Computing Fundamentals 2 Lecture 6 Probability. Lecturer: Patrick Browne
|
|
- Evan Cole
- 7 years ago
- Views:
Transcription
1 Computing Fundamentals 2 Lecture 6 Probability Lecturer: Patrick Browne
2 Probability If a die is thrown we consider it certain that it will land, with a random chance that it will show a 6. With s successes out of n experiments f=s/n is called the relative frequency of success. It becomes stable in the long run. It is this long term stability (limit) that forms the basis of probability.
3 Sample Space and Events The sample space S is the set of all possible outcomes of a given experiment. An element or outcome in S is called a sample point (or sample). An event A is a set of outcomes, it is a subset of the sample space. The singleton {a} where a S is called an elementary event. The empty set,, sometimes represents an impossible event.
4 Sample Space and Events An event gives rise to a set hence we can use set operations to combine events. A B is the event that occurs whenever A occurs or B occurs (or both) A B is the event that occurs whenever A and B both occur. A c is the event that occurs whenever A does not occur (called the complement of A) Two events are mutually exclusive if they are disjoint: A B =.
5 Sample Space and Events Toss a die and observe the top number S={1,2,3,4,5,6} A even number event, B odd number event, C prime number event. A={2,4,6} B={1,3,5} C={2,3,5} A C ={2,3,4,5,6} B C = {3,5} C c = {1,4,6} (1 is not prime) A and B are mutually exclusive.
6 Sample Space and Events Toss 3 coins and observe the H & T sequence S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT} Let A be the two consecutive heads event, B same outcome event. A={HHH,HHT,THH} B={HHH,TTT} A B = {HHH} is the elementary event with only heads.
7 Probability Spaces A probability space is a triple (S, A, P), where: S sample space, all possible outcomes. A event space, sample events/outcomes P is a probability measure. We also have a set of probability axioms e.g. the probability of an event is a nonnegative real number.
8 Probability Spaces A probability space consists of a sample space together with a positive, additive measure, called a probability measure, which sums to one; the points of the sample space represent the different possible outcomes of the phenomenon, and the probability measure assigns probabilities to sets of outcomes.
9 Finite Probability Spaces Let S be a finite sample space S={a 1,a 2,a 3...a n }. A finite probability space is obtained by assigning to each sample point a i S a real number p i, called the probability of a i satisfying the following conditions: Each p i is non-negative. The sum of p i is one. We write P(A) for the sum of the probabilities sample points in A.
10 Finite Probability Spaces Three runners A,B,C; A is twice a likely to win as B, and B is twice as likely to win as C. What is P(A),P(B),P(C) winning? Let P(C) = p P(B) = 2p P(A)=4p p+2p+4p=1 therefore p = 1/7 P(A)=4/7, P(B)=2/7, P(C)=1/7 P({B,C}) = P(B)+P(C)=3/7
11 Equiprobable Spaces If all the sample points within a given finite probability space are equal to each other, then it is known as an equiprobable space. An example would be a fair die, where each number is equally possible P(1) = P(2) = P(3) = P(4) =P(5) = P(6) = 1/6
12 Equiprobable Spaces If S contains n points, then the probability of each point is 1/n. If an event A contains r points then its probability is: r 1/n = r/n P(A) = number of elements in A number of elements in S
13 Equiprobable Spaces S = cards in the deck = 52 A = card is spade B = card is a face P(A) = 13/52 P(B) = 12/52 P(A B) = 3/52
14 Axioms of Finite Probability Spaces 1. For every event A, 0 P(A) 1 2. P(S)=1, where S is sample space, 3. If events A and B are mutually exclusive (or disjoint), then P(A B) = P(A) + P(B)
15 Theorems of Finite Probability 1. P( ) = 0 Spaces 2. P(A c )= 1 P(A) 3. P(A\B)= P(A) - P(A B) 4. A B implies P(A) P(B) 5. P(A) 1 6. P(A B) = P(A) + P(B) - P(A B) 7. P(A B) = P(A) P(B A) Where P(B A) reads the probability B given A
16 Addition P(A B) = P(A) + P(B) - P(A B) Sums are used when we have two events, and we want to know the probability that either event occurs (Event A union Event B). In the Addition Rule, A and B may or may not be disjoint. Mutually exclusive or disjoint events cannot occur together, so we have: P(A B) = 0. Then the addition rule reduces to: P(A U B) = P(A) + P(B)
17 Addition Rule Example Suppose a student is selected at random from 100 students where 30 are taking maths, 20 are taking chemistry, and 10 are taking maths and chemistry. Find the probability p that the student is taking maths or chemistry. P(M) = 30/100, P(C)=20/100 P(M C) = 10/100 P(M C)=P(M) + P(C) - P(M C) P(M C)= 30/100+20/100 10/100=2/5
18 Rule of Multiplication Is used when we want to know the probability that two events occur (Event A intersection Event B). Rule of Multiplication The probability that Events A and B both occur is equal to the probability that Event A occurs times the probability that Event B occurs, given that A has occurred. P(A B) = P(A) P(B A)
19 Rule of Multiplication A bag contains 6 red marbles and 4 blue marbles. Two marbles are drawn without replacement from the bag. What is the probability that both of the marbles are blue? A = first marble is blue, B = second marble is blue. Therefore, P(A) = 4/10, P(B A) = 3/9. Using P(A B) = P(A) P(B A) P(A B) = (4/10) * (3/9) = 12/90 = 2/15
20 Rule of Multiplication A bag contains 6 red marbles and 4 blue marbles. Two marbles are drawn with replacement from the bag. What is the probability that both of the marbles are blue? A = first marble is blue, B = second marble is blue. Therefore, P(A) = 4/10, P(B A) = 4/10. Using P(A B) = P(A) P(B A) P(A B) = (4/10) * (4/10) = 16/100 = 4/25
21 Conditional Probability E is an event in S with P(E)>0. Conditional probability of A is defined as the probability that A has occurred after E has occurred. We say the conditional probability of A given E: P(A E) = P(A E) P(E) P(A E) = number of elements in A E number of elements in E
22 Example: Conditional Probability Alternatively P(A E) = number of ways A and E can occur number of ways E can occur Given the sum of a pair of tossed die is 6. E={sum is 6},5 ways = {(1,5),(2,4),(3,3),(4,2),(5,1)} A= {has at least one two},2 ways= {(2,4),(4,2)} P(A E)=2/5
23 Example 2: Conditional Probability P(A E) = number of ways A and E can occur number of ways E can occur From a class has 12 boys and 4 girls, 3 students are selected. What is the probability that they are all boys? P=Comb(12,3)/Comb(16,3)=11/28 Alternatively P=(12/16)(11/15)(10/14) = 11/28
24 Independence Two events are independent if the occurrence of one of the events gives us no information about whether or not the other event will occur; that is, the events have no influence on each other. We say that two events, A and B, are independent if the probability that they both occur is equal to the product of the probabilities of the two individual events, i.e. P(A B) = P(A) P(B) If two events are independent then they cannot be mutually exclusive (disjoint) and vice versa.
25 Example: Independence Events A and B are independent if P(A B) = P(A) P(B) otherwise they are dependent. A coin tossed three times: S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT} A={first toss head} B={second toss head} C={exactly 2 heads tossed in a row}
26 Example: Independence Continuing, coin tossed three times: P(A)={HHH,HHT,HTH,HTT}=4/8 (1 st head) P(B)={HHH,HHT,THH,THT}=4/8 (2 nd head) P(C)={HHT,THH} = 1/4 (2 heads in row) P(A B)=P({HHH,HHT})= 1/4 P(A C)=P({HHT})= 1/8 P(B C)=P({HHT,THH})= 1/4
27 Example: Independence Continuing, coin tossed three times: P(A B)=P({HHH,HHT})= 1/4 P(A C)=P({HHT})= 1/8 P(B C)=P({HHT,THH})= 1/4 P(A)P(B)=(1/2) (1/2)=(1/4)= P(A B) P(A)P(C)=(1/2) (1/4)=(1/8)= P(A C) P(B)P(C)=(1/2) (1/4)=(1/8) P(B C) Not independent, B and C are dependent.
28 Repeated Trials The Law of Averages states, in the long run, over repeated trials, random fluctuations eventually average out and the average of our observations will approach the expected value. But at the same time with increasing numbers of observations, the number of observations that differ from what we expect will be larger.
29 Repeated Trials Let S* be a finite probability space. By n independent or repeated trials we mean the probability space S consisting of all ordered n-tuples of elements of S*, with the probability of n-tuple defined to be the product of the probabilities of its components. P(s 1,s 2,s 3...s n )=P(s 1 ) P(s 2 ) P(s n )
30 Repeated Trials Let probability space S*={P(a),p(b),P(c)} represents probabilities three runners winning a race. Their probabilities of winning are P(a)=1/2, P(b)=1/3, P(c)=1/6. If there are two races then the sample space S consisting of two repeated trials is: S={aa,ab,ac,ba,bb,bc,ca,cb,cc}
31 Repeated Trials S={aa,ab,ac,ba,bb,bc,ca,cb,cc} The probability of the sample points of S are: P(aa)=(1/2) (1/2)=1/4 P(ab)=(1/2) (1/3)=1/6 P(ac)=(1/2) (1/6)=1/12 P(ba)=(1/3) (1/2)=1/6 P(bb)=(1/3) (1/3)=1/9 P(bc)=(1/3) (1/6)=1/18 P(ca)=(1/6) (1/2)=1/12 P(cb)=(1/6) (1/3)=1/18 P(cc)=(1/6) (1/6)=1/36 The probability of c winning first race and a the second is P(ca)=1/12 EXCEL =(1/4)+(1/6)+(1/12)+(1/6)+(1/9)+(1/18)+(1/12)+(1/18)+(1/36)
32 Bernoulli Trials with 2 possible outcomes. A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure. If p is the probability of success, then q=1-p is the probability of failure. Often we are interested in the number of successes without considering their order. The probability of exactly k successes in n repeated trials is: b(k,n,p)= n p k q n-k k
33 Example: Trials with 2 possible outcomes. A coin is tossed 6 times, H=success,T=failure. n=6, p=q=1/2 The probability of two heads, (k=2) Binomial coefficient 6 b(2,6,1/2)= (1/2) 2 (1/2) 4 =15/64 2
34 Identically Distributed variable Same probability distributions
Section 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 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 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 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 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 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 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 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 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 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 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 informationChapter 13 & 14 - Probability PART
Chapter 13 & 14 - Probability PART IV : PROBABILITY Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Chapter 13 & 14 - Probability 1 / 91 Why Should We Learn Probability Theory? Dr. Joseph
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 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 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 informationProbabilistic 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
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 informationProbabilities. Probability of a event. From Random Variables to Events. From Random Variables to Events. Probability Theory I
Victor Adamchi Danny Sleator Great Theoretical Ideas In Computer Science Probability Theory I CS 5-25 Spring 200 Lecture Feb. 6, 200 Carnegie Mellon University We will consider chance experiments with
More informationProbability. a number between 0 and 1 that indicates how likely it is that a specific event or set of events will occur.
Probability Probability Simple experiment Sample space Sample point, or elementary event Event, or event class Mutually exclusive outcomes Independent events a number between 0 and 1 that indicates how
More informationNotes on Probability. Peter J. Cameron
Notes on Probability Peter J. Cameron ii Preface Here are the course lecture notes for the course MAS108, Probability I, at Queen Mary, University of London, taken by most Mathematics students and some
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 informationAP 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
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 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 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 informationUnit 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,
More informationIAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION
IAM 530 ELEMENTS OF PROBABILITY AND STATISTICS INTRODUCTION 1 WHAT IS STATISTICS? Statistics is a science of collecting data, organizing and describing it and drawing conclusions from it. That is, statistics
More informationLesson Plans for (9 th Grade Main Lesson) Possibility & Probability (including Permutations and Combinations)
Lesson Plans for (9 th Grade Main Lesson) Possibility & Probability (including Permutations and Combinations) Note: At my school, there is only room for one math main lesson block in ninth grade. Therefore,
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 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 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 informationSTATISTICS HIGHER SECONDARY - SECOND YEAR. Untouchability is a sin Untouchability is a crime Untouchability is inhuman
STATISTICS HIGHER SECONDARY - SECOND YEAR Untouchability is a sin Untouchability is a crime Untouchability is inhuman TAMILNADU TEXTBOOK CORPORATION College Road, Chennai- 600 006 i Government of Tamilnadu
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 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 informationMATH 140 Lab 4: Probability and the Standard Normal Distribution
MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes
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 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 informationFor two disjoint subsets A and B of Ω, say that A and B are disjoint events. For disjoint events A and B we take an axiom P(A B) = P(A) + P(B)
Basic probability A probability space or event space is a set Ω together with a probability measure P on it. This means that to each subset A Ω we associate the probability P(A) = probability of A with
More information2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.
Math 142 September 27, 2011 1. How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are
More informationWHERE 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
More informationProbability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2
Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability
More informationProbability & Probability Distributions
Probability & Probability Distributions Carolyn J. Anderson EdPsych 580 Fall 2005 Probability & Probability Distributions p. 1/61 Probability & Probability Distributions Elementary Probability Theory Definitions
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 informationPROBABILITY. The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE
PROBABILITY 53 Chapter 3 PROBABILITY The theory of probabilities is simply the Science of logic quantitatively treated. C.S. PEIRCE 3. Introduction In earlier Classes, we have studied the probability as
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 informationProbability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1 www.math12.com
Probability --QUESTIONS-- Principles of Math - Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..
More informationIntroduction to Probability
3 Introduction to Probability Given a fair coin, what can we expect to be the frequency of tails in a sequence of 10 coin tosses? Tossing a coin is an example of a chance experiment, namely a process which
More informationChapter 5 Section 2 day 1 2014f.notebook. November 17, 2014. Honors Statistics
Chapter 5 Section 2 day 1 2014f.notebook November 17, 2014 Honors Statistics Monday November 17, 2014 1 1. Welcome to class Daily Agenda 2. Please find folder and take your seat. 3. Review Homework C5#3
More informationThe Calculus of Probability
The Calculus of Probability Let A and B be events in a sample space S. Partition rule: P(A) = P(A B) + P(A B ) Example: Roll a pair of fair dice P(Total of 10) = P(Total of 10 and double) + P(Total of
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 informationSolution (Done in class)
MATH 115 CHAPTER 4 HOMEWORK Sections 4.1-4.2 N. PSOMAS 4.6 Winning at craps. The game of craps starts with a come-out roll where the shooter rolls a pair of dice. If the total is 7 or 11, the shooter wins
More informationChapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.
Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive
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 informationMath 141. Lecture 2: More Probability! Albyn Jones 1. jones@reed.edu www.people.reed.edu/ jones/courses/141. 1 Library 304. Albyn Jones Math 141
Math 141 Lecture 2: More Probability! Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Outline Law of total probability Bayes Theorem the Multiplication Rule, again Recall
More informationReview for Test 2. Chapters 4, 5 and 6
Review for Test 2 Chapters 4, 5 and 6 1. You roll a fair six-sided 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 informationSolutions to Self-Help Exercises 1.3. b. E F = /0. d. G c = {(3,4),(4,3),(4,4)}
1.4 Basics of Probability 37 Solutions to Self-Help Exercises 1.3 1. Consider the outcomes as ordered pairs, with the number on the bottom of the red one the first number and the number on the bottom of
More informationSTA 371G: Statistics and Modeling
STA 371G: Statistics and Modeling Decision Making Under Uncertainty: Probability, Betting Odds and Bayes Theorem Mingyuan Zhou McCombs School of Business The University of Texas at Austin http://mingyuanzhou.github.io/sta371g
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 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 informationChapter 7 Probability. Example of a random circumstance. Random Circumstance. What does probability mean?? Goals in this chapter
Homework (due Wed, Oct 27) Chapter 7: #17, 27, 28 Announcements: Midterm exams keys on web. (For a few hours the answer to MC#1 was incorrect on Version A.) No grade disputes now. Will have a chance to
More informationAlgebra 2 C Chapter 12 Probability and Statistics
Algebra 2 C Chapter 12 Probability and Statistics Section 3 Probability fraction Probability is the ratio that measures the chances of the event occurring For example a coin toss only has 2 equally likely
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 informationProbability and Statistics Vocabulary List (Definitions for Middle School Teachers)
Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence
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 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 informationProbability and Venn diagrams UNCORRECTED PAGE PROOFS
Probability and Venn diagrams 12 This chapter deals with further ideas in chance. At the end of this chapter you should be able to: identify complementary events and use the sum of probabilities to solve
More informationV. 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
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 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 informationHow To Understand And Solve A Linear Programming Problem
At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,
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 informationProbability. Section 9. Probability. Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space)
Probability Section 9 Probability Probability of A = Number of outcomes for which A happens Total number of outcomes (sample space) In this section we summarise the key issues in the basic probability
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 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 informationConcepts of Probability
Concepts of Probability Trial question: we are given a die. How can we determine the probability that any given throw results in a six? Try doing many tosses: Plot cumulative proportion of sixes Also look
More informationLecture 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
More informationSection 5-3 Binomial Probability Distributions
Section 5-3 Binomial Probability Distributions Key Concept This section presents a basic definition of a binomial distribution along with notation, and methods for finding probability values. Binomial
More informationSTATISTICS 230 COURSE NOTES. Chris Springer, revised by Jerry Lawless and Don McLeish
STATISTICS 230 COURSE NOTES Chris Springer, revised by Jerry Lawless and Don McLeish JANUARY 2006 Contents 1. Introduction to Probability 1 2. Mathematical Probability Models 5 2.1 SampleSpacesandProbability...
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 informationName: Date: Use the following to answer questions 2-4:
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 informationSample Space and Probability
1 Sample Space and Probability Contents 1.1. Sets........................... p. 3 1.2. Probabilistic Models.................... p. 6 1.3. Conditional Probability................. p. 18 1.4. Total Probability
More informationMath 210 Lecture Notes: Ten Probability Review Problems
Math 210 Lecture Notes: Ten Probability Review Problems Richard Blecksmith Dept. of Mathematical Sciences Northern Illinois University 1. Review Question 1 Face card means jack, queen, or king. You draw
More informationHoover High School Math League. Counting and Probability
Hoover High School Math League Counting and Probability Problems. At a sandwich shop there are 2 kinds of bread, 5 kinds of cold cuts, 3 kinds of cheese, and 2 kinds of dressing. How many different sandwiches
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationMAS113 Introduction to Probability and Statistics
MAS113 Introduction to Probability and Statistics 1 Introduction 1.1 Studying probability theory There are (at least) two ways to think about the study of probability theory: 1. Probability theory is a
More informationMATH 289 PROBLEM SET 4: NUMBER THEORY
MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides
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 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 informationSolutions: Problems for Chapter 3. Solutions: Problems for Chapter 3
Problem A: You are dealt five cards from a standard deck. Are you more likely to be dealt two pairs or three of a kind? experiment: choose 5 cards at random from a standard deck Ω = {5-combinations of
More informationActivities/ Resources for Unit V: Proportions, Ratios, Probability, Mean and Median
Activities/ Resources for Unit V: Proportions, Ratios, Probability, Mean and Median 58 What is a Ratio? A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a
More informationChapter ML:IV. IV. Statistical Learning. Probability Basics Bayes Classification Maximum a-posteriori Hypotheses
Chapter ML:IV IV. Statistical Learning Probability Basics Bayes Classification Maximum a-posteriori Hypotheses ML:IV-1 Statistical Learning STEIN 2005-2015 Area Overview Mathematics Statistics...... Stochastics
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 information2. Three dice are tossed. Find the probability of a) a sum of 4; or b) a sum greater than 4 (may use complement)
Probability Homework Section P4 1. A two-person committee is chosen at random from a group of four men and three women. Find the probability that the committee contains at least one man. 2. Three dice
More informationSample Questions for Mastery #5
Name: Class: Date: Sample Questions for Mastery #5 Multiple Choice Identify the choice that best completes the statement or answers the question.. For which of the following binomial experiments could
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 1040-1 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 informationResponsible Gambling Education Unit: Mathematics A & B
The Queensland Responsible Gambling Strategy Responsible Gambling Education Unit: Mathematics A & B Outline of the Unit This document is a guide for teachers to the Responsible Gambling Education Unit:
More informationSTAT 35A HW2 Solutions
STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },
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 informationSample Term Test 2A. 1. A variable X has a distribution which is described by the density curve shown below:
Sample Term Test 2A 1. A variable X has a distribution which is described by the density curve shown below: What proportion of values of X fall between 1 and 6? (A) 0.550 (B) 0.575 (C) 0.600 (D) 0.625
More information