November 8, Chapter 8: Probability: The Mathematics of Chance


 Adela Underwood
 2 years ago
 Views:
Transcription
1 Chapter 8: Probability: The Mathematics of Chance November 8, 2013
2 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes
3 Crystallographic notation The first symbol is always a p, which indicates that the pattern repeats (is periodic ) in the horizontal direction. The second symbol is m if there is a vertical line of reflection. Otherwise, it is 1. The third symbol is m (for mirror ), if there is a horizontal line of reflection (in which case there is also glide reflection) a (for alternating ), if there is a glide reflection but no horizontal reflection 1 if there is no horizontal reflection or glide reflection The fourth symbol is 2, if there is halfturn rotational symmetry; otherwise, it is 1.
4 Group A group is a collection of elements {A, B, } and an operation between pairs of them such that the following properties hold: Closure: The result of one element operating on another is itself an element of the collection (A B is in the collection). Identity element: There is a special element I, called the identity element, such that the result of an operation involving the identity and any element is that same element (I A = A and A I = A). Inverses: For any element A, there is another element, called its inverse and denoted A 1, such that the result of an operation involving an element and its inverse is the identity element (A A 1 = I and A 1 A = I ). Associativity: The result of several consecutive operations is the same regardless of grouping or parenthesizing, provided that the consecutive order of operations is maintained: A B C = A (B C) = (A B) C.
5 Probability Rules Probability Rules Rule 1. The probability P(A) of any event A satisfies 0 P(A) 1. Rule 2. If S is the sample space in a probability model, then P(S) = 1. Rule 3. The complement rule: P(A C ) = 1 P(A). Rule 4. The multiplication rule for independent events: P(A and B) = P(A) P(B). Rule 5. The general addition rule: P(A or B) = P(A) + P(B) P(A and B). Rule 6. The addition rule for disjoint events: P(A or B) = P(A) + P(B).
6 Counting distinct items Counting Ordered Collections of Distinct Items Rule A. Suppose we have a collection of n distinct items. We want to arrange k of these items in order, and the same item can appear more than once in the arrangement. The number of possible arrangements is n n n = n k Rule B. (Permutations) Suppose we have a collection of n distinct items. We want to arrange k of these items in order, and any item can appear no more than once in the arrangement. The number of possible arrangements is n (n 1) (n k + 1)
7 Counting Distinct Items Counting Unordered Collections of Distinct Items Rule C. Suppose that we have a collection of n distinct items. We want to select k of those items with no regard to order, and any item can appear more than once in the collection. The number of possible collections is (n + k 1)! k!(n 1)! Rule D. (Combinations) Suppose that we have a collection of n distinct items. We want to select k of these items with no regard to order, and any item can appear no more than once in the collection. The number of possible selections is n! k!(n k)!
8 Question Choose a young adult (aged 25 to 34 years) at random. The probability is 0.12 that the person choose did not complete high school, 0.31 that the person has a high school diploma but no further education, and 0.29 that the person has at least a bachelor s degree. (a) What must be the probability that a randomly chosen young adult has some education beyond high school but does not have a bachelor s degree? Answer: =.28 28% (b) What is the probability that a randomly chosen young adult has at least a high school education? Answer: =.88 88%
9 Question You toss a balanced coin 10 times and write down the resulting sequence of heads and tails. (a) How many possible outcomes are there for 10 tosses? Answer: 2 10 (b) What is the probability that your 10toss sequence is either all heads or all tails? Answer: = = (c) What is the probability that your 10toss sequence has a combine 5 heads and a combine 5 tails? Answer: ( 10 )
10 Question In poker, a royal flush is a 5card hand containing an ace, king, queen, jack, and 10, all of the same suit. (a) How many royal flush hands are possible? Answer: 4 (b) What is the number of 5card hands possible from a 52card deck? Answer: ( ) 52 5 (c) What is the probability that 5 cards drawn at random from a 52card deck will yield a royal flush? Answer: 4 ( 52 5 )
11 Question Suppose a monkey is at a type writer and can only press a, r, e. 1 How many possible threeletter words can the monkey type using only these letters? Answer: 3 3 = 27 2 Which of these are words in an English dictionary? Answer: are, era, ear 3 What is the probability that the word the monkey typed is in a English dictionary? Answer:
12 Questions How many different ways can 3 dice sum to 8? How many different ways can 3 dice sum to 9? How many different ways can 3 dice sum to 13? How many different ways can 3 dice sum to 12? How many different ways can 4 dice sum to 9?
13 Questions How many different ways can 3 dice sum to 8?
14 Questions How many different ways can 3 dice sum to 8? Answer: 21 How many different ways can 3 dice sum to 9?
15 Questions How many different ways can 3 dice sum to 8? Answer: 21 How many different ways can 3 dice sum to 9? Answer: 25 How many different ways can 3 dice sum to 13?
16 Questions How many different ways can 3 dice sum to 8? Answer: 21 How many different ways can 3 dice sum to 9? Answer: 25 How many different ways can 3 dice sum to 13? Answer: 21 How many different ways can 3 dice sum to 12?
17 Questions How many different ways can 3 dice sum to 8? Answer: 21 How many different ways can 3 dice sum to 9? Answer: 25 How many different ways can 3 dice sum to 13? Answer: 21 How many different ways can 3 dice sum to 12? Answer: 25 How many different ways can 4 dice sum to 9?
18 Questions How many different ways can 3 dice sum to 8? Answer: 21 How many different ways can 3 dice sum to 9? Answer: 25 How many different ways can 3 dice sum to 13? Answer: 21 How many different ways can 3 dice sum to 12? Answer: 25 How many different ways can 4 dice sum to 9? Answer: 56
19 Question A computer assigns threecharacter login IDs that may contain the digits 0 to 9 as well as the letters a to z, with repeats allowed. 1 What is the probability that your ID contains no x? 2 What is the probability that your ID contains no digits? 3 What is the probability that your ID contains exactly 2 x s?
20 Question What is the probability that two people in this room have the same birthday?
21 Question What is the probability that two people in this room have the same birthday? Answer: 1 365! (365 14)! = 0.223
22 Question How many different ways can you seat 10 couples at a circular table such that everyone is sitting next to their spouse? Assuming we alternate men and women how arrangements have at least one couple not sitting together?
23 Next time Continuous Probability Models
November 6, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern
More informationNovember 11, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 11, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Probability Rules Probability Rules Rule 1.
More informationSuch a description is the basis for a probability model. Here is the basic vocabulary we use.
5.2.1 Probability Models When we toss a coin, we can t know the outcome in advance. What do we know? We are willing to say that the outcome will be either heads or tails. We believe that each of these
More information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,
More informationThe probability setup
CHAPTER 2 The probability setup 2.1. Introduction and basic theory We will have a sample space, denoted S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample
More informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More informationChapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
More information3 The multiplication rule/miscellaneous counting problems
Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is
More informationA Probability Work Sheet
A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair sixsided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we
More informationSection 6.1 #16. Question: What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit?
Section 6.1 #16 What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1
More informationProbability. Ms. Weinstein Probability & Statistics
Probability Ms. Weinstein Probability & Statistics Definitions Sample Space The sample space, S, of a random phenomenon is the set of all possible outcomes. Event An event is a set of outcomes of a random
More informationThe probability setup
CHAPTER The probability setup.1. Introduction and basic theory We will have a sample space, denoted S sometimes Ω that consists of all possible outcomes. For example, if we roll two dice, the sample space
More informationChapter 5: Probability: What are the Chances? Section 5.2 Probability Rules
+ Chapter 5: Probability: What are the Chances? Section 5.2 + TwoWay Tables and Probability When finding probabilities involving two events, a twoway table can display the sample space in a way that
More informationThe next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Fall 2004 Rao Lecture 14 Introduction to Probability The next several lectures will be concerned with probability theory. We will aim to make sense of statements such
More informationSTAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes
STAT 430/510 Probability Lecture 3: Space and Event; Sample Spaces with Equally Likely Outcomes Pengyuan (Penelope) Wang May 25, 2011 Review We have discussed counting techniques in Chapter 1. (Principle
More informationContents 2.1 Basic Concepts of Probability Methods of Assigning Probabilities Principle of Counting  Permutation and Combination 39
CHAPTER 2 PROBABILITY Contents 2.1 Basic Concepts of Probability 38 2.2 Probability of an Event 39 2.3 Methods of Assigning Probabilities 39 2.4 Principle of Counting  Permutation and Combination 39 2.5
More information2.5 Sample Spaces Having Equally Likely Outcomes
Sample Spaces Having Equally Likely Outcomes 3 Sample Spaces Having Equally Likely Outcomes Recall that we had a simple example (fair dice) before on equallylikely sample spaces Since they will appear
More informationCS 237 Fall 2018, Homework SOLUTION
0//08 hw03.solution.lenka CS 37 Fall 08, Homework 03  SOLUTION Due date: PDF file due Thursday September 7th @ :59PM (0% off if up to 4 hours late) in GradeScope General Instructions Please complete
More informationBasic Probability Models. PingShou Zhong
asic Probability Models PingShou Zhong 1 Deterministic model n experiment that results in the same outcome for a given set of conditions Examples: law of gravity 2 Probabilistic model The outcome of the
More information7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count
7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count Probability deals with predicting the outcome of future experiments in a quantitative way. The experiments
More informationToday s Topics. Next week: Conditional Probability
Today s Topics 2 Last time: Combinations Permutations Group Assignment TODAY: Probability! Sample Spaces and Event Spaces Axioms of Probability Lots of Examples Next week: Conditional Probability Sets
More informationSection Introduction to Sets
Section 1.1  Introduction to Sets Definition: A set is a welldefined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase
More informationLecture 6 Probability
Lecture 6 Probability Example: When you toss a coin, there are only two possible outcomes, heads and tails. What if we toss a coin two times? Figure below shows the results of tossing a coin 5000 times
More informationProbability. Dr. Zhang Fordham Univ.
Probability! Dr. Zhang Fordham Univ. 1 Probability: outline Introduction! Experiment, event, sample space! Probability of events! Calculate Probability! Through counting! Sum rule and general sum rule!
More informationINDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2
INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2 WARM UP Students in a mathematics class pick a card from a standard deck of 52 cards, record the suit, and return the card to the deck. The results
More informationMathematical Foundations HW 5 By 11:59pm, 12 Dec, 2015
1 Probability Axioms Let A,B,C be three arbitrary events. Find the probability of exactly one of these events occuring. Sample space S: {ABC, AB, AC, BC, A, B, C, }, and S = 8. P(A or B or C) = 3 8. note:
More informationDiscrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #23: Discrete Probability Based on materials developed by Dr. Adam Lee The study of probability is
More informationWeek 3 Classical Probability, Part I
Week 3 Classical Probability, Part I Week 3 Objectives Proper understanding of common statistical practices such as confidence intervals and hypothesis testing requires some familiarity with probability
More informationIndependent Events. 1. Given that the second baby is a girl, what is the. e.g. 2 The probability of bearing a boy baby is 2
Independent Events 7. Introduction Consider the following examples e.g. E throw a die twice A first thrown is "" second thrown is "" o find P( A) Solution: Since the occurrence of Udoes not dependu on
More informationThe study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability
The study of probability is concerned with the likelihood of events occurring Like combinatorics, the origins of probability theory can be traced back to the study of gambling games Still a popular branch
More information4.3 Rules of Probability
4.3 Rules of Probability If a probability distribution is not uniform, to find the probability of a given event, add up the probabilities of all the individual outcomes that make up the event. Example:
More informationCSCI 2200 Foundations of Computer Science (FoCS) Solutions for Homework 7
CSCI 00 Foundations of Computer Science (FoCS) Solutions for Homework 7 Homework Problems. [0 POINTS] Problem.4(e)(f) [or F7 Problem.7(e)(f)]: In each case, count. (e) The number of orders in which a
More informationChapter 3: Elements of Chance: Probability Methods
Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 34 20142015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,
More informationTEST A CHAPTER 11, PROBABILITY
TEST A CHAPTER 11, PROBABILITY 1. Two fair dice are rolled. Find the probability that the sum turning up is 9, given that the first die turns up an even number. 2. Two fair dice are rolled. Find the probability
More information1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?
1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,
More informationCS 237: Probability in Computing
CS 237: Probability in Computing Wayne Snyder Computer Science Department Boston University Lecture 5: o Independence reviewed; Bayes' Rule o Counting principles and combinatorics; o Counting considered
More informationSection 5.4 Permutations and Combinations
Section 5.4 Permutations and Combinations Definition: nfactorial For any natural number n, n! n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to
More information4.1 Sample Spaces and Events
4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an
More informationMATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG
MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, InclusionExclusion, and Complement. (a An office building contains 7 floors and has 7 offices
More informationn(s)=the number of ways an event can occur, assuming all ways are equally likely to occur. p(e) = n(e) n(s)
The following story, taken from the book by Polya, Patterns of Plausible Inference, Vol. II, Princeton Univ. Press, 1954, p.101, is also quoted in the book by Szekely, Classical paradoxes of probability
More informationIntroductory Probability
Introductory Probability Combinations Nicholas Nguyen nicholas.nguyen@uky.edu Department of Mathematics UK Agenda Assigning Objects to Identical Positions Denitions Committee Card Hands Coin Toss Counts
More informationSection 5.4 Permutations and Combinations
Section 5.4 Permutations and Combinations Definition: nfactorial For any natural number n, n! = n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to
More informationCHAPTER 2 PROBABILITY. 2.1 Sample Space. 2.2 Events
CHAPTER 2 PROBABILITY 2.1 Sample Space A probability model consists of the sample space and the way to assign probabilities. Sample space & sample point The sample space S, is the set of all possible outcomes
More informationBlock 1  Sets and Basic Combinatorics. Main Topics in Block 1:
Block 1  Sets and Basic Combinatorics Main Topics in Block 1: A short revision of some set theory Sets and subsets. Venn diagrams to represent sets. Describing sets using rules of inclusion. Set operations.
More informationSimple Probability. Arthur White. 28th September 2016
Simple Probability Arthur White 28th September 2016 Probabilities are a mathematical way to describe an uncertain outcome. For eample, suppose a physicist disintegrates 10,000 atoms of an element A, and
More information7.1 Experiments, Sample Spaces, and Events
7.1 Experiments, Sample Spaces, and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment
More informationProbability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37
Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete
More informationMath 14 Lecture Notes Ch. 3.3
3.3 Two Basic Rules of Probability If we want to know the probability of drawing a 2 on the first card and a 3 on the 2 nd card from a standard 52card deck, the diagram would be very large and tedious
More informationSection : Combinations and Permutations
Section 11.111.2: Combinations and Permutations Diana Pell A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words
More informationCounting and Probability Math 2320
Counting and Probability Math 2320 For a finite set A, the number of elements of A is denoted by A. We have two important rules for counting. 1. Union rule: Let A and B be two finite sets. Then A B = A
More informationDiscrete probability and the laws of chance
Chapter 8 Discrete probability and the laws of chance 8.1 Multiple Events and Combined Probabilities 1 Determine the probability of each of the following events assuming that the die has equal probability
More informationCompound Probability. Set Theory. Basic Definitions
Compound Probability Set Theory A probability measure P is a function that maps subsets of the state space Ω to numbers in the interval [0, 1]. In order to study these functions, we need to know some basic
More informationMore Probability: Poker Hands and some issues in Counting
More Probability: Poker Hands and some issues in Counting Data From Thursday Everybody flipped a pair of coins and recorded how many times they got two heads, two tails, or one of each. We saw that the
More informationProbability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College
Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical
More information{ a, b }, { a, c }, { b, c }
12 d.) 0(5.5) c.) 0(5,0) h.) 0(7,1) a.) 0(6,3) 3.) Simplify the following combinations. PROBLEMS: C(n,k)= the number of combinations of n distinct objects taken k at a time is COMBINATION RULE It can easily
More informationMath Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.
Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166  Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1  Experiments, Sample Spaces,
More informationMath Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.
Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166  Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1  Experiments, Sample Spaces,
More informationWeek 1: Probability models and counting
Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model
More informationThe topic for the third and final major portion of the course is Probability. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 17 Introduction to Probability The topic for the third and final major portion of the course is Probability. We will aim to make sense of
More informationINDIAN STATISTICAL INSTITUTE
INDIAN STATISTICAL INSTITUTE B1/BVR Probability Home Assignment 1 200707 1. A poker hand means a set of five cards selected at random from usual deck of playing cards. (a) Find the probability that it
More informationMath116Chapter15ProbabilityProbabilityDone.notebook January 08, 2012
15.4 Probability Spaces Probability assignment A function that assigns to each event E a number between 0 and 1, which represents the probability of the event E and which we denote by Pr (E). Probability
More informationProbability I Sample spaces, outcomes, and events.
Probability I Sample spaces, outcomes, and events. When we perform an experiment, the result is called the outcome. The set of possible outcomes is the sample space and any subset of the sample space is
More informationProblem Set 2. Counting
Problem Set 2. Counting 1. (Blitzstein: 1, Q3 Fred is planning to go out to dinner each night of a certain week, Monday through Friday, with each dinner being at one of his favorite ten restaurants. i
More informationMath 166: Topics in Contemporary Mathematics II
Math 166: Topics in Contemporary Mathematics II Xin Ma Texas A&M University September 30, 2017 Xin Ma (TAMU) Math 166 September 30, 2017 1 / 11 Last Time Factorials For any natural number n, we define
More informationLesson 4: Chapter 4 Sections 12
Lesson 4: Chapter 4 Sections 12 Caleb Moxley BSC Mathematics 14 September 15 4.1 Randomness What s randomness? 4.1 Randomness What s randomness? Definition (random) A phenomenon is random if individual
More informationChapter 5  Elementary Probability Theory
Chapter 5  Elementary Probability Theory Historical Background Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling
More informationIntermediate Math Circles November 1, 2017 Probability I
Intermediate Math Circles November 1, 2017 Probability I Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application.
More informationEECS 203 Spring 2016 Lecture 15 Page 1 of 6
EECS 203 Spring 2016 Lecture 15 Page 1 of 6 Counting We ve been working on counting for the last two lectures. We re going to continue on counting and probability for about 1.5 more lectures (including
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More 9.9.3 Practice Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. ) In how many ways can you answer the questions on
More informationKey Concepts. Theoretical Probability. Terminology. Lesson 111
Key Concepts Theoretical Probability Lesson  Objective Teach students the terminology used in probability theory, and how to make calculations pertaining to experiments where all outcomes are equally
More informationProbability and Counting Techniques
Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each
More informationDiscrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 13
CS 70 Discrete Mathematics and Probability Theory Spring 2016 Rao and Walrand Note 13 Introduction to Discrete Probability In the last note we considered the probabilistic experiment where we flipped a
More informationSTAT 430/510 Probability Lecture 1: Counting1
STAT 430/510 Probability Lecture 1: Counting1 Pengyuan (Penelope) Wang May 22, 2011 Introduction In the early days, probability was associated with games of chance, such as gambling. Probability is describing
More informationMath 146 Statistics for the Health Sciences Additional Exercises on Chapter 3
Math 46 Statistics for the Health Sciences Additional Exercises on Chapter 3 Student Name: Find the indicated probability. ) If you flip a coin three times, the possible outcomes are HHH HHT HTH HTT THH
More informationHonors Precalculus Chapter 9 Summary Basic Combinatorics
Honors Precalculus Chapter 9 Summary Basic Combinatorics A. Factorial: n! means 0! = Why? B. Counting principle: 1. How many different ways can a license plate be formed a) if 7 letters are used and each
More informationChapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
More information, the of all of a probability experiment. consists of outcomes. (b) List the elements of the event consisting of a number that is greater than 4.
41 Sample Spaces and Probability as a general concept can be defined as the chance of an event occurring. In addition to being used in games of chance, probability is used in the fields of,, and forecasting,
More informationDefine and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)
12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the
More informationRANDOM EXPERIMENTS AND EVENTS
Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In daytoday life we see that before commencement of a cricket match two captains go for a toss. Tossing of a coin is an activity and getting
More informationThe Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.)
The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you $ that if you give me $, I ll give you $2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If
More informationMath 1111 Math Exam Study Guide
Math 1111 Math Exam Study Guide The math exam will cover the mathematical concepts and techniques we ve explored this semester. The exam will not involve any codebreaking, although some questions on the
More informationAxiomatic Probability
Axiomatic Probability The objective of probability is to assign to each event A a number P(A), called the probability of the event A, which will give a precise measure of the chance thtat A will occur.
More informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 4 Probability and Counting Rules 2 Objectives Determine sample spaces and find the probability of an event using classical probability or empirical
More informationProbability Models. Section 6.2
Probability Models Section 6.2 The Language of Probability What is random? Empirical means that it is based on observation rather than theorizing. Probability describes what happens in MANY trials. Example
More informationPermutations and Combinations
Motivating question Permutations and Combinations A) Rosen, Chapter 5.3 B) C) D) Permutations A permutation of a set of distinct objects is an ordered arrangement of these objects. : (1, 3, 2, 4) is a
More informationSTAT Statistics I Midterm Exam One. Good Luck!
STAT 515  Statistics I Midterm Exam One Name: Instruction: You can use a calculator that has no connection to the Internet. Books, notes, cellphones, and computers are NOT allowed in the test. There are
More informationCSE 312: Foundations of Computing II Quiz Section #1: Counting
CSE 312: Foundations of Computing II Quiz Section #1: Counting Review: Main Theorems and Concepts 1. Product Rule: Suppose there are m 1 possible outcomes for event A 1, then m 2 possible outcomes for
More informationCHAPTERS 14 & 15 PROBABILITY STAT 203
CHAPTERS 14 & 15 PROBABILITY STAT 203 Where this fits in 2 Up to now, we ve mostly discussed how to handle data (descriptive statistics) and how to collect data. Regression has been the only form of statistical
More informationSection continued: Counting poker hands
1 Section 3.1.5 continued: Counting poker hands 2 Example A poker hand consists of 5 cards drawn from a 52card deck. 2 Example A poker hand consists of 5 cards drawn from a 52card deck. a) How many different
More informationFundamentals of Probability
Fundamentals of Probability Introduction Probability is the likelihood that an event will occur under a set of given conditions. The probability of an event occurring has a value between 0 and 1. An impossible
More informationEmpirical (or statistical) probability) is based on. The empirical probability of an event E is the frequency of event E.
Probability and Statistics Chapter 3 Notes Section 31 I. Probability Experiments. A. When weather forecasters say There is a 90% chance of rain tomorrow, or a doctor says There is a 35% chance of a successful
More informationGrade 7/8 Math Circles February 25/26, Probability
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Probability Grade 7/8 Math Circles February 25/26, 2014 Probability Centre for Education in Mathematics and Computing Probability is the study of how likely
More informationMat 344F challenge set #2 Solutions
Mat 344F challenge set #2 Solutions. Put two balls into box, one ball into box 2 and three balls into box 3. The remaining 4 balls can now be distributed in any way among the three remaining boxes. This
More informationMATHEMATICS E102, FALL 2005 SETS, COUNTING, AND PROBABILITY Outline #1 (Probability, Intuition, and Axioms)
MATHEMATICS E102, FALL 2005 SETS, COUNTING, AND PROBABILITY Outline #1 (Probability, Intuition, and Axioms) Last modified: September 19, 2005 Reference: EP(Elementary Probability, by Stirzaker), Chapter
More informationProbability and Randomness. Day 1
Probability and Randomness Day 1 Randomness and Probability The mathematics of chance is called. The probability of any outcome of a chance process is a number between that describes the proportion of
More information2. Combinatorics: the systematic study of counting. The Basic Principle of Counting (BPC)
2. Combinatorics: the systematic study of counting The Basic Principle of Counting (BPC) Suppose r experiments will be performed. The 1st has n 1 possible outcomes, for each of these outcomes there are
More informationLecture 1. Permutations and combinations, Pascal s triangle, learning to count
18.440: Lecture 1 Permutations and combinations, Pascal s triangle, learning to count Scott Sheffield MIT 1 Outline Remark, just for fun Permutations Counting tricks Binomial coefficients Problems 2 Outline
More informationPROBABILITY. 1. Introduction. Candidates should able to:
PROBABILITY Candidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for the total score when two fair dice are thrown), or by calculation
More informationProbability of Independent and Dependent Events. CCM2 Unit 6: Probability
Probability of Independent and Dependent Events CCM2 Unit 6: Probability Independent and Dependent Events Independent Events: two events are said to be independent when one event has no affect on the probability
More informationExam III Review Problems
c Kathryn Bollinger and Benjamin Aurispa, November 10, 2011 1 Exam III Review Problems Fall 2011 Note: Not every topic is covered in this review. Please also take a look at the previous WeekinReviews
More information