# Section 8.1 Properties of Probability

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Section 8. Properties of Probability Section 8. Properties of Probability A probability is a function that assigns a value between 0 and to an event, describing the likelihood of that event happening. Probability retains a strong intuitive appeal: an impossible event is assigned a 0 probability of happening, and probabilities close to zero are unlikely events. As the probability values increase, the events are correspondingly more likely to happen, so that probabilities close to are considered likely events, with absolutely certain events being assigned a probability of (most people would say 00% ). An event that is equally likely to be true as not be true is assigned a probability of 0.5 (again, most people would say 50% ). The formal study of probability assumes a strong knowledge of sets (Preliminary Section 8.0) and their various concepts and operations. In this section it is assumed you have this knowledge of the concepts and of the terminology. Intuitive Examples and Definitions Consider the following example, whose outcome should be obvious: Example. A fair coin is flipped once. What is the probability it lands heads? Discussion. The obvious answer seems to be 50%. Note however the requirement for the coin to be fair, meaning it is not weighted or suspect in any fashion. It is true that under this condition, the probability of the coin landing heads is 50% The following argument supports this claim: a coin can land two possible ways: heads or tails. Assuming this is a fair coin, the two sides are (intuitively) equally likely. We want the specific outcome of heads, which can happen just way out of the 2 possible ways. Hence, the reasonable answer of 50%. This suggests a formal way to define a probability: An experiment generates a set of (equally likely) outcomes called the sample space S. Any subset of the sample space is called an event space (or event for short), usually represented by E. Therefore, the probability that event E occurs is the number of elements in E divided by the number of elements in S: P ( S) where n ( represents the number of elements in E, and n (S) represents the number of elements in S. Since 0 S), then 0 P ( always. In example above, the argument can be made even more formally: S is the sample space (the universe) of outcomes of flipping a coin: S { t, h}, where h heads and t tails. The event we are interested in is heads, or formally, a subset of the sample space E {h}. Since there is just one element in the event space and two elements in the sample space, it follows that the probability of the coin landing heads on one flip is ½, or 0.5, or 50%, all of which are equivalent. 333

2 Section 8. Properties of Probability Example 2: A fair die is rolled once. What is the probability is comes up 2? Discussion. The sample space (universe) is S {,2,3,4,5, } and the event space (subset) is E {2}. Thus, the probability of a 2 is P (" 2") n ( S ), which agrees with intuition Example 3: A roulette wheel has 38 slots, numbered - and two numbered 0 and 00. What is the probability any particular slot comes up as the winner? Discussion. It s not a requirement to list every element in the sample space when we can write n ( S) 38 and n (. Thus, the probability for any particular number coming up, including your lucky number, is P ( n ( S ) 38 Comment: The size of the sample space can become quite large, making it impractical to list every element. However, it is true that we need to know the size of the sample and event spaces exactly, and to be able to generate elements if need be, in order to properly determine probabilities. This is illustrated in the following examples and discussed further in the next section where methods of enumeration are presented to expedite these calculations. Example 4: Two fair dice are rolled once. What is the probability the sum of the two dice is 2? 7? Discussion. It is imperative to identify the correct sample space in which each element is equally likely to occur. Thus, instead of just looking at the sums, we look at every possible roll: S (,) (,2) (,3) (,4) (,5) (,) (2,) (2,2) (2,3) (2,4) (2,5) (2,) (3,) (3,2) (3,3) (3,4) (3,5) (3,) (4,) (4,2) (4,3) (4,4) (4,5) (4,) (5,) (5,2) (5,3) (5,4) (5,5) (5,) (,) (,2) (,3) (,4) (,5) (,) There are possible outcomes. In the case of rolling a 2, there is just one way to do that, so E {(,)} and n (, thus P ( sum 2) S ). The event space for rolling a 7 is E {(,), (2,5), (3,4), (4,3), (5,2), (,)}. Thus, the probability of rolling a 7 is P ( sum 7) S ) Comment: You may see a short-cut to identify the size of S: there are six ways to roll the first die and six ways to roll the second die, hence there are ways to roll two dice simultaneously. This is true and uses the Multiplication Principal (section 8.2) to determine the quantity. 334

3 Section 8. Properties of Probability Comment: Note that rolling two dice simultaneously is the same as rolling one die twice. They produce the exact same sample space so for all intents and purposes are the exact same experiment. This helps explain why the outcomes (5,2) and (2,5) in example 4 are treated as different outcome elements. Basic Properties of Probability Many probability questions can be solved using concepts from sets. Many are quite intuitive and you may find yourself using these concepts without being aware. The next example illustrates the use of complementary sets in calculating probabilities: Example 5: Two fair dice are rolled, and the sample space is the same as that in example 4. What is the probability the sum is not 2? Is not 7? Discussion. Since there is just one way to roll a 2, there must be 35 ways to not roll a 2. Thus the E ) 35 probability of not rolling a 2 is P ( sum 2) S ). Similarly, since there are ways to roll a 7, there must be 30 ways to not roll a 7, and the probability E ) 5 of not rolling a 7 is P ( sum 7) 30 S ) If E is an event, then E is the complement of E, that is, all the elements in the sample space S that are not in E. This suggests a useful property involving an event E and its complement E : Complementary probabilities: P ( E ) Notice that in example 5, there was no need to manually count out every single way to not roll a 2. It s faster to count the ways to roll a 2, and subtract from the known quantity in the sample space. Example : What is the probability that a randomly chosen person has a birthday different from yours, ignoring leap years and assuming all birthdays are equally likely (uniformly distributed)? Discussion. The sample space is the days in a year, thus n ( S) 5. The event space is your birthday itself which is just one day, so n (. It would be time consuming to list every single day that is not your birthday. Just note that your birthday is one day in the year, so there must be 4 4 other days available; that is, n ( E ) 4. Thus, the answer is 5, which is intuitive what you should expect 3 Example 7: If the probability of A is P (, what is the probability of not A? 8 Discussion: Since P ( A ), the probability of not A must be P ( A ) 3 335

4 Section 8. Properties of Probability Example 8: A football team has a probability of 0.52 of winning any given game, and a probability of 0.0 of tying any given game. What is the probability this football team loses any given game? Assume these three options make up the universe of all possible outcomes for a game. Discussion. Since this football team must win, lose or tie a game, we must have that P(win) lose) tie), so P(lose) P(win) P(tie) %. The event {lose} is the complement of the event space {win, tie} Probability of Unions An experiment is performed and a sample space (universe) of outcomes is generated. Consider two different event spaces (subsets) for the experiment; call them A and B. It is then natural to ask, what is the probability of A or B occurring? The key word in this sentence is or, and from sets this suggests we are looking at the union of the two events A and B. We need to be careful to not doublecount some of the elements. The following example is a heuristic approach to developing a formula for this situation. Example 9: Two fair dice are rolled (see the sample space in example 4). Let A the two dice show the same numbers and B at least one of the dice is a. What is P( A? In other words, what is the probability that the two dice show the same number or at least one of the dice is a? Discussion: The two event spaces are viewed separately at first: A {(,), (2,2), (3,3), (4,4), (5,5), (,)} P( (,) (2,) (3,) (4,) (5,) (,) B P( (,2) (,3) (,4) (,5) (,) Note the outcome (,) appears in both event sets A and B, the intersection A B. It gets counted twice, which is not allowed. It should only be counted once, so we subtract its probability of out once to balance the calculation: P ( A + This suggests a formula for the probability of a union: Probability of the Union: P( A P( P( A It is possible that events A and B have no common elements; the events A and B are mutually exclusive. The intersection A B will be empty, and the probability P ( A 0. In other words, it is impossible for events A and B to occur simultaneously. The probability of the union in this case reduces to P ( A P(. Consider the following example (next page): 3

5 Section 8. Properties of Probability Example 0: What is the probability that when two dice are rolled, the sum is exactly 2 or the sum is exactly 7? Discussion. Let A sum is exactly 2 and B sum is exactly 7. It should be clear that it is impossible (probability 0) for these two events to happen simultaneously. Therefore, their intersection is empty, and the probability of rolling exactly a 2 or exactly a 7 is P ( A Comment: Be especially aware of the words and and or in probability questions. The word and requires that the two events A and B occur simultaneously and thus corresponds to the intersection of the two event spaces. The word or requires that at least one of the events occurs and corresponds to the union of the two event spaces. Example : A card is drawn from a deck of 52 well-shuffled cards. Let A the card is a King and B the card is a Diamond. What is the probability (a) the card drawn is a King and a Diamond? What is the probability the card drawn is a King or a Diamond? Discussion. In (a), the and suggest we look for the card(s) that are simultaneously a King and a Diamond card. There is just one such card: the King of Diamonds card. Thus, the probability the card drawn is a King and a Diamond is P ( A 52. In (b) the or suggests the card must be one or the other, possibly both. There are four Kings in the 4 3 deck, so P ( 52, and there are thirteen Diamonds in the deck, so P ( 52. However, the King of Diamond card has been counted twice, once as a King and once as a Diamond. It must be subtracted out once to regain the balance. Thus, P( A P( P( A P( A Example 2: A corner shopping center has two main stores, Bob s Cleaners and Jim s Bait Shop, plus some other stores. A study of traffic entering onto the parking lot shows that 45% of those who pull in shop at Bob s while 37% shop at Jim s. Furthermore, 5% shop at both (this is included in the individual figures for Bob s and Jim s). Based on this information, (a) what percentage shop just at Bob s? (b) What percentage shop at Bob s or Jim s? And (c) what percentage shop at neither store? Discussion. A Venn Diagram is useful to parse the data into strict subdivisions (next page): 337

6 Section 8. Properties of Probability Figure. Venn Diagram for Example 2. The Venn Diagram is filled in first at the intersection of Bob s and Jim s, representing 5% of the shoppers. Working outward, we know that 45% overall shop at Bob s, including the 5% who shop at both stores, so evidently 30% shop just at Bob s, answering part (a) of the question. For part (b), this is the union of those who shop at the two stores. With the Venn Diagram, the union is just the sum of the probabilities within the Bob s and Jim s circles. Thus, 30% + 5% + 22% 7% shop at Bob s or Jim s. This can also be solved using the probability of the union formula: P( B J ) P( J ) P( B J ) P( B J ) For part (c), the Venn Diagram shows the result to be 33%, the complement of all those who shop at one store or the other Empirical Probabilities Empirical probabilities are calculated by direct experimentation as opposed to the theoretical, mathematical approach used in the preceding (and forthcoming) sections. The probability of some events can be found easily by a purely mathematical approach as well as by direct experimentation. For example, if flipping a fair coin, determining that heads comes up with a probability of 0.5 is easily proven to be accurate by actually flipping a coin numerous times and tracking the outcomes. The probability of other scenarios can only be determined either through a mathematical approach, or by direct experimentation. In the dice game Yahtzee, a player throws five dice at once. A yahtzee itself is all five dice showing the same value. The probability of such an event is found mathematically to be, 29. Very few people would have the patience to demonstrate this empirically as it would require many thousands of trials, so the mathematical figure is accepted as fact. On the other hand, a basketball player s shot accuracy (shots made divided by attempts) is essentially a probability of success. This figure can only be determined by past performance; there is no purely mathematical way to determine this figure. 338

7 Section 8. Properties of Probability A key concept in comparing empirical probabilities to theoretical probabilities is a notion known as the Law of Large Numbers. It simply means that the empirical approach must be performed many many times as many times as possible before the probabilities derived from the experiment can be accepted as trustworthy and in agreement with the theoretical figures. Example 3: A coin is flipped twice, and comes up heads both times. Is this proof that the coin is not fair? Discussion: Two flips is hardly enough experimentation to determine the nature of the coin. The answer would be no There is no hard and fast rule as to how many trials must be performed, or when the figures agree or disagree or are proven. The only requirement of the Law of Large Numbers is to perform the experiment many times. If the math and the experiment are done properly and with care, the two should agree after many trials (and stay in agreement after many trials). An interesting avenue of discussion appeals to a person s subjective sense of how much experimentation is enough, and whether the figures are in agreement with the theoretical. Some questions arise: Should you perform more trials? Perhaps the theoretical figure is wrong? Suppose coin is flipped 0 times and comes up heads 3 times, tails 7 times. Most people probably would not be surprised at this, figuring it is within the vagaries of chance. However, after 00 flips of a coin, a split may be suspicious (or maybe not). Everyone has a different sense as to when to feel suspicious about the coin itself, or whether to just flip the coin some more. Example 4: A die is rolled, and the theoretical probability of it showing a 2 is 0. 7 (rounded, see example 2 in this section). How does the actual throwing of a die many times compare to this theoretical figure? Discussion: This experiment was performed by students in three sections of this course at Arizona State University during Spring term, Each student rolled a die 25 times and tallied the number of times each face came up. The figures were collected (in no particular order) and tallied on a spreadsheet. A total of 82 students participated, with a total of 2,050 rolls (trials). The first sheet showed that 2 had come up twice in 25 rolls, a figure. However, two sheets later the student had rolled a 2 seven times in 25 rolls. In a way, these two people balanced one another out. The running tally figures were: Number of rolls: Frequency of 2 : Empirical Probability: After over 2,000 rolls, the empirical figures were in strong alignment with the theoretical figure of 0.7. Figures for the other values were in strong alignment as well. The expected number of 2 s (or any other number) would be (2,050 divided by ). Given we were off by 5 or still seems very reasonable 339

8 Section 8. Properties of Probability Summary An experiment generates a sample space (universe) S of outcomes. An event E is any subset of S. The probability of an event is P ( S) The probability of an impossible event is 0. The probability of a certain event is. Since 0 S), then 0 P (. Given an event E, we can consider its complement E, and their probabilities are related by the complementary probabilities equation P ( E ) The probability of the union of two events A and B is P( A P( P( A If A and B are mutually exclusive events, then these probabilities are true: P ( A 0 P ( A P( An empirical probability is one that is found by direct experimentation. The Law of Large Numbers suggests that an experiment must be performed many times before the empirical probabilities become in strong agreement with the theoretical probabilities. 340

### 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

### 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

### MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

### Ch. 13.2: Mathematical Expectation

Ch. 13.2: Mathematical Expectation Random Variables Very often, we are interested in sample spaces in which the outcomes are distinct real numbers. For example, in the experiment of rolling two dice, we

### 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

### 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

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

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

### Fundamentals 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

### 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

### 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

### 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

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### LET S MAKE A DEAL! ACTIVITY

LET S MAKE A DEAL! ACTIVITY NAME: DATE: SCENARIO: Suppose you are on the game show Let s Make A Deal where Monty Hall (the host) gives you a choice of three doors. Behind one door is a valuable prize.

### 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

### 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.

### 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

### Probabilistic Strategies: Solutions

Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6-sided dice. What s the probability of rolling at least one 6? There is a 1

### Lab 11. Simulations. The Concept

Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that

### Probability --QUESTIONS-- Principles of Math 12 - Probability Practice Exam 1 www.math12.com

Probability --QUESTIONS-- Principles of Math - Probability Practice Exam www.math.com Principles of Math : Probability Practice Exam Use this sheet to record your answers:... 4... 4... 4.. 6. 4.. 6. 7..

### + 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

### Gaming the Law of Large Numbers

Gaming the Law of Large Numbers Thomas Hoffman and Bart Snapp July 3, 2012 Many of us view mathematics as a rich and wonderfully elaborate game. In turn, games can be used to illustrate mathematical ideas.

### Basic 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

### Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

### Introduction and Overview

Introduction and Overview Probability and Statistics is a topic that is quickly growing, has become a major part of our educational program, and has a substantial role in the NCTM Standards. While covering

### Problem of the Month: Fair Games

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

### Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

### Ready, Set, Go! Math Games for Serious Minds

Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 -

### P (A) = lim P (A) = N(A)/N,

1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

### A Few Basics of Probability

A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study

### 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

### 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

### The game of roulette is played by throwing a small ball onto a rotating wheel with thirty seven numbered sectors.

LIVE ROULETTE The game of roulette is played by throwing a small ball onto a rotating wheel with thirty seven numbered sectors. The ball stops on one of these sectors. The aim of roulette is to predict

### That s Not Fair! ASSESSMENT #HSMA20. Benchmark Grades: 9-12

That s Not Fair! ASSESSMENT # Benchmark Grades: 9-12 Summary: Students consider the difference between fair and unfair games, using probability to analyze games. The probability will be used to find ways

### Hoover High School Math League. Counting and Probability

Hoover High School Math League Counting and Probability Problems. At a sandwich shop there are 2 kinds of bread, 5 kinds of cold cuts, 3 kinds of cheese, and 2 kinds of dressing. How many different sandwiches

### 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

### Probability. 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

### Introductory Probability. MATH 107: Finite Mathematics University of Louisville. March 5, 2014

Introductory Probability MATH 07: Finite Mathematics University of Louisville March 5, 204 What is probability? Counting and probability 2 / 3 Probability in our daily lives We see chances, odds, and probabilities

### 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

### 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

### V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay \$ to play. A penny and a nickel are flipped. You win \$ if either

### Section 7C: The Law of Large Numbers

Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half

### Contemporary Mathematics- MAT 130. Probability. a) What is the probability of obtaining a number less than 4?

Contemporary Mathematics- MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than

### 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,

### 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

### Session 8 Probability

Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome

### Minimax Strategies. Minimax Strategies. Zero Sum Games. Why Zero Sum Games? An Example. An Example

Everyone who has studied a game like poker knows the importance of mixing strategies With a bad hand, you often fold But you must bluff sometimes Lectures in Microeconomics-Charles W Upton Zero Sum Games

### Section 6.2 Definition of Probability

Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 7, due Wedneday, March 14 Happy Pi Day! (If any errors are spotted, please email them to morrison at math dot berkeley dot edu..5.10 A croissant

### 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,

### 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

### 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

### 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

### PROBABILITY SECOND EDITION

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

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

Practice Test Chapter 9 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the odds. ) Two dice are rolled. What are the odds against a sum

### Current California Math Standards Balanced Equations

Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.

### Stage 2 - Addition and Subtraction 2

Tighes Hill Public School NSW Syllabus for the Australian Curriculum Number and Algebra Stage 2 - Addition and Subtraction 2 Outcome Teaching and Learning Activities Notes/ Future Directions/Evaluation

### 25 Integers: Addition and Subtraction

25 Integers: Addition and Subtraction Whole numbers and their operations were developed as a direct result of people s need to count. But nowadays many quantitative needs aside from counting require numbers

### 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

### PROBABILITY C A S I N O L A B

A P S T A T S A Fabulous PROBABILITY C A S I N O L A B AP Statistics Casino Lab 1 AP STATISTICS CASINO LAB: INSTRUCTIONS The purpose of this lab is to allow you to explore the rules of probability in the

### Bayesian Tutorial (Sheet Updated 20 March)

Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that

### 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

### 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

### Lecture 13. Understanding Probability and Long-Term Expectations

Lecture 13 Understanding Probability and Long-Term Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).

### 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

### 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

### Jan 31 Homework Solutions Math 151, Winter 2012. Chapter 3 Problems (pages 102-110)

Jan 31 Homework Solutions Math 151, Winter 01 Chapter 3 Problems (pages 10-110) Problem 61 Genes relating to albinism are denoted by A and a. Only those people who receive the a gene from both parents

### Probability. 4.1 Sample Spaces

Probability 4.1 Sample Spaces For a random experiment E, the set of all possible outcomes of E is called the sample space and is denoted by the letter S. For the coin-toss experiment, S would be the results

### 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

### Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80)

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you

### Basic Probability Concepts

page 1 Chapter 1 Basic Probability Concepts 1.1 Sample and Event Spaces 1.1.1 Sample Space A probabilistic (or statistical) experiment has the following characteristics: (a) the set of all possible outcomes

### 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

### STAT 35A HW2 Solutions

STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },

### 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

### 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.

### 6.3 Probabilities with Large Numbers

6.3 Probabilities with Large Numbers In general, we can t perfectly predict any single outcome when there are numerous things that could happen. But, when we repeatedly observe many observations, we expect

### 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,

### Quiz CHAPTER 16 NAME: UNDERSTANDING PROBABILITY AND LONG- TERM EXPECTATIONS

Quiz CHAPTER 16 NAME: UNDERSTANDING PROBABILITY AND LONG- TERM EXPECTATIONS 1. Give two examples of ways that we speak about probability in our every day lives. NY REASONABLE ANSWER OK. EXAMPLES: 1) WHAT

FUN + GAMES = MATHS Sue Fine Linn Maskell Teachers are often concerned that there isn t enough time to play games in maths classes. But actually there is time to play games and we need to make sure that

### A-Level Maths. in a week. Core Maths - Co-ordinate Geometry of Circles. Generating and manipulating graph equations of circles.

A-Level Maths in a week Core Maths - Co-ordinate Geometry of Circles Generating and manipulating graph equations of circles. Statistics - Binomial Distribution Developing a key tool for calculating probability

### 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

### What is the Probability of Pigging Out

What is the Probability of Pigging Out Mary Richardson Susan Haller Grand Valley State University St. Cloud State University richamar@gvsu.edu skhaller@stcloudstate.edu Published: April 2012 Overview of

### Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific

Contemporary Mathematics Online Math 1030 Sample Exam I Chapters 12-14 No Time Limit No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the left-hand margin. You

### Section 6-5 Sample Spaces and Probability

492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)

### 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

Addition and Subtraction Games Odd or Even (Grades 13) Skills: addition to ten, odd and even or more Materials: each player has cards (Ace=1)10, 2 dice 1) Each player arranges their cards as follows. 1

### 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

### 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,

### 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

### 3.2 Roulette and Markov Chains

238 CHAPTER 3. DISCRETE DYNAMICAL SYSTEMS WITH MANY VARIABLES 3.2 Roulette and Markov Chains In this section we will be discussing an application of systems of recursion equations called Markov Chains.

### Question of the Day. Key Concepts. Vocabulary. Mathematical Ideas. QuestionofDay

QuestionofDay Question of the Day What is the probability that in a family with two children, both are boys? What is the probability that in a family with two children, both are boys, if we already know

### Responsible Gambling Education Unit: Mathematics A & B

The Queensland Responsible Gambling Strategy Responsible Gambling Education Unit: Mathematics A & B Outline of the Unit This document is a guide for teachers to the Responsible Gambling Education Unit:

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### Example: Tree Diagram

Foul Or Not? Strategy is certainly crucial to any sport or any situation where we have a confrontation or conflict. In this section, we will see how probability and expected value can help us make better

### 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

### Using games to support. Win-Win Math Games. by Marilyn Burns

4 Win-Win Math Games by Marilyn Burns photos: bob adler Games can motivate students, capture their interest, and are a great way to get in that paperand-pencil practice. Using games to support students

### 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

### 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

### REWARD System For Even Money Bet in Roulette By Izak Matatya

REWARD System For Even Money Bet in Roulette By Izak Matatya By even money betting we mean betting on Red or Black, High or Low, Even or Odd, because they pay 1 to 1. With the exception of the green zeros,

### THE MULTINOMIAL DISTRIBUTION. Throwing Dice and the Multinomial Distribution

THE MULTINOMIAL DISTRIBUTION Discrete distribution -- The Outcomes Are Discrete. A generalization of the binomial distribution from only 2 outcomes to k outcomes. Typical Multinomial Outcomes: red A area1