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

Size: px
Start display at page:

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

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 Non-disjoint 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

Concepts of Probability

Concepts 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 information

Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett

Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett 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 information

Probability: 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 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 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 information

Statistical Inference. Prof. Kate Calder. If the coin is fair (chance of heads = chance of tails) then

Statistical 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 information

AP Stats - Probability Review

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

More information

calculating probabilities

calculating probabilities 4 calculating probabilities Taking Chances What s the probability he s remembered I m allergic to non-precious metals? Life is full of uncertainty. Sometimes it can be impossible to say what will happen

More information

Chapter 4: Probability and Counting Rules

Chapter 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 information

Chapter 2: Systems of Linear Equations and Matrices:

Chapter 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 information

Question 1 Formatted: Formatted: Formatted: Formatted:

Question 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 information

Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty

Basic 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 information

Thursday, October 18, 2001 Page: 1 STAT 305. Solutions

Thursday, 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 information

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 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 information

In the situations that we will encounter, we may generally calculate the probability of an event

In 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

Chapter 13 & 14 - Probability PART

Chapter 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 information

TOPIC 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. 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 information

Basic Probability Theory II

Basic 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 information

STAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia

STAT 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 information

36 Odds, Expected Value, and Conditional Probability

36 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 information

Ch. 13.3: More about Probability

Ch. 13.3: More about Probability Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the

More information

Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. Childers-Day UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum

More information

Expected Value and the Game of Craps

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

More information

Applied Liberal Arts Mathematics MAT-105-TE

Applied Liberal Arts Mathematics MAT-105-TE Applied Liberal Arts Mathematics MAT-105-TE This TECEP tests a broad-based overview of mathematics intended for non-math majors and emphasizes problem-solving modeled on real-life applications. Topics

More information

EXAM. Exam #3. Math 1430, Spring 2002. April 21, 2001 ANSWERS

EXAM. 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 information

Introduction to Discrete Probability. Terminology. Probability definition. 22c:19, section 6.x Hantao Zhang

Introduction 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 information

HOW TO PLAY JustBet. What is Fixed Odds Betting?

HOW 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 information

PROBABILITY 14.3. section. The Probability of an Event

PROBABILITY 14.3. section. The Probability of an Event 4.3 Probability (4-3) 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 information

Topic : Probability of a Complement of an Event- Worksheet 1. Do the following:

Topic : 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 6-sided dice are rolled. What is the 3. Ray and Shan are playing football.

More information

Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability

Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability 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 information

THE LANGUAGE OF SETS AND SET NOTATION

THE 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 information

Probability and Venn diagrams UNCORRECTED PAGE PROOFS

Probability 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 information

Probability and Expected Value

Probability 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 information

MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.

MA 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 information

Betting Terms Explained www.sportsbettingxtra.com

Betting 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 information

1. 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. 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 information

Probability Models.S1 Introduction to Probability

Probability Models.S1 Introduction to Probability Probability Models.S1 Introduction to Probability Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard The stochastic chapters of this book involve random variability. Decisions are

More information

Unit 19: Probability Models

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

More information

Combinatorics: The Fine Art of Counting

Combinatorics: 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 information

Florida Department of Education/Office of Assessment January 2012. Algebra 1 End-of-Course Assessment Achievement Level Descriptions

Florida Department of Education/Office of Assessment January 2012. Algebra 1 End-of-Course Assessment Achievement Level Descriptions Florida Department of Education/Office of Assessment January 2012 Algebra 1 End-of-Course Assessment Achievement Level Descriptions Algebra 1 EOC Assessment Reporting Category Functions, Linear Equations,

More information

The Casino Lab STATION 1: CRAPS

The 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 information

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected

More information

Probability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes

Probability. 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 information

Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University

Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 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 information

2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.

2. 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 information

CmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B

CmSc 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 information

Set operations and Venn Diagrams. COPYRIGHT 2006 by LAVON B. PAGE

Set 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 information

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

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 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 information

Bookmaker Bookmaker Online Bookmaker Offline

Bookmaker Bookmaker Online Bookmaker Offline IN BRIEF Bookmaker is an application suite which will provide you with the necessary tools for setting up a bookmaker s office. Bookmaker offers a complete package which is currently available for trial

More information

Probability OPRE 6301

Probability 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 information

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

6.042/18.062J Mathematics for Computer Science. Expected Value I 6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

More information

Parimutuel Betting. Note by Émile Borel. Translated by Marco Ottaviani

Parimutuel Betting. Note by Émile Borel. Translated by Marco Ottaviani Parimutuel Betting Note by Émile Borel Translated by Marco Ottaviani Comptes Rendus Hebdomadaires des Séances de l Académie des Sciences (Proceedings of the French Academy of Sciences) 1938, 2 nd Semester,

More information

Homework 20: Compound Probability

Homework 20: Compound Probability Homework 20: Compound Probability Definition The probability of an event is defined to be the ratio of times that you expect the event to occur after many trials: number of equally likely outcomes resulting

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level S2 of challenge: B/C S2 Mathematical goals Starting points Materials required Time needed Evaluating probability statements To help learners to: discuss and clarify some common misconceptions about

More information

Statistics 100A Homework 2 Solutions

Statistics 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 information

BRIEF SOLUTIONS. Basic Probability WEEK THREE. This worksheet relates to chapter four of the text book (Statistics for Managers 4 th Edition).

BRIEF SOLUTIONS. Basic Probability WEEK THREE. This worksheet relates to chapter four of the text book (Statistics for Managers 4 th Edition). BRIEF SOLUTIONS Basic Probability WEEK THREE This worksheet relates to chapter four of the text book (Statistics for Managers 4 th Edition). This topic is the one many students find the most difficult.

More information

PROBABILITY SECOND EDITION

PROBABILITY SECOND EDITION PROBABILITY SECOND EDITION Table of Contents How to Use This Series........................................... v Foreword..................................................... vi Basics 1. Probability All

More information

We rst consider the game from the player's point of view: Suppose you have picked a number and placed your bet. The probability of winning is

We rst consider the game from the player's point of view: Suppose you have picked a number and placed your bet. The probability of winning is Roulette: On an American roulette wheel here are 38 compartments where the ball can land. They are numbered 1-36, and there are two compartments labeled 0 and 00. Half of the compartments numbered 1-36

More information

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

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

More information

PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA

PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet

More information

The study of probability has increased in popularity over the years because of its wide range of practical applications.

The 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 information

Probability. 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) 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 information

Definition and Calculus of Probability

Definition 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 information

Probability Using Dice

Probability 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 information

Elementary Statistics and Inference. Elementary Statistics and Inference. 16 The Law of Averages (cont.) 22S:025 or 7P:025.

Elementary 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 information

RULES AND REGULATIONS OF FIXED ODDS BETTING GAMES

RULES AND REGULATIONS OF FIXED ODDS BETTING GAMES RULES AND REGULATIONS OF FIXED ODDS BETTING GAMES Project: Royalhighgate Public Company Ltd. 04.04.2014 Table of contents SECTION I: GENERAL RULES... 6 ARTICLE 1 GENERAL REGULATIONS...6 ARTICLE 2 THE HOLDING

More information

6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0.

6. Let X be a binomial random variable with distribution B(10, 0.6). What is the probability that X equals 8? A) (0.6) (0.4) B) 8! C) 45(0.6) (0. Name: Date:. For each of the following scenarios, determine the appropriate distribution for the random variable X. A) A fair die is rolled seven times. Let X = the number of times we see an even number.

More information

TEAM TENNIS TOURNAMENT *The following rules provided by Purdue Intramural Sports are not meant to be all encompassing.*

TEAM TENNIS TOURNAMENT *The following rules provided by Purdue Intramural Sports are not meant to be all encompassing.* TEAM TENNIS TOURNAMENT *The following rules provided by Purdue Intramural Sports are not meant to be all encompassing.* SECTION 1. LEAGUE FORMAT I. Tournament Overview A. The tournament will consist of

More information

Lesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314

Lesson 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 information

DOUBLES TENNIS LEAGUE *The following rules provided by Purdue Intramural Sports are not meant to be all encompassing.*

DOUBLES TENNIS LEAGUE *The following rules provided by Purdue Intramural Sports are not meant to be all encompassing.* DOUBLES TENNIS LEAGUE *The following rules provided by Purdue Intramural Sports are not meant to be all encompassing.* SECTION 1. TOURNAMENT FORMAT I. League Overview A. The league will consist of a five

More information

Probabilities of Poker Hands with Variations

Probabilities of Poker Hands with Variations Probabilities of Poker Hands with Variations Jeff Duda Acknowledgements: Brian Alspach and Yiu Poon for providing a means to check my numbers Poker is one of the many games involving the use of a 52-card

More information

I. WHAT IS PROBABILITY?

I. WHAT IS PROBABILITY? C HAPTER 3 PROBABILITY Random Experiments I. WHAT IS PROBABILITY? The weatherman on 0 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and

More information

4.4 Conditional Probability

4.4 Conditional Probability 4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school.

More information

Basic Probability Theory I

Basic Probability Theory I A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population

More information

Introduction to Probability

Introduction 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 information

Chapter 4 Probability

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

More information

Mathematical Expectation

Mathematical Expectation Mathematical Expectation Properties of Mathematical Expectation I The concept of mathematical expectation arose in connection with games of chance. In its simplest form, mathematical expectation is the

More information

The New Mexico Lottery

The 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 information

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability 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 information

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

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

More information

E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

More information

Betting systems: how not to lose your money gambling

Betting systems: how not to lose your money gambling Betting systems: how not to lose your money gambling G. Berkolaiko Department of Mathematics Texas A&M University 28 April 2007 / Mini Fair, Math Awareness Month 2007 Gambling and Games of Chance Simple

More information

Binomial Probability Distribution

Binomial Probability Distribution Binomial Probability Distribution In a binomial setting, we can compute probabilities of certain outcomes. This used to be done with tables, but with graphing calculator technology, these problems are

More information

Beginners Guide to Sports Arbitrage Trading and Bonus Scalping

Beginners Guide to Sports Arbitrage Trading and Bonus Scalping Beginners Guide to Sports Arbitrage Trading and Bonus Scalping Contents What Is Sports Arbitrage Trading? What Is Bonus Scalping? Risks of Sports Arbitrage Trading? Betfair/Betdaq The Spreadsheet Worked

More information

Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Problem Set 1 (with solutions)

Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Problem Set 1 (with solutions) Math 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand Problem Set 1 (with solutions) About this problem set: These are problems from Course 1/P actuarial exams that I have collected over the years,

More information

TABLE TENNIS. Rules & Regulations

TABLE TENNIS. Rules & Regulations TABLE TENNIS Rules & Regulations The tournament will be played according to the rules laid by TTFI & BFI. Non-Marking shoes for Badminton is compulsory. No player will be allowed to play without Non-Marking

More information

5. Probability Calculus

5. Probability Calculus 5. Probability Calculus So far we have concentrated on descriptive statistics (deskriptiivinen eli kuvaileva tilastotiede), that is methods for organizing and summarizing data. As was already indicated

More information

Benchmark Test : Algebra 1

Benchmark Test : Algebra 1 1 Benchmark: MA.91.A.3.3 If a ar b r, what is the value of a in terms of b and r? A b + r 1 + r B 1 + b r + b C 1 + b r D b r 1 Benchmark: MA.91.A.3.1 Simplify: 1 g(5 3) 4g 13 4 F 11 4 g 16 G g 1 H 15

More information

DOUBLES TENNIS LEAGUE

DOUBLES TENNIS LEAGUE DOUBLES TENNIS LEAGUE *The following rules provided by Purdue Intramural Sports are not meant to be all encompassing. Please refer to the Participant Manual for comprehensive eligibility guidelines, policies,

More information

The 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. 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 information

William Hill Race and Sports Book House Wagering Rules and Regulations are provided for your information.

William Hill Race and Sports Book House Wagering Rules and Regulations are provided for your information. House Rules William Hill Race and Sports Book House Wagering Rules and Regulations are provided for your information. 1. GENERAL 1.1. MANAGEMENT 1.2. TICKET ACCURACY 1.3. DEFINITION OF "ACTION" 1.4. DEFINITION

More information

rules Everything you need to know, to play a game of TTX

rules Everything you need to know, to play a game of TTX rules Rule! Everything you need to know, to play a game of TTX Version 1 Created 01 August 2016 1. Equipment The great thing about TTX is that it can be played any time, any where 1 2 3 4 Grab yourself

More information

If, under a given assumption, the of a particular observed is extremely. , we conclude that the is probably not

If, under a given assumption, the of a particular observed is extremely. , we conclude that the is probably not 4.1 REVIEW AND PREVIEW RARE EVENT RULE FOR INFERENTIAL STATISTICS If, under a given assumption, the of a particular observed is extremely, we conclude that the is probably not. 4.2 BASIC CONCEPTS OF PROBABILITY

More information

This Method will show you exactly how you can profit from this specific online casino and beat them at their own game.

This Method will show you exactly how you can profit from this specific online casino and beat them at their own game. This Method will show you exactly how you can profit from this specific online casino and beat them at their own game. It s NOT complicated, and you DON T need a degree in mathematics or statistics to

More information

Sections 2.1, 2.2 and 2.4

Sections 2.1, 2.2 and 2.4 SETS Sections 2.1, 2.2 and 2.4 Chapter Summary Sets The Language of Sets Set Operations Set Identities Introduction Sets are one of the basic building blocks for the types of objects considered in discrete

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Ch. 4 Discrete Probability Distributions 4.1 Probability Distributions 1 Decide if a Random Variable is Discrete or Continuous 1) State whether the variable is discrete or continuous. The number of cups

More information

Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

More information

Exam Style Questions. Revision for this topic. Name: Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser

Exam Style Questions. Revision for this topic. Name: Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser Name: Exam Style Questions Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser You may use tracing paper if needed Guidance 1. Read each question carefully before you begin answering

More information

Blood Circulation Game

Blood Circulation Game Blood Circulation Game Developed by June Agar at Rushey Mead School in Leicester. All the bits are here to make a board game, but some work by you and/or your children is needed. You might want to consider

More information

Probability definitions

Probability 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 information

Statistical techniques for betting on football markets.

Statistical techniques for betting on football markets. Statistical techniques for betting on football markets. Stuart Coles Padova, 11 June, 2015 Smartodds Formed in 2003. Originally a betting company, now a company that provides predictions and support tools

More information