REMARKABLE INCIRCLE OF A TRIANGLE Ira Fine and Thomas J. Osler Department of Mathematics Rowan University Glassboro, NJ 0808 osler@rowan.edu 1. Introduction The incircle of a triangle is
More informationChapter 4: Probability and Counting Rules
Chapter 4: Probability and Counting Rules Learning Objectives Upon successful completion of Chapter 4, you will be able to: Determine sample spaces and find the probability of an event using classical
More informationStatistics in Geophysics: Introduction and Probability Theory
Statistics in Geophysics: Introduction and Steffen Unkel Department of Statistics Ludwig-Maximilians-University Munich, Germany Winter Term 2013/14 1/32 What is Statistics? Introduction Statistics is the
More informationEXAM. Exam #3. Math 1430, Spring 2002. April 21, 2001 ANSWERS
EXAM Exam #3 Math 1430, Spring 2002 April 21, 2001 ANSWERS i 60 pts. Problem 1. A city has two newspapers, the Gazette and the Journal. In a survey of 1, 200 residents, 500 read the Journal, 700 read the
More informationFormal Languages and Automata Theory - Regular Expressions and Finite Automata -
Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March
More informationThe sample space for a pair of die rolls is the set. The sample space for a random number between 0 and 1 is the interval [0, 1].
Probability Theory Probability Spaces and Events Consider a random experiment with several possible outcomes. For example, we might roll a pair of dice, flip a coin three times, or choose a random real
More informationMathematical Induction. Mary Barnes Sue Gordon
Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,
More informationHow To Solve The Social Studies Test
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 informationLikewise, we have contradictions: formulas that can only be false, e.g. (p p).
CHAPTER 4. STATEMENT LOGIC 59 The rightmost column of this truth table contains instances of T and instances of F. Notice that there are no degrees of contingency. If both values are possible, the formula
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
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 informationSECTION 10-2 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationElements of probability theory
2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted
More informationProbability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes
Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all
More informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationSection 2.4-2.5 Probability (p.55)
Section 2.4-2.5 Probability (p.55 2.54 Suppose that in a senior college class of 500 students it is found that 210 smoke, 258 drink alcoholic beverage, 216 eat between meals, 122 smoke and drink alcoholic
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 informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such 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 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 informationQuestion: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More 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 informationChapter 8 Integers 8.1 Addition and Subtraction
Chapter 8 Integers 8.1 Addition and Subtraction Negative numbers Negative numbers are helpful in: Describing temperature below zero Elevation below sea level Losses in the stock market Overdrawn checking
More informationLogic in Computer Science: Logic Gates
Logic in Computer Science: Logic Gates Lila Kari The University of Western Ontario Logic in Computer Science: Logic Gates CS2209, Applied Logic for Computer Science 1 / 49 Logic and bit operations Computers
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More informationSection 6-5 Sample Spaces and Probability
492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)
More informationFundamentele Informatica II
Fundamentele Informatica II Answer to selected exercises 1 John C Martin: Introduction to Languages and the Theory of Computation M.M. Bonsangue (and J. Kleijn) Fall 2011 Let L be a language. It is clear
More informationLies My Calculator and Computer Told Me
Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing
More information. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9
Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a
More informationPeople have thought about, and defined, probability in different ways. important to note the consequences of the definition:
PROBABILITY AND LIKELIHOOD, A BRIEF INTRODUCTION IN SUPPORT OF A COURSE ON MOLECULAR EVOLUTION (BIOL 3046) Probability The subject of PROBABILITY is a branch of mathematics dedicated to building models
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 # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected
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