MATHEMATICS 154, SPRING 2010 PROBABILITY THEORY Outline #3 (Combinatorics, bridge, poker)


 Dorothy Marsh
 2 years ago
 Views:
Transcription
1 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 Ex(One Thousand Exercises in Probability, by Grimmett and Stirzaker), page 5. Web pages on poker and bridge from Principles of counting In the following items, use these principles of counting, taken for granted in PRP. Principle : Multiply for sequential counting. If we are forming a set of ntuples where there are k choices for the first element x, k for x, and so on, the number of ntuples x x...x n in the set is k k...k n Principle : Divide to correct systematic overcounting. If M different ntuples all correspond to the same element in our sample space, count the ntuples and then divide by the number of times M that each was overcounted. Principle : Divide and Conquer. To count the number of elements in a union of disjoint subsets, count each subset and sum the results. Principle : Subtract off special cases. To count the number of elements in a difference A \ B where B A, count each set and take the difference. Classic example of principles and : counting the number of kelement subsets of an nelement set. First count the ktuples whose elements are all different: n(n )(n )...(n k + ). Then divide by k! to correct for overcounting. The result is the familiar combinations formula ( ) n n! = k k!(n k)!
2 Using binomial coefficients whenever possible is the safest way to avoid accidental overcounting. For the following topics, the easiest way to get numerical answers is to use Maple or Mathematica. Any Harvard undergraduate can download a copy for free. For details, go to and follow instructions.. Counting poker hands Count the number of ways to get each of the following types of 5card poker hands, using a deck of 5 cards with cards of each of ranks, and use this to calculate the probability of being dealt such a hand. of a kind(four cards of one rank, the fifth of a different rank) a full house(three cards of one rank, two of another) of a kind(three cards of one rank, two others of different ranks) Solutions: How many distinct 5card poker hands can be dealt from a 5card deck? Answer: Select 5 cards sequentially: the number of ways to do this is But this generates each distinct hand 5! = 0 times. So the number of distinct hands is ( 5 5 ) = 5! 7!5! =, 598, 960 How many distinct ways are there to get of a kind, and what is the probability? choices for the rank of the, 8 choices for the other card. P = 8, 58, 960 = A full house ( of one rank, of another) choices for the first rank (with cards), for the suit that is missing. choices for the second rank (with cards), ( ) = 6 for the pair of suits that have the second rank. 6 P =, 58, 960 = Three of a kind ( of one rank, the other two are of different ranks) ( choices for the first rank (with cards), for the suit that is missing. ) = 66 choices for the pair of ranks of the other two cards; 6 choices for the suit of each. So the number is 66 6 = 5, 9 and P = 5, 9, 58, 960 = 0.085
3 . Bridge problems A bridge hand consists of cards from a 5card deck. Count the number of bridge hands with 6 spades, hearts, diamonds, and club. Count the number of bridge hands with 6 suit distribution (6 cards in the longest suit, in the secondlongest, in the thirdlongest) Count the number of bridge hands with  or  suit distribution, and show that the former has a higher probability. Solution: How many ways are there to select 6 cards from the spades? ( ) =! 6 7! 6! How many distinct hands have 6 spades, hearts, diamonds, club? Apply the same analysis to each suit in turn: ( )( )( )( ) =!!!! 6 7! 6! 9!!!!!! How many distinct hands have a 6 distribution? Multiply the preceding number by the number of ways to choose the 6card suit, the card suit, etc, which is!=. How many distinct hands have a  distribution? There are ways to select the card suit followed by ways to select the card suit, so we will multiply by. ( N = )( )( )( ) =!!!! 9!! 9!! 0!!!! How many distinct hands have a  distribution? Now there are ways to select the card suit, so we will multiply only by. ( N = The ratio is )( )( )( ) =!!!! 9!! 0!! 0!! 0!! P = N 0!! 0!! = 5 P N 9!!!!
4 . Dividing up a set with two type of objects Suppose we have a set S with n elements, m of one type and n m of the other. As a concrete example, suppose that Lisa has n = 6 Dunkin Munchkins, of which m = are chocolate and the remaining n m = are plain. She chooses n = of them at random and puts them in a bag for her son Thomas. The remaining three go into a bag for her daughter Catherine. In general we divide up S at random into set S and set S, each with n elements. The number of different ways of doing this is equal to the number of ways of selecting the n elements of S, namely N = ( ) n = (n)! n n!n! We are interested in the probability that k of the special objects (chocolate Munchkins) end up in set S. Assume that k m, k n, and k m n. The number of ways of selecting k special objects and n k nonspecial objects for set is ( )( ) m n m m! (n m)! M = = k n k k!(m k)! (n k)!(n + k m)! If we assume that each way of selecting the elements of S is equally likely, the ratio M gives the probability that set S N contains precisely m special objects. As a practical matter, it is much easier to repeat this reasoning in special cases than to memorize these formulas!
5 5. Waring s Theorem This useful result is proved cleverly in chapter 5 of PRP. Here is a straightforward proof inspired by the solution to problem on page of PRP (solutions on page of 000Ex). It avoids the use of the result from problem. We have n events A i, i = {,,...n}. Examples: Toss coins; A i is the ith coin comes up heads. Place 5 cups of different colors randomly on 5 colored saucers; A i is the ith cup (blue) lands on the blue saucer. Buy 6 boxes of cereal, each containing a bust of a recent president; A i is at least i of your 6 busts is a Barack Obama. Let S be a subset of {,,...n} like the first and fourth coin tosses the red, blue, and yellow cups Clinton and Bush Define events: A S = all A i for i S occur, and maybe others too. B S = no event A i for i outside S occurs. So A S B S = all A i for i S occur, and no others. This is one specific way that precisely k events can occur. For different choices of S, such events are disjoint. Sum over S to get the probability of their union. N k = precisely k events occur. In many cases, P(A S ) is easy to calculate, but P(N k ) is a pain to determine. Waring s theorem relates the two: P(N k ) = ( ) k + P(A S ) k Proof: S =k T =(k+) ( ) k + P(A T )+ k T =(k+) P(A T ) Start, as usual, with a disjoint union: P(A S ) = P(A S B S ) + P(A S B c S) 5
6 P(A S B S ) = P(A S ) P(A S j / S A j ) P(A S B S ) = P(A S ) P( j / S(A S A j )). Now apply generalized inclusionexclusion to the union in the right term: P(A S B S ) = P(A S ) j / S P(A S A j ) + j<k / S P(A S A j A k ) Finally, sum over all subsets of {,,...n}, for sets of size k, k+, k+,, n to get the theorem: P(N k ) = P(A S B S ) = S =k To avoid confusion, I used the generic symbol T for sets with more than k elements. Each set T with k + m elements will arise in the sum ( ) k+m k times, once from each of its kelement subsets that can serve as S. For the problem of placing R, B, Y, and G cups on matching saucers, the probability of getting exactly one cup on the saucer of the same color is, according to Waring s theorem, ( ) ( ) ( ) P(N ) = P(A ) 6 P(A A )+ P(A A A ) P(A A A A ) P(N ) = + = + 6 = 6
7 6. Waring s theorem applied to coin tossing What is the probability of getting exactly one head when you toss three fair coins? Fundamental events: A i is the event that coin i comes up heads. Clearly P(A i ) =. Since the tosses are independent events, P(A i A j ) = for i j, and P(A A A ) =. 8 Since the probability of all heads is the same for any set of coins of a given size, we need only consider probabilities like P(A ) and P(A A ). The summation required by Waring s theorem can be accomplished by multiplying by the number of relevant events. N is the event that exactly one coin comes up heads. Waring says ( ) P(N ) = P(A ) P(A A ) + P(N ) = = 8. ( ) P(A A A ). What is the probability of getting exactly two heads when you toss three fair coins? N is the event that exactly two coins comes up heads. Waring says P(N ) = P(A A ) 7. Waring s theorem applied to cards P(N ) = 8 = 8. ( ) P(A A A ). What is the probability of getting exactly one spade when you select two cards from a shuffled deck? A i is the event that card i is a spade. In this case P(A i ) = for each card. However, P(A A ) = 5 5 = 7. N is the event that exactly one card is a spade. Waring says P(N ) = P(A ) ( ) P(A A ). P(N ) = 7 =. 7
8 8. Waring s theorem applied to dice What is the probability of getting exactly one 6 when you roll three fair dice? A i is the event that die i comes up 6. Clearly P(A i ) =. Since the tosses are independent events, 6 P(A i A j ) = for i j, and P(A 6 A A ) =. 6 Since P(A ) = P(A ) = P(A ) and P(A A ) = P(A A ) = P(A A ), we need only consider probabilities like P(A ) and P(A A ). The summation required by Waring s theorem can be accomplished by multiplying by the number of relevant events. N is the event that exactly one die comes up 6. Waring says ( ) P(N ) = P(A ) P(A A ) + P(N ) = = ( ) P(A A A ). N is the event that exactly two dice come up 6. Waring says P(N ) = P(A A ) ( ) P(A A A ). P(N ) = 6 6 = 5 6. N 0 is the event that no die comes up 6. Waring says ( ) ( ) P(N 0 ) = P(A ) P(A A ) P(N 0 ) = = 5 6. ( ) P(A A A ). 0 8
Remember to leave your answers as unreduced fractions.
Probability Worksheet 2 NAME: Remember to leave your answers as unreduced fractions. We will work with the example of picking poker cards out of a deck. A poker deck contains four suits: diamonds, hearts,
More 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 informationTopic 1 Probability spaces
CSE 103: Probability and statistics Fall 2010 Topic 1 Probability spaces 1.1 Definition In order to properly understand a statement like the chance of getting a flush in fivecard poker is about 0.2%,
More informationStatistics 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 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 informationMath 2001 Homework #10 Solutions
Math 00 Homework #0 Solutions. Section.: ab. For each map below, determine the number of southerly paths from point to point. Solution: We just have to use the same process as we did in building Pascal
More informationNotes on counting finite sets
Notes on counting finite sets Murray Eisenberg February 26, 2009 Contents 0 Introduction 2 1 What is a finite set? 2 2 Counting unions and cartesian products 4 2.1 Sum rules......................................
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 informationMA 1125 Lecture 14  Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.
MA 5 Lecture 4  Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
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 informationQuestion: What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationPoker Probability from Wikipedia. Frequency of 5card poker hands 36 0.00139% 0.00154% 72,192.33 : 1
Poker Probability from Wikipedia Frequency of 5card 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 informationDiscrete 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
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 informationnumber of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.
12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 7, due Wedneday, March 14 Happy Pi Day! (If any errors are spotted, please email them to morrison at math dot berkeley dot edu..5.10 A croissant
More 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 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 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 informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
More informationMAT 1000. Mathematics in Today's World
MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities
More 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 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 informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6sided dice. What s the probability of rolling at least one 6? There is a 1
More 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 informationCombinations and Permutations
Combinations and Permutations What's the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination
More informationHomework 2 Solutions
CSE 21  Winter 2012 Homework #2 Homework 2 Solutions 2.1 In this homework, we will consider ordinary decks of playing cards which have 52 cards, with 13 of each of the four suits (Hearts, Spades, Diamonds
More informationSTAB47S:2003 Midterm Name: Student Number: Tutorial Time: Tutor:
STAB47S:200 Midterm Name: Student Number: Tutorial Time: Tutor: Time: 2hours Aids: The exam is open book Students may use any notes, books and calculators in writing this exam Instructions: Show your reasoning
More informationChapter 4  Practice Problems 1
Chapter 4  Practice Problems SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) Compare the relative frequency formula
More informationWorked examples Basic Concepts of Probability Theory
Worked examples Basic Concepts of Probability Theory Example 1 A regular tetrahedron is a body that has four faces and, if is tossed, the probability that it lands on any face is 1/4. Suppose that one
More informationExam. Name. How many distinguishable permutations of letters are possible in the word? 1) CRITICS
Exam Name How many distinguishable permutations of letters are possible in the word? 1) CRITICS 2) GIGGLE An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm,
More informationP(X = x k ) = 1 = k=1
74 CHAPTER 6. IMPORTANT DISTRIBUTIONS AND DENSITIES 6.2 Problems 5.1.1 Which are modeled with a unifm distribution? (a Yes, P(X k 1/6 f k 1,...,6. (b No, this has a binomial distribution. (c Yes, P(X k
More informationProbabilities of Poker Hands with Variations
Probabilities of Poker Hands with Variations Jeff Duda Acknowledgements: Brian Alspach and Yiu Poon for providing a means to check my numbers Poker is one of the many games involving the use of a 52card
More informationCoin Flip Questions. Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT.
Coin Flip Questions Suppose you flip a coin five times and write down the sequence of results, like HHHHH or HTTHT. 1 How many ways can you get exactly 1 head? 2 How many ways can you get exactly 2 heads?
More informationName: Date: Use the following to answer questions 24:
Name: Date: 1. A phenomenon is observed many, many times under identical conditions. The proportion of times a particular event A occurs is recorded. What does this proportion represent? A) The probability
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 informationProbabilities 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
More informationAP Statistics 7!3! 6!
Lesson 64 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!
More information+ 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
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 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 informationProbability definitions
Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a datagenerating
More informationContemporary Mathematics Online Math 1030 Sample Exam I Chapters 1214 No Time Limit No Scratch Paper Calculator Allowed: Scientific
Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 1214 No Time Limit No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the lefthand margin. You
More informationFind the indicated probability. 1) If a single fair die is rolled, find the probability of a 4 given that the number rolled is odd.
Math 0 Practice Test 3 Fall 2009 Covers 7.5, 8.8.3 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. ) If a single
More informationJan 17 Homework Solutions Math 151, Winter 2012. Chapter 2 Problems (pages 5054)
Jan 17 Homework Solutions Math 11, Winter 01 Chapter Problems (pages 0 Problem In an experiment, a die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample
More 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 informationPROBABILITY. Chapter Overview
Chapter 6 PROBABILITY 6. Overview Probability is defined as a quantitative measure of uncertainty a numerical value that conveys the strength of our belief in the occurrence of an event. The probability
More informationChapter 3: The basic concepts of probability
Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording
More informationDiscrete mathematics
Discrete mathematics Petr Kovář petr.kovar@vsb.cz VŠB Technical University of Ostrava DiM 4702301/01, Winter term 2015/2016 About this file This file is meant to be a guideline for the lecturer. Many
More informationAMS 5 CHANCE VARIABILITY
AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and
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 information1 Introduction to Counting
1 Introduction to Counting 1.1 Introduction In this chapter you will learn the fundamentals of enumerative combinatorics, the branch of mathematics concerned with counting. While enumeration problems can
More informationSection 65 Sample Spaces and Probability
492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)
More 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 informationPROBABILITY 14.3. section. The Probability of an Event
4.3 Probability (43) 727 4.3 PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques
More 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 informationDetermine the empirical probability that a person selected at random from the 1000 surveyed uses Mastercard.
Math 120 Practice Exam II Name You must show work for credit. 1) A pair of fair dice is rolled 50 times and the sum of the dots on the faces is noted. Outcome 2 4 5 6 7 8 9 10 11 12 Frequency 6 8 8 1 5
More information(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball.
Examples for Chapter 3 Probability Math 10401 Section 3.1 1. Draw a tree diagram for each of the following situations. State the size of the sample space. (a) You flip a coin three times. (b) You draw
More 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 informationReview for Test 2. Chapters 4, 5 and 6
Review for Test 2 Chapters 4, 5 and 6 1. You roll a fair sixsided die. Find the probability of each event: a. Event A: rolling a 3 1/6 b. Event B: rolling a 7 0 c. Event C: rolling a number less than
More informationBasic concepts in probability. Sue Gordon
Mathematics Learning Centre Basic concepts in probability Sue Gordon c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Set Notation You may omit this section if you are
More informationDiscrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22
CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette
More informationGaming the Law of Large Numbers
Gaming the Law of Large Numbers Thomas Hoffman and Bart Snapp July 3, 2012 Many of us view mathematics as a rich and wonderfully elaborate game. In turn, games can be used to illustrate mathematical ideas.
More informationFormula for Theoretical Probability
Notes Name: Date: Period: Probability I. Probability A. Vocabulary is the chance/ likelihood of some event occurring. Ex) The probability of rolling a for a sixfaced die is 6. It is read as in 6 or out
More informationSession 8 Probability
Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome
More informationProbability. Chapter Introduction to Probability. Why do we study probability?
Chapter 5 Probability 5.1 Introduction to Probability Why do we study probability? Youhave likely studied hashing as a way to store data (or keys to find data) in a way that makes it possible to access
More informationStats Review Chapters 56
Stats Review Chapters 56 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test
More informationProbability. 4.1 Sample Spaces
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 cointoss experiment, S would be the results
More informationQuestion of the Day. Key Concepts. Vocabulary. Mathematical Ideas. QuestionofDay
QuestionofDay Question of the Day What is the probability that in a family with two children, both are boys? What is the probability that in a family with two children, both are boys, if we already know
More informationREPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.
REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game
More informationChapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.
Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More informationhttps://assessment.casa.uh.edu/assessment/printtest.htm PRINTABLE VERSION Quiz 10
1 of 8 4/9/2013 8:17 AM PRINTABLE VERSION Quiz 10 Question 1 Let A and B be events in a sample space S such that P(A) = 0.34, P(B) = 0.39 and P(A B) = 0.19. Find P(A B). a) 0.4872 b) 0.5588 c) 0.0256 d)
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 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 OPRE 6301
Probability OPRE 6301 Random Experiment... Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.
More informationLECTURE 3. Probability Computations
LECTURE 3 Probability Computations Pg. 67, #42 is one of the hardest problems in the course. The answer is a simple fraction there should be a simple way to do it. I don t know a simple way I used the
More 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 informationWhat is the probability of throwing a fair die and receiving a six? Introduction to Probability. Basic Concepts
Basic Concepts Introduction to Probability A probability experiment is any experiment whose outcomes relies purely on chance (e.g. throwing a die). It has several possible outcomes, collectively called
More informationSECTION 105 Multiplication Principle, Permutations, and Combinations
105 Multiplication Principle, Permutations, and Combinations 761 54. Can you guess what the next two rows in Pascal s triangle, shown at right, are? Compare the numbers in the triangle with the binomial
More 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 is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty.
Chapter 1 Probability Spaces 11 What is Probability? Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty Example 111
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 informationProbability. 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
More informationIntroductory 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
More informationContemporary Mathematics MAT 130. Probability. a) What is the probability of obtaining a number less than 4?
Contemporary Mathematics MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than
More information6.042/18.062J 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
More informationAn Introduction to Combinatorics and Graph Theory. David Guichard
An Introduction to Combinatorics and Graph Theory David Guichard This work is licensed under the Creative Commons AttributionNonCommercialShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/byncsa/3.0/
More informationA natural introduction to probability theory. Ronald Meester
A natural introduction to probability theory Ronald Meester ii Contents Preface v 1 Experiments 1 1.1 Definitions and examples........................ 1 1.2 Counting and combinatorics......................
More information4.3. Addition and Multiplication Laws of Probability. Introduction. Prerequisites. Learning Outcomes. Learning Style
Addition and Multiplication Laws of Probability 4.3 Introduction When we require the probability of two events occurring simultaneously or the probability of one or the other or both of two events occurring
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 informationProbability And Odds Examples
Probability And Odds Examples. Will the Cubs or the Giants be more likely to win the game? What is the chance of drawing an ace from a deck of cards? What are the possibilities of rain today? What are
More informationCurriculum Design for Mathematic Lesson Probability
Curriculum Design for Mathematic Lesson Probability This curriculum design is for the 8th grade students who are going to learn Probability and trying to show the easiest way for them to go into this class.
More informationCombinatorial Probability
Chapter 2 Combinatorial Probability 2.1 Permutations and combinations As usual we begin with a question: Example 2.1. The New York State Lottery picks 6 numbers out of 54, or more precisely, a machine
More information3. Discrete Probability. CSE 312 Autumn 2011 W.L. Ruzzo
3. Discrete Probability CSE 312 Autumn 2011 W.L. Ruzzo sample spaces Sample space: S is the set of all possible outcomes of an experiment (Ω in your text book Greek uppercase omega) Coin flip: S = {Heads,
More informationDiscrete Math for Computer Science Students
Discrete Math for Computer Science Students Ken Bogart Dept. of Mathematics Dartmouth College Scot Drysdale Dept. of Computer Science Dartmouth College Cliff Stein Dept. of Industrial Engineering and Operations
More informationIn the situations that we will encounter, we may generally calculate the probability of an event
What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead
More informationECE302 Spring 2006 HW1 Solutions January 16, 2006 1
ECE302 Spring 2006 HW1 Solutions January 16, 2006 1 Solutions to HW1 Note: These solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in italics
More information4. 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
More informationProbability and Expected Value
Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are
More informationGenerating 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
More information