# Business Statistics. Chapter 4 Using Probability and Probability Distributions QMIS 120. Dr. Mohammad Zainal

Size: px
Start display at page:

Download "Business Statistics. Chapter 4 Using Probability and Probability Distributions QMIS 120. Dr. Mohammad Zainal"

Transcription

1 Department of Quantitative Methods & Information Systems Business Statistics Chapter 4 Using Probability and Probability Distributions QMIS 120 Dr. Mohammad Zainal

2 Chapter Goals After completing this chapter, you should be able to: Explain basic probability concepts and definitions Use a Venn diagram or tree diagram to illustrate simple probabilities Apply common rules of probability Explain three approaches to assessing probabilities Apply common rules of probability, including the Addition Rule and the Multiplication Rule QMIS 120, by Dr. M. Zainal Chap 4-2

3 Chapter Goals (continued) Compute conditional probabilities Determine whether events are statistically independent Use Bayes Theorem for conditional probabilities QMIS 120, by Dr. M. Zainal Chap 4-3

4 Important Terms Random Experiment a process leading to an uncertain outcome Basic Outcome (Simple Event ) a possible outcome of a random experiment Sample Space the collection of all possible outcomes of a random experiment Event (Compound Event) any subset of basic outcomes from the sample space Probability the chance that an uncertain event will occur (always between 0 and 1) QMIS 120, by Dr. M. Zainal Chap 4-4

5 Sample Space The Sample Space is the collection of all possible outcomes e.g., All 6 faces of a die: e.g., All 52 cards of a bridge deck: QMIS 120, by Dr. M. Zainal Chap 4-5

6 Sample Space e.g., All 2 faces of a coin: e.g., All 4 colors in the jar: QMIS 120, by Dr. M. Zainal Chap 4-6

7 Examples of Sample Spaces Rolling one die: S = [1, 2, 3, 4, 5, 6] Tossing one coin: S = [Head, Tail] QMIS 120, by Dr. M. Zainal Chap 4-7

8 Examples of Sample Spaces Choosing one ball: S = [Red, Blue, Green, Yellow] Playing a game: S = [Win, Lose, Tie] QMIS 120, by Dr. M. Zainal Chap 4-8

9 Events Experimental outcome An outcome from a sample space with one characteristic Example: A red card from a deck of cards Event May involve two or more outcomes simultaneously Example: An ace that is also red from a deck of cards QMIS 120, by Dr. M. Zainal Chap 4-9

10 Visualizing Events Contingency Tables Ace Not Ace Total Black Red Total Sample Space Tree Diagrams Full Deck of 52 Cards Sample Space QMIS 120, by Dr. M. Zainal Chap 4-10

11 Experimental Outcomes A automobile consultant records fuel type and vehicle type for a sample of vehicles 2 Fuel types: Gasoline, Diesel 3 Vehicle types: Truck, Car, SUV 6 possible experimental outcomes: e 1 Gasoline, Truck e 2 Gasoline, Car e 3 Gasoline, SUV e 4 Diesel, Truck e 5 Diesel, Car Diesel, SUV e 6 QMIS 120, by Dr. M. Zainal Chap 4-11

12 Important Terms Intersection of Events If A and B are two events in a sample space S, then the intersection, A B, is the set of all outcomes in S that belong to both A and B S A A B B QMIS 120, by Dr. M. Zainal Chap 4-12

13 Important Terms A and B are Mutually Exclusive Events if they have no basic outcomes in common i.e., the set A B is empty If A occurs, then B cannot occur A and B have no common elements (continued) S A B A ball cannot be Green and Red at the same time. QMIS 120, by Dr. M. Zainal Chap 4-13

14 Important Terms (continued) Union of Events If A and B are two events in a sample space S, then the union, A U B, is the set of all outcomes in S that belong to either A or B S The entire shaded area represents A U B A B QMIS 120, by Dr. M. Zainal Chap 4-14

15 Important Terms (continued) Events E 1, E 2, E k are Collectively Exhaustive events if E 1 U E 2 U... U E k = S i.e., the events completely cover the sample space S The Complement of an event A is the set of all basic outcomes in the sample space that do not belong to A. The complement is denoted A A A A = + A A QMIS 120, by Dr. M. Zainal Chap 4-15

16 Examples Rolling a die S = [1, 2, 3, 4, 5, 6] Let A be the event Number rolled is even Let B be the event Number rolled is at least 4 Then: QMIS 120, by Dr. M. Zainal Chap 4-16

17 Examples (continued) Complements: Intersections: Unions: QMIS 120, by Dr. M. Zainal Chap 4-17

18 Examples (continued) Mutually exclusive: Collectively exhaustive: QMIS 120, by Dr. M. Zainal Chap 4-18

19 Probability Probability the chance that an uncertain event will occur (always between 0 and 1) Also, it is a numerical measure of the likelihood that an event will occur 1.5 Certain 0 P(A) 1 For any event A 0 Impossible QMIS 120, by Dr. M. Zainal Chap 4-19

20 Assessing Probability There are three approaches to assessing the probability of an uncertain event: 1. classical probability N probability of event A A N numberof outcomesthat satisfythe event total numberof outcomesinthe samplespace Assumes all outcomes in the sample space are equally likely to occur QMIS 120, by Dr. M. Zainal Chap 4-20

21 Classical Probability Examples Example: Find the probability of obtaining an even number in one roll of a die and find the probability of obtaining a number greater than 4. S = [1, 2, 3, 4, 5, 6] A = [2, 4, 6] C = [5, 6] QMIS 120, by Dr. M. Zainal Chap 4-21

22 Counting the Possible Outcomes Use the Combinations formula to determine the number of distinct combinations of n distinct objects that can be formed, taking them r at a time (No ordering) C n n C () n r r r n! r!( n r)! where n! = n(n-1)(n-2) (1) 0! = 1 by definition QMIS 120, by Dr. M. Zainal Chap 4-22

23 Counting the Possible Outcomes Combinations: If one has 5 different objects (e.g. A, B, C, D, and E), how many ways can they be grouped as 3 objects when position does not matter (e.g. ABC, ABD, ABE, ACD, ACE, ADE are correct but CBA is not ok as is equal to ABC) QMIS 120, by Dr. M. Zainal Chap 4-23

24 Counting the Possible Outcomes Use the Permutations formula to determine the number of ways we can arrange n distinct objects, taking them r at a time (with ordering) where P n! = n(n-1)(n-2) (1) 0! = 1 by definition n n P () n r r r n! ( n r)! QMIS 120, by Dr. M. Zainal Chap 4-24

25 Counting the Possible Outcomes Permutations: Given that position (order) is important, if one has 5 different objects (e.g. A, B, C, D, and E), how many unique ways can they be placed in 3 positions (e.g. ADE, AED, DEA, DAE, EAD, EDA, ABC, ACB, BCA, BAC etc.) QMIS 120, by Dr. M. Zainal Chap 4-25

26 Assessing Probability Three approaches (continued) 2. relative frequency probability n probability of event A A n number of events inthe populationthat satisfyevent total number of events inthe population A the limit of the proportion of times that an event A occurs in a large number of trials, n 3. subjective probability an individual opinion or belief about the probability of occurrence QMIS 120, by Dr. M. Zainal Chap 4-26

27 Relative Frequency Probability Examples Example: In a group of 500 women, 80 have played football at least once in their life. Suppose one of these 500 woman is selected. What is the probability that she played football at least once? Solution: n = 500 and f = 80 Women Did not play football Played football f n = 500 Relative frequency 420 / 500 = / 500 = 0.16 Sum = 1.0 QMIS 120, by Dr. M. Zainal Chap 4-27

28 Probability Postulates 1. If A is any event in the sample space S, then 0 P(A) 2. Let A be an event in S, and let E i denote the basic outcomes. Then A (the notation means that the summation is over all the basic outcomes in A) 1 P(A) P(E i) 3. P(S) = 1 QMIS 120, by Dr. M. Zainal Chap 4-28

29 Probability Rules The Complement rule: P( A) 1 P(A) i.e., P(A) P(A) 1 A A QMIS 120, by Dr. M. Zainal Chap 4-29

30 Joint and Marginal Probabilities The probability of a joint event, A B: P(A B) numberof outcomessatisfyinga andb total number of elementaryoutcomes Computing a marginal probability: P(A) P(A B 1) P(A B2) P(A B k ) Where B 1, B 2,, B k are k mutually exclusive and collectively exhaustive events QMIS 120, by Dr. M. Zainal Chap 4-30

31 Marginal Probability Marginal probability is the probability of a single event without consideration of any other event Suppose one employee is selected out of a sample of 100, he/she maybe classified either on the bases of gender alone or on the bases of a marital status. Married Single Total Male Female Total QMIS 120, by Dr. M. Zainal Chap 4-31

32 Marginal Probability Example The probability of each of the following event is called marginal probability P(male) = 60/100 P(female) = 40/100 P(married) = 19/100 P(single) = 81/100 Male Female Married 15 4 Single Total Total QMIS 120, by Dr. M. Zainal Chap 4-32

33 Conditional Probability This probability is called conditional probability and is written as the probability that the employee selected is married given that he is a male. P(married male ) This event has already occurred The event whose probability is to be determined Read as given QMIS 120, by Dr. M. Zainal Chap 4-33

34 Conditional Probability A conditional probability is the probability of one event, given that another event has occurred: P(A B) P(A B) P(B) The conditional probability of A given that B has occurred P(B A) P(A B) P(A) The conditional probability of B given that A has occurred QMIS 120, by Dr. M. Zainal Chap 4-34

35 Conditional Probability Example Example: Find the conditional probability P(married male) for the data on 100 employees Solution Married Single Total Male Female Total QMIS 120, by Dr. M. Zainal Chap 4-35

36 Conditional Probability Example (continued) QMIS 120, by Dr. M. Zainal Chap 4-36

37 Conditional Probability Example Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. What is the probability that a car has a CD player, given that it has AC? i.e., we want to find P(CD AC) QMIS 120, by Dr. M. Zainal Chap 4-37

38 Conditional Probability Example (continued) Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. AC No AC CD No CD Total Total 1.0 QMIS 120, by Dr. M. Zainal Chap 4-38

39 Using a Tree Diagram Given AC or no AC: All Cars QMIS 120, by Dr. M. Zainal Chap 4-39

40 Conditional Probability Example Example: Consider the experiment of tossing a fair die. Denote by A and B the following events: A={Observing an even number}, B={Observing a number of dots less than or equal to 3}. Find the probability of the event A, given the event B. Solution QMIS 120, by Dr. M. Zainal Chap 4-40

41 Multiplication Rule Multiplication rule for two events A and B: P(A B) P(A B)P(B) also P(A B) P(B A) P(A) QMIS 120, by Dr. M. Zainal Chap 4-41

42 Probability Concepts Independent and Dependent Events Independent: Occurrence of one does not influence the probability of occurrence of the other E 1 = heads on one flip of fair coin E 2 = heads on second flip of same coin Result of second flip does not depend on the result of the first flip. QMIS 120, by Dr. M. Zainal Chap 4-42

43 Probability Concepts Independent and Dependent Events Dependent: Occurrence of one affects the probability of the other E 1 = rain forecasted on the news E 2 = take umbrella to work Probability of the second event is affected by the occurrence of the first event QMIS 120, by Dr. M. Zainal Chap 4-43

44 Statistical Independence Two events are statistically independent if and only if: P(A B) P(A)P(B) Events A and B are independent when the probability of one event is not affected by the other event If A and B are independent, then P(A B) P(A) if P(B)>0 P(B A) P(B) if P(A)>0 QMIS 120, by Dr. M. Zainal Chap 4-44

45 Statistical Independence Example Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD player (CD). 20% of the cars have both. CD No CD Total AC No AC Total Are the events AC and CD statistically independent? QMIS 120, by Dr. M. Zainal Chap 4-45

46 Statistical Independence Example CD No CD Total AC No AC Total (continued) QMIS 120, by Dr. M. Zainal Chap 4-46

47 For Independent Events: Conditional probability for independent events E 1, E 2 : P(E E ) P(E ) where P(E ) 2 0 where P(E 1) 0 2 P(E E ) P(E ) 1 2 QMIS 120, by Dr. M. Zainal Chap 4-47

48 Probability Rules The Addition rule: The probability of the union of two events is P(A B) P(A) P(B) P(A B) A + = B A B P(A or B) = P(A) + P(B) - P(A and B) Don t count common elements twice! QMIS 120, by Dr. M. Zainal Chap 4-48

49 Addition Rule for Mutually Exclusive Events If A and B are mutually exclusive, then P(A B) = 0 A B P(A or B) = P(A) + P(B) - P(A and B) So P(A U B) = P(A) + P(B) QMIS 120, by Dr. M. Zainal Chap 4-49

50 A Probability Table Probabilities and joint probabilities for two events A and B are summarized in this table: B B A P(A B) P(A B) P(A) A P( A B) P( A B) P(A) P(B) P(B) P(S) 1.0 QMIS 120, by Dr. M. Zainal Chap 4-50

51 Law of Total Probability Consider the following Venn diagram B B 1 B 2 B n-1 B n A A 1 A 2 A n-1 A n Can we find the area (probability) of A assuming that we know the probability of each Bi & P(A Bi)? QMIS 120, by Dr. M. Zainal Chap 4-51

52 Bayes Theorem P(B A) i P(A B ) P(A) i P(A B )P(B ) i P(A B )P(B ) P(A B )P(B ) i where: B i = i th event of k mutually exclusive and collectively exhaustive events A = new event that might impact P(B i ) QMIS 120, by Dr. M. Zainal Chap 4-52

53 Bayes Theorem Example A drilling company has estimated a 40% chance of striking oil for their new well. A detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have had detailed tests. Given that this well has been scheduled for a detailed test, what is the probability that the well will be successful? QMIS 120, by Dr. M. Zainal Chap 4-53

54 Bayes Theorem Example (continued) Let S = successful well U = unsuccessful well P(S) =, P(U) = (prior probabilities) Define the detailed test event as D Conditional probabilities: Goal is to find QMIS 120, by Dr. M. Zainal Chap 4-54

55 Bayes Theorem Example Apply Bayes Theorem: (continued) QMIS 120, by Dr. M. Zainal Chap 4-55

56 Bayes Theorem Example (continued) Given the detailed test, the revised probability of a successful well has from the original estimate of QMIS 120, by Dr. M. Zainal Chap 4-56

57 Complete Example Consider a standard deck of card that we see normally being played on computer or table. It consists of 52 cards with 4 suits and so 13 cards in a suit. Consider a standard deck of 52 cards, with four suits: Spades and Clubs are traditionally in black color, while Hearts and Diamonds are in red. Spade (S): Heart (H): Club (C): Diamond (D): Let event A = card is an Ace Let event B = card is from a Red suit QMIS 120, by Dr. M. Zainal Chap 4-57

58 Complete Example What is the probability of Red given it was Ace (Cond.) Type Color Red Black Total Ace Non-Ace Total QMIS 120, by Dr. M. Zainal Chap 4-58

59 Complete Example What is the probability of Red and Ace (Intersection) Type Color Red Black Total Ace Non-Ace Total QMIS 120, by Dr. M. Zainal Chap 4-59

60 Complete Example What is the probability of Red or Ace (Union) Type Color Red Black Total Ace Non-Ace Total QMIS 120, by Dr. M. Zainal Chap 4-60

61 Complete Example What is the probability of Ace (Marginal) Type Color Red Black Total Ace Non-Ace Total QMIS 120, by Dr. M. Zainal Chap 4-61

62 Complete Example Are the events Red and Ace statistically independent? Type Color Red Black Total Ace Non-Ace Total QMIS 120, by Dr. M. Zainal Chap 4-62

63 Complete Example What is the complement of the event Ace? Type Color Red Black Total Ace Non-Ace Total QMIS 120, by Dr. M. Zainal Chap 4-63

64 Bayes Theorem Example Example: Manufacturing firm that receives shipment of parts from two different suppliers. Currently, 65 percent of the parts purchased by the company are from supplier 1 and the remaining 35 percent are from supplier 2. Historical Data suggest the quality rating of the two supplier are shown in the table: Good Parts Bad Parts Supplier Supplier a) Draw a tree diagram for this experiment with the probability of all outcomes b) Given the information the part is bad, What is the probability the part came from supplier 1? QMIS 120, by Dr. M. Zainal Chap 4-64

65 Bayes Theorem Example QMIS 120, by Dr. M. Zainal Chap 4-65

66 Bayes Theorem Example (continued) Apply Bayes Theorem: QMIS 120, by Dr. M. Zainal Chap 4-66

67 Bayes Theorem Example Example: An insurance company rents 35% of the cars for its customers from Avis and the rest from Hertz. From past records they know that 8% of Avis cars break down and 5% of Hertz cars break down. A customer calls and complains that his rental car broke down. What is the probability that his car was rented from Avis? QMIS 120, by Dr. M. Zainal Chap 4-67

68 Bayes Theorem Example QMIS 120, by Dr. M. Zainal Chap 4-68

69 Chapter Summary Defined basic probability concepts Sample spaces and events, intersection and union of events, mutually exclusive and collectively exhaustive events, complements Examined basic probability rules Complement rule, addition rule, multiplication rule Defined conditional, joint, and marginal probabilities Reviewed odds and the overinvolvement ratio Defined statistical independence Discussed Bayes theorem QMIS 120, by Dr. M. Zainal Chap 4-69

70 Copyright The materials of this presentation were mostly taken from the PowerPoint files accompanied Business Statistics: A Decision-Making Approach, 7e 2008 Prentice-Hall, Inc. QMIS 120, by Dr. M. Zainal Chap 4-70

### Introduction to Probability and Statistics I Lecture 7 and 8

Introduction to Probability and Statistics I Lecture 7 and 8 Basic Probability and Counting Methods Computing theoretical probabilities:counting methods Great for gambling! Fun to compute! If outcomes

### Chapter 4 Student Lecture Notes 4-1

Chapter 4 Student Lecture Notes 4-1 Basic Business Statistics (9 th Edition) Chapter 4 Basic Probability 2004 Prentice-Hall, Inc. Chap 4-1 Chapter Topics Basic Probability Concepts Sample spaces and events,

### Textbook: pp Chapter 2: Probability Concepts and Applications

1 Textbook: pp. 39-80 Chapter 2: Probability Concepts and Applications 2 Learning Objectives After completing this chapter, students will be able to: Understand the basic foundations of probability analysis.

### Contents 2.1 Basic Concepts of Probability Methods of Assigning Probabilities Principle of Counting - Permutation and Combination 39

CHAPTER 2 PROBABILITY Contents 2.1 Basic Concepts of Probability 38 2.2 Probability of an Event 39 2.3 Methods of Assigning Probabilities 39 2.4 Principle of Counting - Permutation and Combination 39 2.5

### Probability Concepts and Counting Rules

Probability Concepts and Counting Rules Chapter 4 McGraw-Hill/Irwin Dr. Ateq Ahmed Al-Ghamedi Department of Statistics P O Box 80203 King Abdulaziz University Jeddah 21589, Saudi Arabia ateq@kau.edu.sa

### 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,

### Probability. Ms. Weinstein Probability & Statistics

Probability Ms. Weinstein Probability & Statistics Definitions Sample Space The sample space, S, of a random phenomenon is the set of all possible outcomes. Event An event is a set of outcomes of a random

### 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

### CHAPTER 2 PROBABILITY. 2.1 Sample Space. 2.2 Events

CHAPTER 2 PROBABILITY 2.1 Sample Space A probability model consists of the sample space and the way to assign probabilities. Sample space & sample point The sample space S, is the set of all possible outcomes

### 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

### Probability. Engr. Jeffrey T. Dellosa.

Probability Engr. Jeffrey T. Dellosa Email: jtdellosa@gmail.com Outline Probability 2.1 Sample Space 2.2 Events 2.3 Counting Sample Points 2.4 Probability of an Event 2.5 Additive Rules 2.6 Conditional

### 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

### CHAPTER 8 Additional Probability Topics

CHAPTER 8 Additional Probability Topics 8.1. Conditional Probability Conditional probability arises in probability experiments when the person performing the experiment is given some extra 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 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical

### 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

### 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

### Chapter 5 - Elementary Probability Theory

Chapter 5 - Elementary Probability Theory Historical Background Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling

### Chapter 6: Probability and Simulation. The study of randomness

Chapter 6: Probability and Simulation The study of randomness 6.1 Randomness Probability describes the pattern of chance outcomes. Probability is the basis of inference Meaning, the pattern of chance outcomes

### , -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,

### Probability and Randomness. Day 1

Probability and Randomness Day 1 Randomness and Probability The mathematics of chance is called. The probability of any outcome of a chance process is a number between that describes the proportion of

### Probability - Chapter 4

Probability - Chapter 4 In this chapter, you will learn about probability its meaning, how it is computed, and how to evaluate it in terms of the likelihood of an event actually happening. A cynical person

### Statistics Intermediate Probability

Session 6 oscardavid.barrerarodriguez@sciencespo.fr April 3, 2018 and Sampling from a Population Outline 1 The Monty Hall Paradox Some Concepts: Event Algebra Axioms and Things About that are True Counting

### 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

### 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

### 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:

### 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

### Chapter 5: Probability: What are the Chances? Section 5.2 Probability Rules

+ Chapter 5: Probability: What are the Chances? Section 5.2 + Two-Way Tables and Probability When finding probabilities involving two events, a two-way table can display the sample space in a way that

### 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

### Week 3 Classical Probability, Part I

Week 3 Classical Probability, Part I Week 3 Objectives Proper understanding of common statistical practices such as confidence intervals and hypothesis testing requires some familiarity with probability

### 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

### 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.

### Probability. Dr. Zhang Fordham Univ.

Probability! Dr. Zhang Fordham Univ. 1 Probability: outline Introduction! Experiment, event, sample space! Probability of events! Calculate Probability! Through counting! Sum rule and general sum rule!

### 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

### Grade 6 Math Circles Fall Oct 14/15 Probability

1 Faculty of Mathematics Waterloo, Ontario Centre for Education in Mathematics and Computing Grade 6 Math Circles Fall 2014 - Oct 14/15 Probability Probability is the likelihood of an event occurring.

### 2. The value of the middle term in a ranked data set is called: A) the mean B) the standard deviation C) the mode D) the median

1. An outlier is a value that is: A) very small or very large relative to the majority of the values in a data set B) either 100 units smaller or 100 units larger relative to the majority of the values

### Chapter 6: Probability and Simulation. The study of randomness

Chapter 6: Probability and Simulation The study of randomness Introduction Probability is the study of chance. 6.1 focuses on simulation since actual observations are often not feasible. When we produce

### Probability (Devore Chapter Two)

Probability (Devore Chapter Two) 1016-351-01 Probability Winter 2011-2012 Contents 1 Axiomatic Probability 2 1.1 Outcomes and Events............................... 2 1.2 Rules of 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.

### Review Questions on Ch4 and Ch5

Review Questions on Ch4 and Ch5 1. Find the mean of the distribution shown. x 1 2 P(x) 0.40 0.60 A) 1.60 B) 0.87 C) 1.33 D) 1.09 2. A married couple has three children, find the probability they are all

### 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,

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

Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even

### Probability is the likelihood that an event will occur.

Section 3.1 Basic Concepts of is the likelihood that an event will occur. In Chapters 3 and 4, we will discuss basic concepts of probability and find the probability of a given event occurring. Our main

### 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:

### 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

### 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

### 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

### 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

### Probability and Counting Techniques

Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each

### 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

### STOR 155 Introductory Statistics. Lecture 10: Randomness and Probability Model

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 10: Randomness and Probability Model 10/6/09 Lecture 10 1 The Monty Hall Problem Let s Make A Deal: a game show

### "Well, statistically speaking, you are for more likely to have an accident at an intersection, so I just make sure that I spend less time there.

6.2 Probability Models There was a statistician who, when driving his car, would always accelerate hard before coming to an intersection, whiz straight through it, and slow down again once he was beyond

### Empirical (or statistical) probability) is based on. The empirical probability of an event E is the frequency of event E.

Probability and Statistics Chapter 3 Notes Section 3-1 I. Probability Experiments. A. When weather forecasters say There is a 90% chance of rain tomorrow, or a doctor says There is a 35% chance of a successful

### 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

### Raise your hand if you rode a bus within the past month. Record the number of raised hands.

166 CHAPTER 3 PROBABILITY TOPICS Raise your hand if you rode a bus within the past month. Record the number of raised hands. Raise your hand if you answered "yes" to BOTH of the first two questions. Record

### Unit 1 Day 1: Sample Spaces and Subsets. Define: Sample Space. Define: Intersection of two sets (A B) Define: Union of two sets (A B)

Unit 1 Day 1: Sample Spaces and Subsets Students will be able to (SWBAT) describe events as subsets of sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions,

### CH 13. Probability and Data Analysis

11.1: Find Probabilities and Odds 11.2: Find Probabilities Using Permutations 11.3: Find Probabilities Using Combinations 11.4: Find Probabilities of Compound Events 11.5: Analyze Surveys and Samples 11.6:

### 3 The multiplication rule/miscellaneous counting problems

Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,

### Name: Class: Date: 6. An event occurs, on average, every 6 out of 17 times during a simulation. The experimental probability of this event is 11

Class: Date: Sample Mastery # Multiple Choice Identify the choice that best completes the statement or answers the question.. One repetition of an experiment is known as a(n) random variable expected value

### STAT 155 Introductory Statistics. Lecture 11: Randomness and Probability Model

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 11: Randomness and Probability Model 10/5/06 Lecture 11 1 The Monty Hall Problem Let s Make A Deal: a game show

### 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,

### 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,

### 12 Probability. Introduction Randomness

2 Probability Assessment statements 5.2 Concepts of trial, outcome, equally likely outcomes, sample space (U) and event. The probability of an event A as P(A) 5 n(a)/n(u ). The complementary events as

### RANDOM EXPERIMENTS AND EVENTS

Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In day-to-day life we see that before commencement of a cricket match two captains go for a toss. Tossing of a coin is an activity and getting

### Chapter 3: PROBABILITY

Chapter 3 Math 3201 1 3.1 Exploring Probability: P(event) = Chapter 3: PROBABILITY number of outcomes favourable to the event total number of outcomes in the sample space An event is any collection of

### I. WHAT IS PROBABILITY?

C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and

### 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

### Applied Statistics I

Applied Statistics I Liang Zhang Department of Mathematics, University of Utah June 12, 2008 Liang Zhang (UofU) Applied Statistics I June 12, 2008 1 / 29 In Probability, our main focus is to determine

### Introduction to probability

Introduction to probability Suppose an experiment has a finite set X = {x 1,x 2,...,x n } of n possible outcomes. Each time the experiment is performed exactly one on the n outcomes happens. Assign each

### Lecture 6 Probability

Lecture 6 Probability Example: When you toss a coin, there are only two possible outcomes, heads and tails. What if we toss a coin two times? Figure below shows the results of tossing a coin 5000 times

### 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.

### 4.3 Rules of Probability

4.3 Rules of Probability If a probability distribution is not uniform, to find the probability of a given event, add up the probabilities of all the individual outcomes that make up the event. Example:

### 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

### Before giving a formal definition of probability, we explain some terms related to probability.

probability 22 INTRODUCTION In our day-to-day life, we come across statements such as: (i) It may rain today. (ii) Probably Rajesh will top his class. (iii) I doubt she will pass the test. (iv) It is unlikely

### A Probability Work Sheet

A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair six-sided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we

### 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

### 1. How to identify the sample space of a probability experiment and how to identify simple events

Statistics Chapter 3 Name: 3.1 Basic Concepts of Probability Learning objectives: 1. How to identify the sample space of a probability experiment and how to identify simple events 2. How to use the Fundamental

### 5 Elementary Probability Theory

5 Elementary Probability Theory 5.1 What is Probability? The Basics We begin by defining some terms. Random Experiment: any activity with a random (unpredictable) result that can be measured. Trial: one

### Section : Combinations and Permutations

Section 11.1-11.2: Combinations and Permutations Diana Pell A construction crew has three members. A team of two must be chosen for a particular job. In how many ways can the team be chosen? How many words

### Applications of Probability

Applications of Probability CK-12 Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive

### Probability as a general concept can be defined as the chance of an event occurring.

3. Probability In this chapter, you will learn about probability its meaning, how it is computed, and how to evaluate it in terms of the likelihood of an event actually happening. Probability as a general

### 3 The multiplication rule/miscellaneous counting problems

Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1 Suppose P (A 0, P (B 05 (a If A and B are independent, what is P (A B? What is P (A B? (b If A and B are disjoint, what is

### 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

### Independent and Mutually Exclusive Events

Independent and Mutually Exclusive Events By: OpenStaxCollege Independent and mutually exclusive do not mean the same thing. Independent Events Two events are independent if the following are true: P(A

### Probability and Counting Rules. Chapter 3

Probability and Counting Rules Chapter 3 Probability as a general concept can be defined as the chance of an event occurring. Many people are familiar with probability from observing or playing games of

### 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

### Chapter 2. Permutations and Combinations

2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find

### 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:

### 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)

### Chapter 5 Probability

Chapter 5 Probability Math150 What s the likelihood of something occurring? Can we answer questions about probabilities using data or experiments? For instance: 1) If my parking meter expires, I will probably

### Probability - Grade 10 *

OpenStax-CNX module: m32623 1 Probability - Grade 10 * Rory Adams Free High School Science Texts Project Sarah Blyth Heather Williams This work is produced by OpenStax-CNX and licensed under the Creative

### Bell Work. Warm-Up Exercises. Two six-sided dice are rolled. Find the probability of each sum or 7

Warm-Up Exercises Two six-sided dice are rolled. Find the probability of each sum. 1. 7 Bell Work 2. 5 or 7 3. You toss a coin 3 times. What is the probability of getting 3 heads? Warm-Up Notes Exercises

### 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.

### 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

### Class XII Chapter 13 Probability Maths. Exercise 13.1

Exercise 13.1 Question 1: Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E F) = 0.2, find P (E F) and P(F E). It is given that P(E) = 0.6, P(F) = 0.3, and P(E F) = 0.2 Question 2:

### (a) Suppose you flip a coin and roll a die. Are the events obtain a head and roll a 5 dependent or independent events?

Unit 6 Probability Name: Date: Hour: Multiplication Rule of Probability By the end of this lesson, you will be able to Understand Independence Use the Multiplication Rule for independent events Independent

### 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

### Exam 2 Review F09 O Brien. Finite Mathematics Exam 2 Review

Finite Mathematics Exam Review Approximately 5 0% of the questions on Exam will come from Chapters, 4, and 5. The remaining 70 75% will come from Chapter 7. To help you prepare for the first part of the