4.3 Rules of Probability

Size: px
Start display at page:

Download "4.3 Rules of Probability"

Transcription

1 4.3 Rules of Probability If a probability distribution is not uniform, to find the probability of a given event, add up the probabilities of all the individual outcomes that make up the event. Example: Suppose you are given the following probability distribution for a sample space S = {s 1, s 2, s 3, s 4, s 5, s 6 } Outcome s 1 s 2 s 3 s 4 s 5 s Probability Supppose E = {s 1, s 4, s 5 }, F = {s 2, s 3 }, and G = {s 2, s 5 }. Calculate the following. P (s 4 ) P (E) P (F G) P (E F ) P (E c ) P (E G) Example: Suppose an experiment has a sample space S = {s 1, s 2, s 3 } where P (s 1 ) + P (s 2 ) = 0.35 and P (s 2 ) + P (s 3 ) = Find the probability distribution for S. 1

2 What happens if you cannot list out all the outcomes and their probalities (or do not want to)? Or worse, what if we don t even know what the specific outcomes in the events are? We can use the following more general rules. Rules of Probability: 1. 0 P (E) 1 for any event E in a sample space S. In particular P ( ) = 0 and P (S) = Union rule for probability: If E and F are ANY two events, then P (E F ) = P (E) + P (F ) P (E F ) Note: If E and F are mutually exclusive, then E F =, which means P (E F ) = 0 and the formula reduces to just P (E F ) = P (E) + P (F ). 3. Complement Principle: P (E c ) = 1 P (E) or P (E) = 1 P (E c ) Example: Let E and F be two events of an experiment with sample space S. Suppose P (E) = 0.5, P (F ) = 0.4, and P (E F ) = 0.1. Compute the following. P (F c ) P (E c ) P (E F ) Example: If P (E c ) = 0.3 and P (F ) = 0.2 with E and F mutually exclusive, what is P (E F )? Example: In a survey of a group of people it was found that the probability someone did not like Dr. Pepper was 0.65, the probability someone liked Dr. Pepper was 0.45, and the probability that someone liked both Dr. Pepper and Coke was What is the probability that someone in this group likes Dr. Pepper or likes Coke? 2

3 Example: An experiment consists of selecting a card at random from a 52-card deck. Find the probability that a red face card is drawn. Find the probability that a face card is not drawn. Find the probability that a diamond or a club is drawn. Find the probability that a spade or a queen is drawn. Find the probability that a 3 or a red card is drawn. Example: The table below gives the number of students of each classification who are majoring and not majoring in business in a class of 110 students. Freshmen Sophomores Juniors Seniors Total Business Non-Business Total A student is randomly selected from this class. What is the probability that... The student is not a junior? The student is a freshman and a non-business major? The student is a business major or a sophomore? The student is a non-business major or is not a junior? 3

4 4.4 Random Variables and Expected Value A random variable is a rule that assigns a number to each outcome of an experiment. We usually denote a random variable by X. Example: A coin is tossed three times and the sequence of heads and tails is observed. List the outcomes of the experiment. Let the random variable X denote the number of tails that occur. What are the possible values of X? Find the probability distribution of X. What is P (1 X 2)? What is P (X > 0)? 4

5 The expected value of a random variable X, denoted E(X), is given by E(X) = x 1 p 1 + x 2 p x n p n where x 1, x 2,..., x n are the values that X may assume, and p 1, p 2,..., p n are the probabilities of each of these values. Example: Find the expected value of the random variable X given below. X Probability Consider the experiment of rolling 2 fair 5-sided dice. Let X be the sum of the numbers rolled. Find the probability distribution of X. What is E(X)? Example: A survey was conducted of families to determine the distribution of families by size. The results are: Family Size Number of Families Let the random variable X be the number of people in a randomly chosen family. Find the probability distribution for X. What is the expected number of people in a randomly chosen family? 5

6 Expected values are often used in games to determine whether the game is fair. A game is considered fair when the expected NET winnings are 0. You are playing a game at a carnival. The game costs $1. You select a card from a standard deck. If the card is an ace, then you win $3. If the card is a face card, you win $2. Otherwise you win nothing. Find the expected net winnings. Example: A raffle is held people buy a ticket for $3 each. First prize is $1500. Second prize is $750. Then, four $100 consolation prizes will be given. What are the expected net winnings for one person who buys a $3 ticket. A game consists of rolling a fair 5-sided die. The game costs $3 to play. If you roll an odd number, you win an amount of money equal to the number rolled. Otherwise you win nothing. What are your expected net winnings? 6

7 A game consists of rolling a pair of fair 6-sided dice. The game costs $4 to play. If you roll a double, you win $a. Otherwise, you get nothing. What value of a would make this game fair? Example: A man purchased a $25,000 life insurance policy from his employer for $200/yr. (The cost of $200 is called the premium.) The probability that he lives another year is What is the life insurance company s expected net gain? If the probability that the man lives drops to 0.98, what is the minimum amount of money, $a, he can expect to pay for his policy? Note: The insurance company will charge at minimum an amount of money so that their expected net gain is 0. They would probably want to charge more than that to have a positive expected net gain. 7

4.1 Sample Spaces and Events

4.1 Sample Spaces and Events 4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an

More information

7.1 Experiments, Sample Spaces, and Events

7.1 Experiments, Sample Spaces, and Events 7.1 Experiments, Sample Spaces, and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment

More information

Chapter 1: Sets and Probability

Chapter 1: Sets and Probability Chapter 1: Sets and Probability Section 1.3-1.5 Recap: Sample Spaces and Events An is an activity that has observable results. An is the result of an experiment. Example 1 Examples of experiments: Flipping

More information

Math 1313 Section 6.2 Definition of Probability

Math 1313 Section 6.2 Definition of Probability Math 1313 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

More information

Chapter 11: Probability and Counting Techniques

Chapter 11: Probability and Counting Techniques Chapter 11: Probability and Counting Techniques Diana Pell Section 11.3: Basic Concepts of Probability Definition 1. A sample space is a set of all possible outcomes of an experiment. Exercise 1. An experiment

More information

8.2 Union, Intersection, and Complement of Events; Odds

8.2 Union, Intersection, and Complement of Events; Odds 8.2 Union, Intersection, and Complement of Events; Odds Since we defined an event as a subset of a sample space it is natural to consider set operations like union, intersection or complement in the context

More information

Unit 9: Probability Assignments

Unit 9: Probability Assignments Unit 9: Probability Assignments #1: Basic Probability In each of exercises 1 & 2, find the probability that the spinner shown would land on (a) red, (b) yellow, (c) blue. 1. 2. Y B B Y B R Y Y B R 3. Suppose

More information

Chapter 1. Probability

Chapter 1. Probability Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.

More information

0-5 Adding Probabilities. 1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins.

0-5 Adding Probabilities. 1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins. 1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins. d. a. Copy the table and add a column to show the experimental probability of the spinner landing on

More information

Mathematical Foundations HW 5 By 11:59pm, 12 Dec, 2015

Mathematical Foundations HW 5 By 11:59pm, 12 Dec, 2015 1 Probability Axioms Let A,B,C be three arbitrary events. Find the probability of exactly one of these events occuring. Sample space S: {ABC, AB, AC, BC, A, B, C, }, and S = 8. P(A or B or C) = 3 8. note:

More information

Chapter 11: Probability and Counting Techniques

Chapter 11: Probability and Counting Techniques Chapter 11: Probability and Counting Techniques Diana Pell Section 11.1: The Fundamental Counting Principle Exercise 1. How many different two-letter words (including nonsense words) can be formed when

More information

Intermediate Math Circles November 1, 2017 Probability I

Intermediate Math Circles November 1, 2017 Probability I Intermediate Math Circles November 1, 2017 Probability I Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application.

More information

Mutually Exclusive Events Algebra 1

Mutually Exclusive Events Algebra 1 Name: Mutually Exclusive Events Algebra 1 Date: Mutually exclusive events are two events which have no outcomes in common. The probability that these two events would occur at the same time is zero. Exercise

More information

Exam III Review Problems

Exam III Review Problems c Kathryn Bollinger and Benjamin Aurispa, November 10, 2011 1 Exam III Review Problems Fall 2011 Note: Not every topic is covered in this review. Please also take a look at the previous Week-in-Reviews

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. 6.1 Practice Problems Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. 1) The probability of rolling an even number on a

More information

Probability Rules. 2) The probability, P, of any event ranges from which of the following?

Probability Rules. 2) The probability, P, of any event ranges from which of the following? Name: WORKSHEET : Date: Answer the following questions. 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if. 2) The probability,

More information

Chapter 1. Probability

Chapter 1. Probability Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.

More information

Important Distributions 7/17/2006

Important Distributions 7/17/2006 Important Distributions 7/17/2006 Discrete Uniform Distribution All outcomes of an experiment are equally likely. If X is a random variable which represents the outcome of an experiment of this type, then

More information

Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Section 6.1 #16. Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? Section 6.1 #16 What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1

More information

PROBABILITY. 1. Introduction. Candidates should able to:

PROBABILITY. 1. Introduction. Candidates should able to: PROBABILITY Candidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for the total score when two fair dice are thrown), or by calculation

More information

Probability: introduction

Probability: introduction May 6, 2009 Probability: introduction page 1 Probability: introduction Probability is the part of mathematics that deals with the chance or the likelihood that things will happen The probability of an

More information

Quiz 2 Review - on Notebook Paper Are You Ready For Your Last Quiz In Honors Math II??

Quiz 2 Review - on Notebook Paper Are You Ready For Your Last Quiz In Honors Math II?? Quiz 2 Review - on Notebook Paper Are You Ready For Your Last Quiz In Honors Math II?? Some things to Know, Memorize, AND Understand how to use are n What are the formulas? Pr ncr Fill in the notation

More information

Grade 7/8 Math Circles February 25/26, Probability

Grade 7/8 Math Circles February 25/26, Probability Faculty of Mathematics Waterloo, Ontario N2L 3G1 Probability Grade 7/8 Math Circles February 25/26, 2014 Probability Centre for Education in Mathematics and Computing Probability is the study of how likely

More information

Such a description is the basis for a probability model. Here is the basic vocabulary we use.

Such a description is the basis for a probability model. Here is the basic vocabulary we use. 5.2.1 Probability Models When we toss a coin, we can t know the outcome in advance. What do we know? We are willing to say that the outcome will be either heads or tails. We believe that each of these

More information

Section 7.1 Experiments, Sample Spaces, and Events

Section 7.1 Experiments, Sample Spaces, and Events Section 7.1 Experiments, Sample Spaces, and Events Experiments An experiment is an activity with observable results. 1. Which of the follow are experiments? (a) Going into a room and turning on a light.

More information

Math 227 Elementary Statistics. Bluman 5 th edition

Math 227 Elementary Statistics. Bluman 5 th edition Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 4 Probability and Counting Rules 2 Objectives Determine sample spaces and find the probability of an event using classical probability or empirical

More information

Section Introduction to Sets

Section Introduction to Sets Section 1.1 - Introduction to Sets Definition: A set is a well-defined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase

More information

The Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.)

The Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.) The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you $ that if you give me $, I ll give you $2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If

More information

Math 1324 Finite Mathematics Sections 8.2 and 8.3 Conditional Probability, Independent Events, and Bayes Theorem

Math 1324 Finite Mathematics Sections 8.2 and 8.3 Conditional Probability, Independent Events, and Bayes Theorem Finite Mathematics Sections 8.2 and 8.3 Conditional Probability, Independent Events, and Bayes Theorem What is conditional probability? It is where you know some information, but not enough to get a complete

More information

Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance Chapter 8: Probability: The Mathematics of Chance Free-Response 1. A spinner with regions numbered 1 to 4 is spun and a coin is tossed. Both the number spun and whether the coin lands heads or tails is

More information

Chapter 3: Elements of Chance: Probability Methods

Chapter 3: Elements of Chance: Probability Methods Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 3-4 2014-2015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,

More information

Probability Test Review Math 2. a. What is? b. What is? c. ( ) d. ( )

Probability Test Review Math 2. a. What is? b. What is? c. ( ) d. ( ) Probability Test Review Math 2 Name 1. Use the following venn diagram to answer the question: Event A: Odd Numbers Event B: Numbers greater than 10 a. What is? b. What is? c. ( ) d. ( ) 2. In Jason's homeroom

More information

Simple Probability. Arthur White. 28th September 2016

Simple Probability. Arthur White. 28th September 2016 Simple Probability Arthur White 28th September 2016 Probabilities are a mathematical way to describe an uncertain outcome. For eample, suppose a physicist disintegrates 10,000 atoms of an element A, and

More information

Define and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)

Define and Diagram Outcomes (Subsets) of the Sample Space (Universal Set) 12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the

More information

Section 11.4: Tree Diagrams, Tables, and Sample Spaces

Section 11.4: Tree Diagrams, Tables, and Sample Spaces Section 11.4: Tree Diagrams, Tables, and Sample Spaces Diana Pell Exercise 1. Use a tree diagram to find the sample space for the genders of three children in a family. Exercise 2. (You Try!) A soda machine

More information

Probability Review before Quiz. Unit 6 Day 6 Probability

Probability Review before Quiz. Unit 6 Day 6 Probability Probability Review before Quiz Unit 6 Day 6 Probability Warm-up: Day 6 1. A committee is to be formed consisting of 1 freshman, 1 sophomore, 2 juniors, and 2 seniors. How many ways can this committee be

More information

13-6 Probabilities of Mutually Exclusive Events

13-6 Probabilities of Mutually Exclusive Events Determine whether the events are mutually exclusive or not mutually exclusive. Explain your reasoning. 1. drawing a card from a standard deck and getting a jack or a club The jack of clubs is an outcome

More information

Probability and Statistics. Copyright Cengage Learning. All rights reserved.

Probability and Statistics. Copyright Cengage Learning. All rights reserved. Probability and Statistics Copyright Cengage Learning. All rights reserved. 14.2 Probability Copyright Cengage Learning. All rights reserved. Objectives What Is Probability? Calculating Probability by

More information

Chapter 4: Probability and Counting Rules

Chapter 4: Probability and Counting Rules Chapter 4: Probability and Counting Rules Before we can move from descriptive statistics to inferential statistics, we need to have some understanding of probability: Ch4: Probability and Counting Rules

More information

MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #1 - SPRING DR. DAVID BRIDGE

MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #1 - SPRING DR. DAVID BRIDGE MATH 205 - CALCULUS & STATISTICS/BUSN - PRACTICE EXAM # - SPRING 2006 - DR. DAVID BRIDGE TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. Tell whether the statement is

More information

Conditional Probability Worksheet

Conditional Probability Worksheet Conditional Probability Worksheet EXAMPLE 4. Drug Testing and Conditional Probability Suppose that a company claims it has a test that is 95% effective in determining whether an athlete is using a steroid.

More information

6) A) both; happy B) neither; not happy C) one; happy D) one; not happy

6) A) both; happy B) neither; not happy C) one; happy D) one; not happy MATH 00 -- PRACTICE TEST 2 Millersville University, Spring 202 Ron Umble, Instr. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all natural

More information

Simulations. 1 The Concept

Simulations. 1 The Concept 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 can be

More information

CHAPTERS 14 & 15 PROBABILITY STAT 203

CHAPTERS 14 & 15 PROBABILITY STAT 203 CHAPTERS 14 & 15 PROBABILITY STAT 203 Where this fits in 2 Up to now, we ve mostly discussed how to handle data (descriptive statistics) and how to collect data. Regression has been the only form of statistical

More information

Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.

Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1 - Experiments, Sample Spaces,

More information

Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.

Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1 - Experiments, Sample Spaces,

More information

Def: The intersection of A and B is the set of all elements common to both set A and set B

Def: The intersection of A and B is the set of all elements common to both set A and set B Def: Sample Space the set of all possible outcomes Def: Element an item in the set Ex: The number "3" is an element of the "rolling a die" sample space Main concept write in Interactive Notebook Intersection:

More information

Mathematics 3201 Test (Unit 3) Probability FORMULAES

Mathematics 3201 Test (Unit 3) Probability FORMULAES Mathematics 3201 Test (Unit 3) robability Name: FORMULAES ( ) A B A A B A B ( A) ( B) ( A B) ( A and B) ( A) ( B) art A : lace the letter corresponding to the correct answer to each of the following in

More information

PROBABILITY Case of cards

PROBABILITY Case of cards WORKSHEET NO--1 PROBABILITY Case of cards WORKSHEET NO--2 Case of two die Case of coins WORKSHEET NO--3 1) Fill in the blanks: A. The probability of an impossible event is B. The probability of a sure

More information

CHAPTER 7 Probability

CHAPTER 7 Probability CHAPTER 7 Probability 7.1. Sets A set is a well-defined collection of distinct objects. Welldefined means that we can determine whether an object is an element of a set or not. Distinct means that we can

More information

FALL 2012 MATH 1324 REVIEW EXAM 4

FALL 2012 MATH 1324 REVIEW EXAM 4 FALL 01 MATH 134 REVIEW EXAM 4 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the sample space for the given experiment. 1) An ordinary die

More information

19.4 Mutually Exclusive and Overlapping Events

19.4 Mutually Exclusive and Overlapping Events Name Class Date 19.4 Mutually Exclusive and Overlapping Events Essential Question: How are probabilities affected when events are mutually exclusive or overlapping? Resource Locker Explore 1 Finding the

More information

Ex 1: A coin is flipped. Heads, you win $1. Tails, you lose $1. What is the expected value of this game?

Ex 1: A coin is flipped. Heads, you win $1. Tails, you lose $1. What is the expected value of this game? AFM Unit 7 Day 5 Notes Expected Value and Fairness Name Date Expected Value: the weighted average of possible values of a random variable, with weights given by their respective theoretical probabilities.

More information

S = {(1, 1), (1, 2),, (6, 6)}

S = {(1, 1), (1, 2),, (6, 6)} Part, MULTIPLE CHOICE, 5 Points Each An experiment consists of rolling a pair of dice and observing the uppermost faces. The sample space for this experiment consists of 6 outcomes listed as pairs of numbers:

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. Mathematical Ideas Chapter 2 Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) In one town, 2% of all voters are Democrats. If two voters

More information

Section 6.5 Conditional Probability

Section 6.5 Conditional Probability Section 6.5 Conditional Probability Example 1: An urn contains 5 green marbles and 7 black marbles. Two marbles are drawn in succession and without replacement from the urn. a) What is the probability

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. Statistics Homework Ch 5 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) A coin is tossed. Find the probability

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. MATH 00 -- PRACTICE EXAM 3 Millersville University, Fall 008 Ron Umble, Instr. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. For the given question,

More information

Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations

Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations Algebra 2 Notes Section 10.1: Apply the Counting Principle and Permutations Objective(s): Vocabulary: I. Fundamental Counting Principle: Two Events: Three or more Events: II. Permutation: (top of p. 684)

More information

Instructions: Choose the best answer and shade in the corresponding letter on the answer sheet provided. Be sure to include your name and student ID.

Instructions: Choose the best answer and shade in the corresponding letter on the answer sheet provided. Be sure to include your name and student ID. Math 3201 Unit 3 Probability Test 1 Unit Test Name: Part 1 Selected Response: Instructions: Choose the best answer and shade in the corresponding letter on the answer sheet provided. Be sure to include

More information

November 6, Chapter 8: Probability: The Mathematics of Chance

November 6, Chapter 8: Probability: The Mathematics of Chance Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern

More information

Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results:

Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results: Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results: Outcome Frequency 1 8 2 8 3 12 4 7 5 15 8 7 8 8 13 9 9 10 12 (a) What is the experimental probability

More information

CHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY

CHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY CHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY Probability is the Probability is used in many real-world fields, such as insurance, medical research, law enforcement, and political science. Objectives:

More information

Stat 20: Intro to Probability and Statistics

Stat 20: Intro to Probability and Statistics Stat 20: Intro to Probability and Statistics Lecture 12: More Probability Tessa L. Childers-Day UC Berkeley 10 July 2014 By the end of this lecture... You will be able to: Use the theory of equally likely

More information

INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2

INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2 INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2 WARM UP Students in a mathematics class pick a card from a standard deck of 52 cards, record the suit, and return the card to the deck. The results

More information

Probability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37

Probability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37 Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete

More information

Section 5.4 Permutations and Combinations

Section 5.4 Permutations and Combinations Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to

More information

Name (Place your name here and on the Scantron form.)

Name (Place your name here and on the Scantron form.) MATH 053 - CALCULUS & STATISTICS/BUSN - CRN 0398 - EXAM # - WEDNESDAY, FEB 09 - DR. BRIDGE Name (Place your name here and on the Scantron form.) MULTIPLE CHOICE. Choose the one alternative that best completes

More information

Section 5.4 Permutations and Combinations

Section 5.4 Permutations and Combinations Section 5.4 Permutations and Combinations Definition: n-factorial For any natural number n, n! = n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to

More information

TEST A CHAPTER 11, PROBABILITY

TEST A CHAPTER 11, PROBABILITY TEST A CHAPTER 11, PROBABILITY 1. Two fair dice are rolled. Find the probability that the sum turning up is 9, given that the first die turns up an even number. 2. Two fair dice are rolled. Find the probability

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) 1 6

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) 1 6 Math 300 Exam 4 Review (Chapter 11) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Give the probability that the spinner shown would land on

More information

Suppose Y is a random variable with probability distribution function f(y). The mathematical expectation, or expected value, E(Y) is defined as:

Suppose Y is a random variable with probability distribution function f(y). The mathematical expectation, or expected value, E(Y) is defined as: Suppose Y is a random variable with probability distribution function f(y). The mathematical expectation, or expected value, E(Y) is defined as: E n ( Y) y f( ) µ i i y i The sum is taken over all values

More information

Module 4 Project Maths Development Team Draft (Version 2)

Module 4 Project Maths Development Team Draft (Version 2) 5 Week Modular Course in Statistics & Probability Strand 1 Module 4 Set Theory and Probability It is often said that the three basic rules of probability are: 1. Draw a picture 2. Draw a picture 3. Draw

More information

If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics

If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics probability that you get neither? Class Notes The Addition Rule (for OR events) and Complements

More information

Permutations: The number of arrangements of n objects taken r at a time is. P (n, r) = n (n 1) (n r + 1) =

Permutations: The number of arrangements of n objects taken r at a time is. P (n, r) = n (n 1) (n r + 1) = Section 6.6: Mixed Counting Problems We have studied a number of counting principles and techniques since the beginning of the course and when we tackle a counting problem, we may have to use one or a

More information

Section 7.2 Definition of Probability

Section 7.2 Definition of Probability Section 7.2 Definition of Probability Question: Suppose we have an experiment that consists of flipping a fair 2-sided coin and observing if the coin lands on heads or tails? From section 7.1 weshouldknowthatthereare

More information

Conditional Probability Worksheet

Conditional Probability Worksheet Conditional Probability Worksheet P( A and B) P(A B) = P( B) Exercises 3-6, compute the conditional probabilities P( AB) and P( B A ) 3. P A = 0.7, P B = 0.4, P A B = 0.25 4. P A = 0.45, P B = 0.8, P A

More information

AP Statistics Ch In-Class Practice (Probability)

AP Statistics Ch In-Class Practice (Probability) AP Statistics Ch 14-15 In-Class Practice (Probability) #1a) A batter who had failed to get a hit in seven consecutive times at bat then hits a game-winning home run. When talking to reporters afterward,

More information

7 5 Compound Events. March 23, Alg2 7.5B Notes on Monday.notebook

7 5 Compound Events. March 23, Alg2 7.5B Notes on Monday.notebook 7 5 Compound Events At a juice bottling factory, quality control technicians randomly select bottles and mark them pass or fail. The manager randomly selects the results of 50 tests and organizes the data

More information

Probability Models. Section 6.2

Probability Models. Section 6.2 Probability Models Section 6.2 The Language of Probability What is random? Empirical means that it is based on observation rather than theorizing. Probability describes what happens in MANY trials. Example

More information

1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?

1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,

More information

MATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG

MATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, Inclusion-Exclusion, and Complement. (a An office building contains 7 floors and has 7 offices

More information

, -the of all of a probability experiment. consists of outcomes. (b) List the elements of the event consisting of a number that is greater than 4.

, -the of all of a probability experiment. consists of outcomes. (b) List the elements of the event consisting of a number that is greater than 4. 4-1 Sample Spaces and Probability as a general concept can be defined as the chance of an event occurring. In addition to being used in games of chance, probability is used in the fields of,, and forecasting,

More information

Fundamentals of Probability

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

More information

Fall (b) Find the event, E, that a number less than 3 is rolled. (c) Find the event, F, that a green marble is selected.

Fall (b) Find the event, E, that a number less than 3 is rolled. (c) Find the event, F, that a green marble is selected. Fall 2018 Math 140 Week-in-Review #6 Exam 2 Review courtesy: Kendra Kilmer (covering Sections 3.1-3.4, 4.1-4.4) (Please note that this review is not all inclusive) 1. An experiment consists of rolling

More information

Discrete Random Variables Day 1

Discrete Random Variables Day 1 Discrete Random Variables Day 1 What is a Random Variable? Every probability problem is equivalent to drawing something from a bag (perhaps more than once) Like Flipping a coin 3 times is equivalent to

More information

Mutually Exclusive Events

Mutually Exclusive Events Mutually Exclusive Events Suppose you are rolling a six-sided die. What is the probability that you roll an odd number and you roll a 2? Can these both occur at the same time? Why or why not? Mutually

More information

n(s)=the number of ways an event can occur, assuming all ways are equally likely to occur. p(e) = n(e) n(s)

n(s)=the number of ways an event can occur, assuming all ways are equally likely to occur. p(e) = n(e) n(s) The following story, taken from the book by Polya, Patterns of Plausible Inference, Vol. II, Princeton Univ. Press, 1954, p.101, is also quoted in the book by Szekely, Classical paradoxes of probability

More information

CSC/MATA67 Tutorial, Week 12

CSC/MATA67 Tutorial, Week 12 CSC/MATA67 Tutorial, Week 12 November 23, 2017 1 More counting problems A class consists of 15 students of whom 5 are prefects. Q: How many committees of 8 can be formed if each consists of a) exactly

More information

EE 126 Fall 2006 Midterm #1 Thursday October 6, 7 8:30pm DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO

EE 126 Fall 2006 Midterm #1 Thursday October 6, 7 8:30pm DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO EE 16 Fall 006 Midterm #1 Thursday October 6, 7 8:30pm DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO You have 90 minutes to complete the quiz. Write your solutions in the exam booklet. We will

More information

More Probability: Poker Hands and some issues in Counting

More Probability: Poker Hands and some issues in Counting More Probability: Poker Hands and some issues in Counting Data From Thursday Everybody flipped a pair of coins and recorded how many times they got two heads, two tails, or one of each. We saw that the

More information

1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 100 calculators is tested.

1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 100 calculators is tested. 1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 0 calculators is tested. Write down the expected number of faulty calculators in the sample. Find

More information

Probability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College

Probability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical

More information

Day 5: Mutually Exclusive and Inclusive Events. Honors Math 2 Unit 6: Probability

Day 5: Mutually Exclusive and Inclusive Events. Honors Math 2 Unit 6: Probability Day 5: Mutually Exclusive and Inclusive Events Honors Math 2 Unit 6: Probability Warm-up on Notebook paper (NOT in notes) 1. A local restaurant is offering taco specials. You can choose 1, 2 or 3 tacos

More information

MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #1 - SPRING DR. DAVID BRIDGE

MATH CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #1 - SPRING DR. DAVID BRIDGE MATH 205 - CALCULUS & STATISTICS/BUSN - PRACTICE EXAM #1 - SPRING 2009 - DR. DAVID BRIDGE TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. Tell whether the statement is

More information

ABC High School, Kathmandu, Nepal. Topic : Probability

ABC High School, Kathmandu, Nepal. Topic : Probability BC High School, athmandu, Nepal Topic : Probability Grade 0 Teacher: Shyam Prasad charya. Objective of the Module: t the end of this lesson, students will be able to define and say formula of. define Mutually

More information

The probability set-up

The probability set-up CHAPTER 2 The probability set-up 2.1. Introduction and basic theory We will have a sample space, denoted S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample

More information

ECON 214 Elements of Statistics for Economists

ECON 214 Elements of Statistics for Economists ECON 214 Elements of Statistics for Economists Session 4 Probability Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education School of Continuing

More information

Contemporary Mathematics Math 1030 Sample Exam I Chapters Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific

Contemporary Mathematics Math 1030 Sample Exam I Chapters Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific Contemporary Mathematics Math 1030 Sample Exam I Chapters 13-15 Time Limit: 90 Minutes No Scratch Paper Calculator Allowed: Scientific Name: The point value of each problem is in the left-hand margin.

More information

Probability. The Bag Model

Probability. The Bag Model Probability The Bag Model Imagine a bag (or box) containing balls of various kinds having various colors for example. Assume that a certain fraction p of these balls are of type A. This means N = total

More information