# Combinatorics: The Fine Art of Counting

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli Process All of the probability measures we have looked at so far have used the uniform probability measure on a finite sample space. Today we will look at an example of a random process that may involve a non-uniform probability measure as well as a potentially infinite sample space. A Bernoulli process consists of a sequence of repeated experiments, called trials. The canonical example is repeated coin tosses, but there are many other random processes that can be treated as a Bernoulli process. The three key requirements of any Bernoulli process are: 1. Each trial has exactly two possible outcomes. These are typically designated success or failure. 2. The trials are all identical the probability of success is always p, where 0 < p < 1, and the probability of failure is q = 1 - p. 3. The trials are all mutually independent. This means that the probability of any finite intersection of events relating to the different individual trials is simply the product of the probabilities of the events. Some examples of Bernoulli trials include: o a series of games e.g. a best-of-7 world-series, a football season, a gambling session. o a 1-dimensional random walk e.g. stock market price changes o a particular outcome e.g. rolling a six on a die, shooting a hole-in-one, a baseball player getting a hit o a medical trial conducted on a group of people o a repeatable science experiment o a consumer s response to an advertisement or survey question As long as each trial can be categorized as a success or failure, the Bernoulli process may be applicable. It is critical, however, that the trials satisfy the requirements above. Note that when modeling a real world event that is not strictly random (e.g, a human response) the question of independence is always a tricky one, but in many cases it is a reasonable assumption. A Bernoulli process may be limited to a particular number of trials (e.g. 7 games, or 10 cointosses), or it may go on indefinitely, in which case we may regard it as an infinite process. A finite Bernoulli sample space consists of all binary sequences of some particular length n (1 denotes success, 0 failure). In the infinite case, the sample space consists of all infinite binary sequences. The probability measure on this sample space is simply defined in terms of the probability of the elementary events H i which represent the set of sample points where the i th trial was successful. P(H i ) = p, and P(H i c ) = 1-p = q. 1

2 The probability of a finite intersection of any of these elementary events (with distinct i s) is simply the product of the probabilities. Unless p = ½, this probability measure is not uniform. To take a concrete example, if we consider a sequence of 3 tosses of a coin which comes up heads 2/3 of the time, the probability of the event containing the sequence HHT is (2/3) 2 *1/3 = 8/27, while the probability of the event containing the sequence TTH is (1/3) 3 *2/3 = 2/27 and is thus 4 times less likely. For an infinite Bernoulli process, it worth noting that the individual sample points are not defined as events the finite intersection of any of the elementary events as defined above is still going to contain many infinite sequences (uncountably many in fact). If an event were defined which contained only one sample point (or even a countably infinite number of sample points), it would necessarily have probability 0. Note that this does not mean that such an event is cannot exist or that it is the empty set, it simply means that the probability measure assigns the number 0 to it this is the 1-d analog of the fact that a line segment has zero area it s still a perfectly good line segment, it just has no area and doesn t cover a measurable portion of the plane. Many subsets of an infinite sample space may not cover enough of the space to have non-zero probability, and we are not really interested in considering such subsets as events. If the notion of an infinite process seems unsettling, simply regard it as having an indefinite length. There are many games that could in theory go on indefinitely (e.g. baseball, or the dice game craps ), but in practice will always end in a reasonable short number of steps. It is convenient in such cases to not have to set an arbitrary limit to the number of trials, but rather to regard the process as going on forever this ensures that our sample space can handle all possible outcomes. For a finite Bernoulli process, we are often interested in the probability of getting a specific number of successful trials. This probability is easy to compute. Let S k be the event that exactly k successes occur in a Bernoulli process with n trials and success probability p. The probability of the event S k is given by: P(S k ) = (n k)*p k *q n-k k successes in n trials In general, the probability of any event which is determined simply by the number of successes can be computed using binomial coefficients and appropriate powers of p and q. Problem #1: What is the probability of getting exactly 5 heads in 10 tosses of a fair coin? How about 50 heads in 100 tosses? Solution #1: (10 5)*(1/2) 5 *(1/2) 5 = 252/1024 = 63/256 < 1/4. (100 50)*(1/2) 50 *(1/2) 50 ~ 0.08 Problem #2: If teams A and B play in a best of 7 series, what are the probabilities that the series ends in 4, 5, 6, or 7 games if team A wins each game with independent probability p? Given an explicit answer for the case where the teams are evenly matched. Solution #2: We will assume the outcomes of the games are independent and treat this as a Bernoulli process with 7 trials and p = 1/2. The probability of the series ending after 4 games is simply the probability of getting either 4 heads or 4 tails in a row which is (4 4) * (p 4 + q 4 ) = 2/16 when p=q. For a five game series, we are interested in sequences where exactly 3 of the first 4 trials are heads (or tails) and the 5 th trial is a head (or a tail). This is (4 3) * (p 3 q*p + q 3 p*q) = 4/16 when p=q. (Note the difference in the two terms when p q). The computations for six and seven games series are similar. For six games we get (5 3) * (p 3 q 2 *p + q 3 p 2 *q) = 5/16 when p=q and for seven games we get (6 3) * (p 3 q 3 *p + q 3 p 3 *q) = 5/16 when p=q. Problem #3: If teams A and B play in a best of 3 series and if team A wins each game with independent probability p > ½, what is the probability that team A wins the series in terms of p? What about a best of 5 series? Compute actual values for p = Solution #3: This problem can be solved by partitioning based A winning in either 2 or 3 games, but a simpler approach (especially for longer series) is to simply consider all sequences of three tosses of a coin with bias p that have more heads than tails (if the series really only lasts 2 games, we can imagine them playing a third consolation game, it won t change the answer). This 2

3 approach gives (3 2)*p 2 *(1-p) + (3 3)*p 3 = 3p 2-2p 3. Note that this value is equal to ½ for p=1/2 but is always greater than p when p > ½. Applying the same approach for a best of 5 series gives: (5 3)*p 3 *(1-p) 2 + (5 4)*p 4 *(1-p) + (5 5)*p 5 = 10p 3-15p 4 +6p 5. Plugging in p = 0.75 gives probability of ~0.84 for team A to win a best of 3 series and ~0.90 for a best of 7 series. Infinite Bernoulli Processes We now want to take a look at some problems involving infinite Bernoulli processes. Consider a game where two players take turns flipping a fair coin. If a player gets heads on their turn the win and the game is over, otherwise they give the coin to the other player who then takes his or her turn. What is the probablility that the player who goes first wins? First note that while this game will end on the first turn half the time, it could take an arbitrarily long sequence of turns before someone gets a head. Note also that the player who goes first clearly has an advantage - half the time the second player never even gets to flip the coin, and when they do there is no guarantee they will win before the first player has another flip. To model this game we will use the infinite Bernoulli sample space with p = ½. Let s assume the players are named Alice and Bob (A and B) and that Alice goes first. Let A k be the event that Alice wins on her k th turn: P(A 1 ) = ½ P(A 2 ) = [(½)*(½)]*(½) P(A 2 ) = [(½)*(½)] 2 *(½) P(A k ) = [(½)*(½)] k-1 *(½) A gets heads on the first flip A gets tails, B gets tails, then A gets heads A and B both get tails twice, then A gets heads A and B get tails k times, then A gets heads These events are all disjoint, so the probability that A wins is just: P(A) = P(A 1 ) + P(A 2 ) + P(A 3 ) + P(A 4 ) + (Note that this infinite sum is bounded (it is a probability so it is at most 1) and it converges we won t worry about the details behind this or any other issues of convergence necessary to justify the steps below - they are all straight-forward). P(A) = ½ + (½)*(¼) + (½)*(¼) 2 + (½)*(¼) 3 + P(A) = ½ * (1 + ¼ + (¼) 2 + (¼) 3 + ) The infinite sum in brackets is known as a geometric series and there is a simple way to solve such infinite sums which we will review briefly here for the benefit of those who may not have seen it before. Recall the following facts from algebra for factoring the difference of two squares or two cubes and it s generalization to higher powers: 1 2 x 2 = (1 - x) * (1 + x) 1 + x = (1 2 x 2 ) / (1 - x) 1 3 x 3 = (1 - x) * (1 + x + x 2 ) 1 + x + x 2 = (1 3 x 3 ) / (1 - x) 1 4 x 4 = (1 - x) * (1 + x + x 2 + x 3 ) 1 + x + x 2 + x 3 = (1 3 x 3 ) / (1 - x) 1 n x n = (1 - x) * (1 + x + x x n-1 ) 1 + x + x x n-1 = (1 n x n ) / (1 - x) These equations hold for any x not equal to 1. Now when x happens to be a probability less than 1 (or any number with absolute value less than 1), as n gets large, x n gets very small and rapidly becomes indistinguishable from 0. Pick any number between 0 and 1, punch it in to your calculator and then press the x 2 button over and over. It won t take long before your calculator display is nothing but 0 s. Thus to compute the infinite sum 1 + x + x 2 +, provided x < 1 we can assume that x n is zero in the equation on the right to obtain a very simple solution: 1 + x + x 2 + x 3 + = 1 / (1 x) provided that x < 1 3

4 We can now use this handy equation to solve our problem by plugging in ¼ for x. which gives us P(A) = ½ * (1 / (1- ¼)) = ½ * 4/3 = 2/3. Thus Alice will win 2/3 of the time. The analysis above can easily be generalized to a non-fair coin simply replace ½ by p and ¼ by (1-p) 2. The last problem we will look at is one that is often considered a difficult problem, but can be solved quite easily using the approach above. Problem #5: A fair coin is flipped repeatedly. What is the probability of getting 3 heads in a row before 5 tails in a row. Non-Solution #5: At first glance it might seem that since the probability of getting 3 heads in a row is 1/8 and the probability of getting 5 tails in a row is 1/32, that getting 3 heads in a row first should occur 4 times as often. This is not quite right however consider for example that there are 8 ways to get a run of 3 heads in the first 5 flips but only 1 way to get 5 tails. Trying to compute the exact probability of runs of various lengths quickly becomes complex, especially since we don t know exactly how many flips may be required to produce a run of 3 heads or 5 tails. Another approach is to try to count sequences of binary strings of a given length that end in a run of 3 heads or 5 tails but don t contain any other runs of either type. This can be done, but it requires finding and solving the right recurrence relation. Fortunately there is an easier way. Solution #5: Let s look at the situation as a game between two players Alice and Bob, where Alice is trying to get a run of 3 heads and Bob is trying for a run of 5 tails. They use the first coin toss to decide who goes first, and then that player continues flipping until either they win, or the get the wrong toss in which case they hand the coin to the other person who takes their turn. For example if the first toss is heads, Alice then flips the coin hoping to get 2 more heads in a row, but if she flips a tail before then, Bob takes over and tries to get 4 more tails (the first tail occurred when Alice s turn ended). Note that after the initial flip, every turn looks the same Alice tries to get 2 heads in a row, and Bob tries to get 4 tails in a row. Alice always has a 1/4 th chance of winning on her turn, and Bob always has a 1/16 th chance. We can now compute the probabilities using an analysis similar to the one above, except there are two cases depending on the outcome of the first flip. As above, let Let A k be the event that Alice wins on her k th turn, and let H be the event that the first flip is heads and let T be the complementary event. P(A 1 H) = ¼ P(A 2 H) = (3/4)*(15/16)*(1/4) P(A 3 H) = [(3/4)*(15/16)] 2 *(1/4) P(A k H) = [(3/4)*(15/16)] k *(1/4) A wins on her first turn A doesn t win, B doesn t win, A wins A and B don t win twice, A wins A and B don t win twice, A wins As above, these events are disjoint and we have: P(A H) = P(A 1 H) + P(A 2 H) + P(A 3 H) + P(A H) = ¼ * (1 + 45/64 + (45/64) 2 + (45/64) 3 + ) P(A H) = ¼ * [1 / (1 45/64)] = ¼ * 64/19 = 16/19 If Bob goes first, the situation is identical except that Bob needs to lose on his first turn: P(A 1 T) = (15/16)*¼ B doesn t win then A wins on her first turn P(A k T) = (15/16)[(3/4)*(15/16)] k *(1/4) B doesn t win, A and B don t win, then A wins Every equation simply has the factor 15/16 in it, so we have P(A T) = 15/16 * P(A H) = 15/19. Putting it all together using the Law of Alternatives we have: P(A) = P(H)P(A H) + P(T)P(A T) = (½) * (16/19) + (½) * (15/19) = 31/38 Note that this is slightly higher than we would get if we used our initial erroneous assumption that Alice wins 4 times as often as Bob which would have given 4/5. 4

5 MIT OpenCourseWare Combinatorics: The Fine Art of Counting Summer 2007 For information about citing these materials or our Terms of Use, visit:

### Examples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form

Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,

### Massachusetts Institute of Technology

n (i) m m (ii) n m ( (iii) n n n n (iv) m m Massachusetts Institute of Technology 6.0/6.: Probabilistic Systems Analysis (Quiz Solutions Spring 009) Question Multiple Choice Questions: CLEARLY circle the

### Toss a coin twice. Let Y denote the number of heads.

! Let S be a discrete sample space with the set of elementary events denoted by E = {e i, i = 1, 2, 3 }. A random variable is a function Y(e i ) that assigns a real value to each elementary event, e i.

### Chapter 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,

### Probabilistic Strategies: Solutions

Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6-sided dice. What s the probability of rolling at least one 6? There is a 1

### JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability)

JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability) Author: Daniel Cooney Reviewer: Rachel Zax Last modified: January 4, 2012 Reading from Meyer, Mathematics

### Lecture 2: Probability

Lecture 2: Probability Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics İstanbul Kültür University Outline 1 Introduction 2 Sample Spaces 3 Event 4 The Probability

### Probability distributions

Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.14-2.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,

### Probability: 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

### Lecture 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.

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### Question: 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

### Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

### Basics of Probability

Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.

### Discrete 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

### + Section 6.2 and 6.3

Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

### 6.042/18.062J Mathematics for Computer Science. Expected Value I

6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

### 6.3 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

### P (A) = lim P (A) = N(A)/N,

1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

### Lecture 4: Probability Spaces

EE5110: Probability Foundations for Electrical Engineers July-November 2015 Lecture 4: Probability Spaces Lecturer: Dr. Krishna Jagannathan Scribe: Jainam Doshi, Arjun Nadh and Ajay M 4.1 Introduction

### Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution

Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.

### Sums of Independent Random Variables

Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables

### Discrete 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,

### 4. Joint Distributions

Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose

### MATHEMATICS 117/E-217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions )

MATHEMATICS 117/E-217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions ) Last modified: May 1, 2012 Reading Dineen, pages 57-67 Proof

### Math/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

### Expected Value and the Game of Craps

Expected Value and the Game of Craps Blake Thornton Craps is a gambling game found in most casinos based on rolling two six sided dice. Most players who walk into a casino and try to play craps for the

### Introduction to probability theory in the Discrete Mathematics course

Introduction to probability theory in the Discrete Mathematics course Jiří Matoušek (KAM MFF UK) Version: Oct/18/2013 Introduction This detailed syllabus contains definitions, statements of the main results

### MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables

MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you

### Discrete Mathematics Lecture 5. Harper Langston New York University

Discrete Mathematics Lecture 5 Harper Langston New York University Empty Set S = {x R, x 2 = -1} X = {1, 3}, Y = {2, 4}, C = X Y (X and Y are disjoint) Empty set has no elements Empty set is a subset of

### PROBABILITY NOTIONS. Summary. 1. Random experiment

PROBABILITY NOTIONS Summary 1. Random experiment... 1 2. Sample space... 2 3. Event... 2 4. Probability calculation... 3 4.1. Fundamental sample space... 3 4.2. Calculation of probability... 3 4.3. Non

### Finite and discrete probability distributions

8 Finite and discrete probability distributions To understand the algorithmic aspects of number theory and algebra, and applications such as cryptography, a firm grasp of the basics of probability theory

### Unit 19: Probability Models

Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

A Bit About the St. Petersburg Paradox Jesse Albert Garcia March 0, 03 Abstract The St. Petersburg Paradox is based on a simple coin flip game with an infinite expected winnings. The paradox arises by

### 6.3 Probabilities with Large Numbers

6.3 Probabilities with Large Numbers In general, we can t perfectly predict any single outcome when there are numerous things that could happen. But, when we repeatedly observe many observations, we expect

### 3. Examples of discrete probability spaces.. Here α ( j)

3. EXAMPLES OF DISCRETE PROBABILITY SPACES 13 3. Examples of discrete probability spaces Example 17. Toss n coins. We saw this before, but assumed that the coins are fair. Now we do not. The sample space

### MASSACHUSETTS 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

### Study Manual for Exam P/Exam 1. Probability

Study Manual for Exam P/Exam 1 Probability Eleventh Edition by Krzysztof Ostaszewski Ph.D., F.S.A., CFA, M.A.A.A. Note: NO RETURN IF OPENED TO OUR READERS: Please check A.S.M. s web site at www.studymanuals.com

### Probabilities and Random Variables

Probabilities and Random Variables This is an elementary overview of the basic concepts of probability theory. 1 The Probability Space The purpose of probability theory is to model random experiments so

### Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80)

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you

### It is not immediately obvious that this should even give an integer. Since 1 < 1 5

Math 163 - Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some

### AP Stats - Probability Review

AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose

### V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay \$ to play. A penny and a nickel are flipped. You win \$ if either

### Study Manual for Exam P/Exam 1. Probability

Study Manual for Exam P/Exam 1 Probability Seventh Edition by Krzysztof Ostaszewski Ph.D., F.S.A., CFA, M.A.A.A. Note: NO RETURN IF OPENED TO OUR READERS: Please check A.S.M. s web site at www.studymanuals.com

### Events. Independence. Coin Tossing. Random Phenomena

Random Phenomena Events A random phenomenon is a situation in which we know what outcomes could happen, but we don t know which particular outcome did or will happen For any random phenomenon, each attempt,

### 1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let

Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as

### Discrete probability and the laws of chance

Chapter 8 Discrete probability and the laws of chance 8.1 Introduction In this chapter we lay the groundwork for calculations and rules governing simple discrete probabilities. These steps will be essential

### Basic 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

### Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event

### E3: 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............................

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Lecture Notes Counting 101 Note to improve the readability of these lecture notes, we will assume that multiplication takes precedence over division, i.e. A / B*C

### Lab 11. Simulations. The Concept

Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that

### Economics 1011a: Intermediate Microeconomics

Lecture 11: Choice Under Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 11: Choice Under Uncertainty Tuesday, October 21, 2008 Last class we wrapped up consumption over time. Today we

### Probability 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

### WORKED EXAMPLES 1 TOTAL PROBABILITY AND BAYES THEOREM

WORKED EXAMPLES 1 TOTAL PROBABILITY AND BAYES THEOREM EXAMPLE 1. A biased coin (with probability of obtaining a Head equal to p > 0) is tossed repeatedly and independently until the first head is observed.

### Lecture 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

### The Binomial Distribution

The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing

### Contents. TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics. Yuming Jiang. Basic Concepts Random Variables

TTM4155: Teletraffic Theory (Teletrafikkteori) Probability Theory Basics Yuming Jiang 1 Some figures taken from the web. Contents Basic Concepts Random Variables Discrete Random Variables Continuous Random

### Lecture 2 Binomial and Poisson Probability Distributions

Lecture 2 Binomial and Poisson Probability Distributions Binomial Probability Distribution l Consider a situation where there are only two possible outcomes (a Bernoulli trial) H Example: u flipping a

### You flip a fair coin four times, what is the probability that you obtain three heads.

Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

### Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014

Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities

### Basic concepts in probability. Sue Gordon

Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are

### Random 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

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### Geometric Series. A Geometric Series is an infinite sum whose terms follow a geometric progression that is, each pair of terms is in the same ratio.

Introduction Geometric Series A Geometric Series is an infinite sum whose terms follow a geometric progression that is, each pair of terms is in the same ratio. A simple example wound be: where the first

### Generating Functions Count

2 Generating Functions Count 2.1 Counting from Polynomials to Power Series Consider the outcomes when a pair of dice are thrown and our interest is the sum of the numbers showing. One way to model the

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### 2. Discrete Random Variables Part III: St Petersburg Paradox. ECE 302 Fall 2009 TR 3 4:15pm Purdue University, School of ECE Prof.

2. Discrete Random Variables Part III: St Petersburg Paradox ECE 302 Fall 2009 TR 3 4:15pm Purdue University, School of ECE Prof. Ilya Pollak Problem 2.21: St Petersburg Paradox A coin is flipped repeatedly

### FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL

FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint

### Probability, statistics and football Franka Miriam Bru ckler Paris, 2015.

Probability, statistics and football Franka Miriam Bru ckler Paris, 2015 Please read this before starting! Although each activity can be performed by one person only, it is suggested that you work in groups

### MAT 1000. Mathematics in Today's World

MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities

### The Advantage Testing Foundation Solutions

The Advantage Testing Foundation 013 Problem 1 The figure below shows two equilateral triangles each with area 1. The intersection of the two triangles is a regular hexagon. What is the area of the union

### Chapter 4 Probability

The Big Picture of Statistics Chapter 4 Probability Section 4-2: Fundamentals Section 4-3: Addition Rule Sections 4-4, 4-5: Multiplication Rule Section 4-7: Counting (next time) 2 What is probability?

### A 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

### , for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0

Chapter 4 The Poisson Distribution 4.1 The Fish Distribution? The Poisson distribution is named after Simeon-Denis Poisson (1781 1840). In addition, poisson is French for fish. In this chapter we will

### Ch. 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

### 6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games

6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4, 2009 1 Introduction Outline Decisions, utility maximization Strategic form games Best responses

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### Prediction Markets, Fair Games and Martingales

Chapter 3 Prediction Markets, Fair Games and Martingales Prediction markets...... are speculative markets created for the purpose of making predictions. The current market prices can then be interpreted

### Chapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.

MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.

### An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event

An event is any set of outcomes of a random experiment; that is, any subset of the sample space of the experiment. The probability of a given event is the sum of the probabilities of the outcomes in the

### Lecture 13. Understanding Probability and Long-Term Expectations

Lecture 13 Understanding Probability and Long-Term Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).

### Grade 7/8 Math Circles Fall 2012 Probability

1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics

### Section 1.3 P 1 = 1 2. = 1 4 2 8. P n = 1 P 3 = Continuing in this fashion, it should seem reasonable that, for any n = 1, 2, 3,..., = 1 2 4.

Difference Equations to Differential Equations Section. The Sum of a Sequence This section considers the problem of adding together the terms of a sequence. Of course, this is a problem only if more than

### Ching-Han Hsu, BMES, National Tsing Hua University c 2014 by Ching-Han Hsu, Ph.D., BMIR Lab

Lecture 2 Probability BMIR Lecture Series in Probability and Statistics Ching-Han Hsu, BMES, National Tsing Hua University c 2014 by Ching-Han Hsu, Ph.D., BMIR Lab 2.1 1 Sample Spaces and Events Random

### 1 Gambler s Ruin Problem

Coyright c 2009 by Karl Sigman 1 Gambler s Ruin Problem Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial fortune of \$i and then on each successive gamble either wins

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### Chapter 1 Axioms of Probability. Wen-Guey Tzeng Computer Science Department National Chiao University

Chapter 1 Axioms of Probability Wen-Guey Tzeng Computer Science Department National Chiao University Introduction Luca Paccioli(1445-1514), Studies of chances of events Niccolo Tartaglia(1499-1557) Girolamo

### Chapter 31 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M.

31 Geometric Series Motivation (I hope) Geometric series are a basic artifact of algebra that everyone should know. 1 I am teaching them here because they come up remarkably often with Markov chains. The

### Lecture 2 : Basics of Probability Theory

Lecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### Definition of Random Variable A random variable is a function from a sample space S into the real numbers.

.4 Random Variable Motivation example In an opinion poll, we might decide to ask 50 people whether they agree or disagree with a certain issue. If we record a for agree and 0 for disagree, the sample space

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover

### The basics of probability theory. Distribution of variables, some important distributions

The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a

### The 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

### MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS THE BINOMIAL THEOREM. (p + q) 0 =1,

THE BINOMIAL THEOREM Pascal s Triangle and the Binomial Expansion Consider the following binomial expansions: (p + q) 0 1, (p+q) 1 p+q, (p + q) p +pq + q, (p + q) 3 p 3 +3p q+3pq + q 3, (p + q) 4 p 4 +4p