Complement. If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A.


 Marjory Austin
 2 years ago
 Views:
Transcription
1 Complement If A is an event, then the complement of A, written A c, means all the possible outcomes that are not in A. For example, if A is the event UNC wins at least 5 football games, then A c is the event UNC wins less than 5 football games. 1
2 We can represent this by a Venn diagram, as follows: S A c A Figure 1. Venn diagram for a single event A and its complement A c. 2
3 The Law of Complementary Events states that P (A c ) = 1 P (A). Example. In the game involving two throws of a die, if A is the event the total is 10 or greater, then A c is the event the total is 9 or smaller. We know P (A) = 1 6, so P (Ac ) =
4 Disjoint events, Intersection and Union Two events A and B are said to be disjoint if they cannot both occur. This is represented by the following Venn diagram: S A B Figure 2. Venn diagram for two disjoint events A and B. 4
5 For example, in two throws of a die, if A is the event the total is 10 or larger and B is the event the total is 3 or smaller, it is clear that A and B cannot both be true, so they are disjoint events. With any two events A and B, we define the intersection of A and B, also written A and B, to be the event that A and B both occur. With any two events A and B, we define the union of A and B, also written A or B, to be the event that at least one of A and B occurs. Note that in common English, if we say A or B, that s often taken as excluding the possibility of both A and B occurring. In the language of probability, A or B always includes the possibility that both A and B might occur unless they are disjoint, in which case it is impossible. 5
6 So another definition of disjoint events is: two events A and B are disjoint if the intersection A and B is an impossible event. The law of addition for disjoint events states that: If two events A and B are disjoint, then P (A or B) = P (A) + P (B). Example. Consider the toss of two dice where A is the event the total is 10 or larger and B is the event the total is 3 or smaller. We have already seen that P (A) = 6 1 and it is easy to see by similar reasoning that P (B) = Therefore P (A or B) = P (A) + P (B) = = 1 4. We could also figure this out directly, by noting that the event A or B consists of 9 outcomes of the sample space ((1,1),(1,2),(2,1) plus the 6 outcomes that comprise B) so the probability is 9 36 =
7 The Law of Addition for Nondisjoint Events S A B Figure 3. Venn diagram for two disjoint events A and B. 7
8 In this case the Law of Addition reads P (A or B) = P (A) + P (B) P (A and B). Example. In a certain university, 52% of all students take a statistics class, 23% take a computing course, and 7% take both. What percentage of students take at least one of computing or statistics? 8
9 For a randomly chosen student let A be the event the student takes statistics, and let B be the event the student takes computing. The Venn diagram to represent this situation is: S A B Figure 4. Venn diagram for this problem. 9
10 Applying the Law of Addition, P (A or B) = = In other words, 68% of students take at least one of Statistics or Computing. 10
11 Independent Events Two events are said to be independent if the outcome of one of them does not influence the other. For example, in sporting events, the outcomes of different games are usually considered independent even though that may not be true in a completely strict and literal sense. The multiplication rule for independent events says that if A and B are independent, P (A and B) = P (A) P (B). 11
12 Example: A football pundit states that the probability that UNC will beat NC State is 0.4, while the probability that UNC will beat Duke is 0.8. What is the probability that 1. UNC wins both games? 2. UNC wins at least one game? 3. UNC loses both games? 12
13 Solution: 1. If A is the event UNC beats State and B is the event UNC beats Duke, and if we assume these are independent events, then the probability of A and B is = Apply the Law of Addition: P (A or B) = P (A)+P (B) P (A and B) = = Apply the Law of Complementary Events: UNC loses both games is the complement of UNC wins at least one game, so its probability is =
14 Warning: Don t confuse the notions of independent events and disjoint events. Independence means that the outcome of one event does not influence the outcome of the other. Disjoint means that if one event occurs then the other cannot occur the very opposite of independence! 14
15 Conditional Probabilities Consider the example (page 218 of text, referring to the Wimbledon tennis tournament), A: Federer misses his first serve B: Federer misses his second serve We are told that Federer misses his first serve 36% of the time, and that of all the times he misses his first serve, he also misses his second serve 6% of the time. What, then, is the probability he has a double fault? Logically, the answer is 6% of 36%, or , which is about
16 Now let us rephrase this in the language of conditional probability. We are told that the event A occurs 36% of the time, or in other words P (A) = We are also told that, given that A has occurred, the event B occurs 6% of the time. This is written in probability notation as P (B A) = The left hand side is read as the probability of B given A. In this particular context, it would not make sense to talk about the probability of B given A c, though in other contexts, that would make sense (e.g. free throws in basketball). 16
17 The law of multiplication for conditional probabilities says P (A and B) = P (A) P (B A). Note that if we just interchange the role of A and B, we also get P (A and B) = P (B) P (A B). Finally, if A and B are independent, we get P (A B) = P (A) and P (B A) = P (B) that formalizes what is meant by saying that the outcome of one event does not influence the outcome of the other. But in that case, either of the last two formulas reduces to P (A and B) = P (A) P (B) as in our earlier formulation of the multiplication rule for independent events. 17
18 Here is another (more complicated) example. Consider the game in which a player tosses a die twice, and we want to calculate the probability that the total of the two tosses is at least 10. Define the events A: The first throw is a 6. B: The first throw is a 5. C: The first throw is a 4. D: The total of the two throws is at least 10. Note that if the first throw is less than 4, it s impossible for the total to be 10 or higher. So P (D) = P (A and D) + P (B and D) + P (C and D). (1) 18
19 Now P (A) = 6 1. Given that A has occurred, D will occur if the second throw produces any of 4, 5 or 6, and the probability of one of those outcomes is 3 6 or 1 2. So we have P (A) = 1 6, P (D A) = 1 2, P (A and D) = = Similarly P (B) = 1 6, P (D B) = 1 3, P (B and D) = = 1 18, P (C) = 1 6, P (D C) = 1 6, P (C and D) = = Therefore, (1) leads us to P (D) = = 1 6 giving the same answer as in our earlier calculation. 19
PROBABILITY NOTIONS. Summary. 1. Random experiment
PROBABILITY NOTIONS Summary 1. Random experiment... 1 2. Sample space... 2 3. Event... 2 4. Probability calculation... 3 4.1. Fundamental sample space... 3 4.2. Calculation of probability... 3 4.3. Non
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 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 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 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 informationName Date. Goal: Understand and represent the intersection and union of two sets.
F Math 12 3.3 Intersection and Union of Two Sets p. 162 Name Date Goal: Understand and represent the intersection and union of two sets. A. intersection: The set of elements that are common to two or more
More informationChapter 4: Probabilities and Proportions
Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 4: Probabilities and Proportions Section 4.1 Introduction In the real world,
More informationA Simple Example. Sample Space and Event. Tree Diagram. Tree Diagram. Probability. Probability  1. Probability and Counting Rules
Probability and Counting Rules researcher claims that 10% of a large population have disease H. random sample of 100 people is taken from this population and examined. If 20 people in this random sample
More informationMath 117 Chapter 7 Sets and Probability
Math 117 Chapter 7 and Probability Flathead Valley Community College Page 1 of 15 1. A set is a welldefined collection of specific objects. Each item in the set is called an element or a member. Curly
More informationStatistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then
Probability Statistical Inference Question: How often would this method give the correct answer if I used it many times? Answer: Use laws of probability. 1 Example: Tossing a coin If the coin is fair (chance
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 informationEvents. Independence. Coin Tossing. Random Phenomena
Random Phenomena Events A random phenomenon is a situation in which we know what outcomes could happen, but we don t know which particular outcome did or will happen For any random phenomenon, each attempt,
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 informationBasics of Probability
Basics of Probability August 27 and September 1, 2009 1 Introduction A phenomena is called random if the exact outcome is uncertain. The mathematical study of randomness is called the theory of probability.
More informationcalculating probabilities
4 calculating probabilities Taking Chances What s the probability he s remembered I m allergic to nonprecious metals? Life is full of uncertainty. Sometimes it can be impossible to say what will happen
More informationSome Definitions about Sets
Some Definitions about Sets Definition: Two sets are equal if they contain the same elements. I.e., sets A and B are equal if x[x A x B]. Notation: A = B. Recall: Sets are unordered and we do not distinguish
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 informationChapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.
MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.
More informationQuestion 1 Formatted: Formatted: Formatted: Formatted:
In many situations in life, we are presented with opportunities to evaluate probabilities of events occurring and make judgments and decisions from this information. In this paper, we will explore four
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 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 information33 Probability: Some Basic Terms
33 Probability: Some Basic Terms In this and the coming sections we discuss the fundamental concepts of probability at a level at which no previous exposure to the topic is assumed. Probability has been
More informationChapter 2: Systems of Linear Equations and Matrices:
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 informationAn approach to Calculus of Probabilities through real situations
MaMaEuSch Management Mathematics for European Schools http://www.mathematik.unikl.de/ mamaeusch An approach to Calculus of Probabilities through real situations Paula Lagares Barreiro Federico Perea RojasMarcos
More informationAQA Statistics 1. Probability. Section 2: Tree diagrams
Notes and Examples AQA Statistics Probability Section 2: Tree diagrams These notes include subsections on; Reminder of the addition and multiplication rules Probability tree diagrams Problems involving
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 information**Chance behavior is in the short run but has a regular and predictable pattern in the long run. This is the basis for the idea of probability.
AP Statistics Chapter 5 Notes 5.1 Randomness, Probability,and Simulation In tennis, a coin toss is used to decide which player will serve first. Many other sports use this method because it seems like
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 informationA (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes.
Chapter 7 Probability 7.1 Experiments, Sample Spaces, and Events A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes. Each outcome
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 informationThursday, October 18, 2001 Page: 1 STAT 305. Solutions
Thursday, October 18, 2001 Page: 1 1. Page 226 numbers 2 3. STAT 305 Solutions S has eight states Notice that the first two letters in state n +1 must match the last two letters in state n because they
More informationBetting Terms Explained www.sportsbettingxtra.com
Betting Terms Explained www.sportsbettingxtra.com To most people betting has a language of its own, so to help, we have explained the main terms you will come across when betting. STAKE The stake is the
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 informationSTAT 270 Probability Basics
STAT 270 Probability Basics Richard Lockhart Simon Fraser University Spring 2015 Surrey 1/28 Purposes of These Notes Jargon: experiment, sample space, outcome, event. Set theory ideas and notation: intersection,
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 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 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 informationTOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE. Topic P2: Sample Space and Assigning Probabilities
TOPIC P2: SAMPLE SPACE AND ASSIGNING PROBABILITIES SPOTLIGHT: THE CASINO GAME OF ROULETTE Roulette is one of the most popular casino games. The name roulette is derived from the French word meaning small
More informationExpected Value and the Game of Craps
Expected Value and the Game of Craps Blake Thornton Craps is a gambling game found in most casinos based on rolling two six sided dice. Most players who walk into a casino and try to play craps for the
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 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 informationIntroduction to Discrete Probability. Terminology. Probability definition. 22c:19, section 6.x Hantao Zhang
Introduction to Discrete Probability 22c:19, section 6.x Hantao Zhang 1 Terminology Experiment A repeatable procedure that yields one of a given set of outcomes Rolling a die, for example Sample space
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 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 informationStat 20: Intro to Probability and Statistics
Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. ChildersDay UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum
More informationExample: If we roll a dice and flip a coin, how many outcomes are possible?
12.5 Tree Diagrams Sample space Sample point Counting principle Example: If we roll a dice and flip a coin, how many outcomes are possible? TREE DIAGRAM EXAMPLE: Use a tree diagram to show all the possible
More informationProbability Models.S1 Introduction to Probability
Probability Models.S1 Introduction to Probability Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard The stochastic chapters of this book involve random variability. Decisions are
More informationMassachusetts Institute of Technology
n (i) m m (ii) n m ( (iii) n n n n (iv) m m Massachusetts Institute of Technology 6.0/6.: Probabilistic Systems Analysis (Quiz Solutions Spring 009) Question Multiple Choice Questions: CLEARLY circle the
More informationLecture 2 : Basics of Probability Theory
Lecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different
More informationElementary Statistics and Inference. Elementary Statistics and Inference. 16 The Law of Averages (cont.) 22S:025 or 7P:025.
Elementary Statistics and Inference 22S:025 or 7P:025 Lecture 20 1 Elementary Statistics and Inference 22S:025 or 7P:025 Chapter 16 (cont.) 2 D. Making a Box Model Key Questions regarding box What numbers
More informationApplied Liberal Arts Mathematics MAT105TE
Applied Liberal Arts Mathematics MAT105TE This TECEP tests a broadbased overview of mathematics intended for nonmath majors and emphasizes problemsolving modeled on reallife applications. Topics
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 informationGCSE Revision Notes Mathematics Probability
GCSE Revision Notes Mathematics Probability irevise.com 2014. All revision notes have been produced by mockness ltd for irevise.com. Email: info@irevise.com Copyrighted material. All rights reserved; no
More informationEXAM. Exam #3. Math 1430, Spring 2002. April 21, 2001 ANSWERS
EXAM Exam #3 Math 1430, Spring 2002 April 21, 2001 ANSWERS i 60 pts. Problem 1. A city has two newspapers, the Gazette and the Journal. In a survey of 1, 200 residents, 500 read the Journal, 700 read the
More informationLast Snowman Standing Sum of 2 Dice. Last Snowman Standing Game 1
Sum of 2 Dice 2 3 4 5 6 7 8 9 10 11 12 Last Snowman Standing Game 1 tkawas@mathwire.com Difference of 2 Dice 0 1 2 3 4 5 Last Snowman Standing Game 2 tkawas@mathwire.com One Die Toss 1 2 3 4 5 6 Last Snowman
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 informationTopic : Probability of a Complement of an Event Worksheet 1. Do the following:
Topic : Probability of a Complement of an Event Worksheet 1 1. You roll a die. What is the probability that 2 will not appear 2. Two 6sided dice are rolled. What is the 3. Ray and Shan are playing football.
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 information4.5 Finding Probability Using Tree Diagrams and Outcome Tables
4.5 Finding Probability Using ree Diagrams and Outcome ables Games of chance often involve combinations of random events. hese might involve drawing one or more cards from a deck, rolling two dice, or
More information7.1 Sample space, events, probability
7.1 Sample space, events, probability In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.
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 informationIf a tennis player was selected at random from the group, find the probability that the player is
Basic Probability. The table below shows the number of left and right handed tennis players in a sample of 0 males and females. Left handed Right handed Total Male 3 29 32 Female 2 6 8 Total 4 0 If a tennis
More informationHOW TO PLAY JustBet. What is Fixed Odds Betting?
HOW TO PLAY JustBet What is Fixed Odds Betting? Fixed Odds Betting is wagering or betting against on odds of any event or activity, with a fixed outcome. In this instance we speak of betting on the outcome
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 informationProbability Using Dice
Using Dice One Page Overview By Robert B. Brown, The Ohio State University Topics: Levels:, Statistics Grades 5 8 Problem: What are the probabilities of rolling various sums with two dice? How can you
More informationTHE LANGUAGE OF SETS AND SET NOTATION
THE LNGGE OF SETS ND SET NOTTION Mathematics is often referred to as a language with its own vocabulary and rules of grammar; one of the basic building blocks of the language of mathematics is the language
More informationSample Space, Events, and PROBABILITY
Sample Space, Events, and PROBABILITY In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.
More informationNumber in Probability
Number in Probability We are solving problems in probability contexts We are exploring probability concepts and language We are exploring the properties of numbers in probability contexts Exercise 1 Trials
More information1. The sample space S is the set of all possible outcomes. 2. An event is a set of one or more outcomes for an experiment. It is a sub set of S.
1 Probability Theory 1.1 Experiment, Outcomes, Sample Space Example 1 n psychologist examined the response of people standing in line at a copying machines. Student volunteers approached the person first
More informationProbability. Vocabulary
MAT 142 College Mathematics Probability Module #PM Terri L. Miller & Elizabeth E. K. Jones revised January 5, 2011 Vocabulary In order to discuss probability we will need a fair bit of vocabulary. Probability
More informationLesson 48 Conditional Probability
(A) Opening Example #1: A survey of 500 adults asked about college expenses. The survey asked questions about whether or not the person had a child in college and about the cost of attending college. Results
More informationTopic 5 Review [81 marks]
Topic 5 Review [81 marks] A foursided die has three blue faces and one red face. The die is rolled. Let B be the event a blue face lands down, and R be the event a red face lands down. 1a. Write down
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 informationAP Stats Fall Final Review Ch. 5, 6
AP Stats Fall Final Review 2015  Ch. 5, 6 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
More informationComment on the Tree Diagrams Section
Comment on the Tree Diagrams Section The reversal of conditional probabilities when using tree diagrams (calculating P (B A) from P (A B) and P (A B c )) is an example of Bayes formula, named after the
More informationTopic 6: Conditional Probability and Independence
Topic 6: September 1520, 2011 One of the most important concepts in the theory of probability is based on the question: How do we modify the probability of an event in light of the fact that something
More informationCombinatorics: The Fine Art of Counting
Combinatorics: The Fine Art of Counting Week 7 Lecture Notes Discrete Probability Continued Note Binomial coefficients are written horizontally. The symbol ~ is used to mean approximately equal. The Bernoulli
More informationThe New Mexico Lottery
The New Mexico Lottery 26 February 2014 Lotteries 26 February 2014 1/27 Today we will discuss the various New Mexico Lottery games and look at odds of winning and the expected value of playing the various
More informationLecture 2: Probability
Lecture 2: Probability Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics İstanbul Kültür University Outline 1 Introduction 2 Sample Spaces 3 Event 4 The Probability
More informationGrade Level Year Total Points Core Points % At Standard %
Performance Assessment Task Marble Game task aligns in part to CCSSM HS Statistics & Probability Task Description The task challenges a student to demonstrate an understanding of theoretical and empirical
More informationC.4 Tree Diagrams and Bayes Theorem
A26 APPENDIX C Probability and Probability Distributions C.4 Tree Diagrams and Bayes Theorem Find probabilities using tree diagrams. Find probabilities using Bayes Theorem. Tree Diagrams A type of diagram
More informationGrade 7/8 Math Circles Fall 2012 Probability
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics
More informationThe Casino Lab STATION 1: CRAPS
The Casino Lab Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will
More informationFlorida Department of Education/Office of Assessment January 2012. Algebra 1 EndofCourse Assessment Achievement Level Descriptions
Florida Department of Education/Office of Assessment January 2012 Algebra 1 EndofCourse Assessment Achievement Level Descriptions Algebra 1 EOC Assessment Reporting Category Functions, Linear Equations,
More informationCmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B
CmSc 175 Discrete Mathematics Lesson 10: SETS Sets: finite, infinite, : empty set, U : universal set Describing a set: Enumeration = {a, b, c} Predicates = {x P(x)} Recursive definition, e.g. sequences
More informationA set of maths games provided by the Wiltshire Primary Maths Team. These can be used at home as a fun way of practising the bare necessities in maths
A set of maths games provided by the Wiltshire Primary Maths Team. These can be used at home as a fun way of practising the bare necessities in maths skills that children will need to be confident with
More informationThe Normal Approximation to Probability Histograms. Dice: Throw a single die twice. The Probability Histogram: Area = Probability. Where are we going?
The Normal Approximation to Probability Histograms Where are we going? Probability histograms The normal approximation to binomial histograms The normal approximation to probability histograms of sums
More informationSet operations and Venn Diagrams. COPYRIGHT 2006 by LAVON B. PAGE
Set operations and Venn Diagrams Set operations and Venn diagrams! = { x x " and x " } This is the intersection of and. # = { x x " or x " } This is the union of and. n element of! belongs to both and,
More informationStatistics. Probability 2
Statistics Probability 2 Grades 8 and 9 Teacher Document Malati staff involved in developing these materials: Kate Bennie Kate Hudson Karen Newstead We acknowledge the valuable comments of Heleen Verhage
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 informationLesson 1: Experimental and Theoretical Probability
Lesson 1: Experimental and Theoretical Probability Probability is the study of randomness. For instance, weather is random. In probability, the goal is to determine the chances of certain events happening.
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 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 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 informationElements of probability theory
The role of probability theory in statistics We collect data so as to provide evidentiary support for answers we give to our many questions about the world (and in our particular case, about the business
More informationCarrom. Power lifting
Power lifting 1. Athletes in the sport are divided in five weight classes and placing is based on the total weight lifted on the three main lifts: a. Squat b. Bench Press c. Deadlift 2. Five bodyweight
More information15 Chances, Probabilities, and Odds
15 Chances, Probabilities, and Odds 15.1 Random Experiments and Sample Spaces 15.2 Counting Outcomes in Sample Spaces 15.3 Permutations and Combinations 15.4 Probability Spaces 15.5 Equiprobable Spaces
More informationProbability and Random Variables (Rees: )
Probability and Random Variables (Rees:. .) Earlier in this course, we looked at methods of describing the data in a sample. Next we would like to have models for the ways in which data can arise. Before
More informationA set of maths games based on ideas provided by the Wiltshire Primary Maths Team. These can be used at home as a fun way of practising the bare
A set of maths games based on ideas provided by the Wiltshire Primary Maths Team. These can be used at home as a fun way of practising the bare necessities in maths skills that children will need to be
More informationHey, That s Not Fair! (Or is it?)
Concept Probability and statistics Number sense Activity 9 Hey, That s Not Fair! (Or is it?) Students will use the calculator to simulate dice rolls to play two different games. They will decide if the
More information