Math 227 Elementary Statistics. Bluman 5 th edition


 Curtis Todd
 3 years ago
 Views:
Transcription
1 Math 227 Elementary Statistics Bluman 5 th edition
2 CHAPTER 4 Probability and Counting Rules 2
3 Objectives Determine sample spaces and find the probability of an event using classical probability or empirical probability. Find the probability of compound events using the addition rules. Find the probability of compound events using the multiplication rules. Find the conditional probability of an event. 3
4 Objectives (cont.) Determine the number of outcomes of a sequence of events using a tree diagram. Find the total number of outcomes in a sequence of events using the fundamental counting rule. Find the number of ways r objects can be selected from n objects using the permutation rule. Find the number of ways r objects can be selected from n objects without regard to order using the combination rule. Find the probability of an event using the counting rules. 4
5 Introduction 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 insurance, investments, and weather forecasting, and in various areas. Rules such as the fundamental counting rule, combination rule and permutation rules allow us to count the number of ways in which events can occur. Counting rules and probability rules can be used together to solve a wide variety of problems. 5
6 Section 4.1 Sample Spaces and Probability I. Basic Concepts A probability experiment is a chance process that leads to welldefined results called outcomes. An outcome is the result of a single trial of a probability experiment. A sample space is the set of all possible outcomes of a probability experiment. 6
7 Sample Spaces and Events Examples of some sample space: Experiment Sample Space Toss one coin H(head), T(tail) Roll a die 1, 2, 3, 4, 5, 6 Answer a true/false question True, false Toss two coins HH, TT, HT, TH 7
8 Sample Spaces and Events An event consists of a set of outcomes of a probability experiment. Simple event  an event with one outcome Compound event  an event with two or more outcomes. 8
9 Basic Concepts (cont.) Equally likely events are events that have the same probability of occurring. A tree diagram is a device consisting of line segments emanating from a starting point and also from the outcome point. It is used to determine all possible outcomes of a probability experiment. 9
10 Example: A tree diagram to find the sample space for tossing two coins. H T H T H T Outcomes H H H T T H T T 10
11 Example 1 : A die is tossed one time. (a) List the elements of the sample space S. S = {1, 2, 3, 4, 5, 6} (b) List the elements of the event consisting of a number that is greater than 4. Event A: x > 4 A = {5, 6} 11
12 Example 2 : A coin is tossed twice. List the elements of the sample space S, and list the elements of the event consisting of at least one head. H T H T H T S = {HH, HT, TH, TT} Event A : At least one head (H) A = {HT, TH, HH} 12
13 Example 3 : A randomly selected citizen is interviewed and the following information is recorded: employment status and level of education. The symbols for employment status are Y = employed and N = unemployed, and the symbols for level of education are 1 = did not complete high school, 2 = completed high school but did not complete college, and 3 = completed college. List the elements of sample space, and list the elements of the following events. Y N S = {Y1, Y2, Y3, N1, N2, N3} (a) Did not complete high school (b) Is unemployed {Y1, N1} {N1, N2, N3} 13
14 II. Calculating Probabilities A. Empirical Probability (Relative Frequency Approximation of Probability) Empirical probability relies on actual experience to determine the likelihood of outcomes. Given a frequency distribution, the probability of an event being in a given class is: frequency for the class P( E) total frequencies in the distribution f n 14
15 Example 1 : The age distribution of employees for this college is shown below: Age # of employees Under and over 10 n = 130 If an employee is selected at random, find the probability that he or she is in the following age groups. (a) Between 30 and 39 years of age (b) Under 20 or over 49 years of age 15
16 Example 2 : During a sale at men s store, 16 white sweaters, 3 red sweaters, 9 blue sweaters, and 7 yellow sweaters were purchased. If a customer is selected at random, find the probability that he bought a sweater that was not white. 16
17 B. Classical Probability Classical probability uses sample spaces to determine the numerical probability that an event will happen. Classical probability assumes that all outcomes in the sample space are equally likely to occur. PE ( ) ne ( ) Number of outcomes in E ns ( ) total number of outcomes in the sample space 17
18 Example 1 : A statistics class contains 14 males and 20 females. A student is to be selected by chance and the gender of the student recorded. (Ref: General Statistics by Chase/Bown, 4 th Ed.) (a) Give a sample space S for the experiment. S = {M, F} (b) Is each outcome equally likely? Explain. P(M) P(F). It is not equally likely because we have more females than males. (c) Assign probabilities to each outcome. 18
19 Example 2 : Two dice are tossed. Find the probability that the sum of two dice is greater than 8? S = { (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) } 19
20 Example set of 52 playing cards; 13 of each suit clubs, diamonds, hearts, and spades 20
21 Example 3 : If one card is drawn from a deck, find the probability of getting (a) a club 52 cards in a deck, 13 of those are clubs. (b) a 4 and a club 21
22 Example 4 : Three equally qualified runners, Mark (M), Bill (B) and Alan (A), run a 100meter, sprint, and the order of finish is recorded. (a) Give a sample space S. B A M A B B M A A M S = {MBA, MAB, BMA, BAM, AMB, ABM} M B A B M (b) What is the probability that Mark will finish last? {BAM, ABM} 22
23 III. Rounding Rule for Probabilities Probabilities should be expressed as reduced fractions or rounded to two or three decimal places. 23
24 Probability Rules 1. The probability of an event E is a number (either a fraction or decimal) between and including 0 and 1. This is denoted by: 0 P( E) 1 *Rule 1 states that probabilities cannot be negative or greater than one. 24
25 Probability Rules (cont.) 2. If an event E cannot occur (i.e., the event contains no members in the sample space), the probability is zero. 3. If an event E is certain, then the probability of E is The sum of the probabilities of the outcomes in the sample space is 1. 25
26 Example 1: A probability experiment is conducted. Which of these can be considered a probability of an outcome? a) 2/5 Yes b) No c) 1.09 No 26
27 Example 2 : Given : S = {E 1, E 2, E 3, E 4 } Find : P(E 4 ) P(E 1 ) = P(E 2 ) = 0.2 and P(E 3 ) = 0.5 P(E 1 ) + P(E 2 ) + P(E 3 ) +P(E 4 ) = P(E 4 ) = 1 P(E 4 ) = P(E 4 ) =
28 IV. Complementary Events The complement of an event E is the set of outcomes in the sample space that are not included in the outcomes of event E. The complement of E is denoted by. Rule for Complementary Events P( E ) 1 P( E), OR P( E) P( E) 1 E P( E) 1 P( E) 28
29 Example 1 : The chance of raining tomorrow is 70%. What is the probability that it will not rain tomorrow? P (No Rain) = = % chance it will not rain tomorrow. 29
30 Section 4.2 The Addition Rules for Probability Mutually Exclusive Events Two events are mutually exclusive if they cannot occur at the same time (i.e., they have no outcomes in common). The probability of two or more events can be determined by the addition rules. 30
31 I. Addition Rules Addition Rule 1 When two events A and B are mutually exclusive, the probability that A or B will occur is: P( A or B) P( A) P( B) 31
32 Ex1) A single card is drawn from a deck. Find the probability of selecting a club or a diamond. P (club or diamond) mutually exclusive P (club or diamond) = P (club) + P (diamond)
33 Ex2) In a large department store, there are 2 managers, 4 department heads, 16 clerks, and 4 stock persons. If a person is selected at random, find the probability that the person is either a clerk or a manager. P (clerk or manager) mutually exclusive P (clerk or manager) = P (clerk) + P (manager)
34 I. Addition Rules (cont.) Addition Rule 2 If A and B are not mutually exclusive, then: P( A or B) P( A) P( B) P( A and B) 34
35 Example1 : A single card is drawn from a deck. Find the probability of selecting a jack or a black card. P (Jack or Black) = P (Jack) + P (Black) P (Jack and Black) 35
36 Example 2 : In a certain geographic region, newspapers are classified as being published daily morning, daily evening, and weekly. Some have a comics section and other do not. The distribution is shown here. Having comics Section Morning Evening Weekly Yes No If a newspaper is selected at random, find these probabilities. (a) The newspaper is weekly publication. S = = 15 (b) The newspaper is a daily morning publication or has comics. P (Daily morning or has comics) = P (Daily morning) + P (Has comics) P (Daily morning and has comics) 36
37 (c) The newspaper is published weekly or does not have comics. P (Weekly or No comics) = P (Weekly) + P (No comics) P (Weekly and no comics) 37
38 Section 4.3 The Multiplication Rules and Conditional Probability Independent Events Two events A and B are independent if the fact that A occurs does not affect the probability of B occurring. When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed, the events are said to be dependent. 38
39 I. Multiplication Rules The multiplication rules can be used to find the probability of two or more events that occur in sequence. Multiplication Rule 1 When two events are independent, the probability of both occurring is: P( A and B) P( A) P( B) 39
40 Example 1 : If 36% of college students are overweight, find the probability that if three college students are selected at random, all will be overweight. P (1st overweight and 2nd overweight and 3rd overweight) = =
41 Example 2 : If 25% of U.S. federal prison inmates are not U.S. citizens, find the probability that two randomly selected federal prison inmates will be U.S. citizens. P (1 st U.S. citizen and 2 nd U.S. citizen) = Example 3 : = Suppose the probability of remaining with a particular company 10 years or longer is 1/6. A man and a woman start work at the company on the same day. (a) What is the probability that the man will work there less than 10 years? (b) What is the probability that both the man and woman will work there less than 10 years? 41
42 Example 4 : A smokedetector system uses two devices, A and B. If smoke is present, the probability that it will be detected by device A is 0.95; by device B, 0.98; and by both devices, (a) If smoke is present, find the probability that the smoke will be detected by device A or device B or by both devices. P ( A or B) = P (A ) + P (B) P (A and B) = = 0.99 (b) Find the probability that the smoke will not be detected. P (Neither ) = =
43 Example 5 : If you make random guesses for four multiplechoice test questions (each with five possible answers), what is the probability of getting at least one correct? P (All wrong) = P (1 st wrong & 2 nd wrong & 3 rd wrong & 4 th wrong ) P (At least 1 correct) = 1 P (All wrong) 43
44 Example 6 : There are 2000 voters in a town. Consider the experiment of randomly selecting a voter to be interviewed. The event A consists of being in favor of more stringent building codes; the event B consists of having lived in the town less than 10 years. The following table gives the number of voters in various categories. (Ref: General statistics by Chase/Bown, 4 th Ed.) Favor more Stringent codes Do not favor more stringent codes Less than 10 years At least 10 years Find each of the following: (a) (b) (c) 44
45 Multiplication Rules (cont.) Multiplication Rule 2 When two events are dependent (e.g. the outcome of event A affects the outcomes of event B), the probability of both occurring is: P( A and B) P( A) P( B A) 45
46 Example 1 : A box has 5 red balls and 2 white balls. If two balls are randomly selected (one after the other), what is the probability that they both are red? (a) With replacement independent case (b) Without replacement dependent case 46
47 Example 2 : Three cards are drawn from a deck without replacement. Find the probability that all are Jacks. 47
48 Multiplication Rules (cont.) The conditional probability of an event B in relationship to an event A is the probability that event B occurs after event A has already occurred. The notation for conditional probability is P(B A). 48
49 Formula for Conditional Probability The probability that the second event B occurs given that the first event A has occurred can be found dividing the probability that both events occurred by the probability that the first event has occurred. The formula is: P( B A) P( A and B) P( A) 49
50 Example 1: Two fair dice are tossed. Consider the following events. A = sum is 7 or more, B = sum is even, and C = a match (both numbers are the same). A = { (1,6) (2,5) (2,6) (3,4) (3,5) (3, 6) (4, 3) (4,4) (4,5) (4,6) (5,2) (5,3) (5, 4) (5, 5) (5,6) (6,1) (6,2) (6,3) (6,4) (6, 5) (6, 6) } B = { (1,1) (1,3) (1,5) (2,2) (2,4) (2,6) (3,1) (3,3) (3,5) (4,2) (4,4) (4,6) (5,1) (5,3) (5,5) (6,2) (6,4) (6,6) } C = { (1,1) (2,2) (3,3) (4,4) (5,5) (6,6) } 50
51 (a) P (A and B) = A = { (1,6) (2,5) (2,6) (3,4) (3,5) (3, 6) (4, 3) (4,4) (4,5) (4,6) (5,2) (5,3) (5, 4) (5, 5) (5,6) (6,1) (6,2) (6,3) (6,4) (6, 5) (6, 6) } B = { (1,1) (1,3) (1,5) (2,2) (2,4) (2,6) (3,1) (3,3) (3,5) (4,2) (4,4) (4,6) (5,1) (5,3) (5,5) (6,2) (6,4) (6,6) } (b) P (A or B) = P (A) + P (B) P (A and B) 51
52 (c) P (A B) = (d) P (B C) = Example 2 : At a local Country Club, 65% of the members play bridge and swim, and 72% play bridge. If a member is selected at random, find the probability that the member swims, given that the member plays bridge. 52
53 Example 3 : Eighty students in a school cafeteria were asked if they favored a ban on smoking in the cafeteria. The results of the survey are shown in the table. Class Favor Oppose No opinion Freshman Sophomore If a student is selected at random, find these probabilities. (a) The student is a freshman or favors the ban. P (Freshman or Favor ban) = P (Freshman) + P (Favor ban) P (Freshman and Favor ban) (b) Given that the student favors the ban, the student is a sophomore. 53
54 Section 4.4 Counting Rules I. The Fundamental Counting Rule In a sequence of n events in which the first one has k 1 possibilities and the second event has k 2 and the third has k 3, and so forth, the total number of possibilities of the sequence will be: k 1 k 2 k 3 k n Note: And in this case means to multiply. 54
55 Example 1 : Two dice are tossed. How many outcomes are in S. How many outcomes for Task 1? How many outcome for Task 2? 6 6 = 36 outcomes Example 2 : A password consists of two letters followed by one digit. How many different passwords can be created? (Note: Repetitions are allowed) 1st letter 26 outcomes (A Z) 2 nd letter 26 outcomes (A Z) One digit 10 outcomes (09) = 6,760 ways 55
56 Example 3 : Suppose four digits are to be randomly selected (with repetitions allowed). (Note: the set of digits is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.) If the first digit must be a 2 and repetitions are permitted, how many different 4digit can be made? 1 st digit has to be a 2 1 outcome 2 nd digit any number (0 9) 10 outcomes 3 rd digit any number (0 9) 10 outcomes 4 th digit any number (0 9) 10 outcomes = 1,000 ways 56
57 II. Permutations and Combinations Factorial Notation 5! = ! = In general, n! = n (n1) (n2) (n3) 1 0!=1 57
58 Permutation A permutation is an arrangement of n objects in a specific order. The arrangement of n objects in a specific order using r objects at a time is called a permutation of n objects taking r objects at a time. Note: Order does matter. It is written as n P r, and the formula is: n Pr n! ( n r)! 58
59 Combinations A selection of distinct objects without regard to order is called a combination. (Order does NOT matter!) The number of combinations of r objects selected from n objects is denoted n C r and is given by the formula: n C r n! ( n r)! r! Note: Combinations are always less than permutations for the same n and r. 59
60 Example 1 : (a) 5! = 120 (b) 25P8 (c) 12P4 (d) 25C8 60
61 Example 2 : A television news director wishes to use three news stories on an evening show. One story will be the lead story, one will be the second story, and the last will be a closing story. If the director has a total of eight stories to choose from, how many possible ways can the program be set up? Example 3 : If a person can select 3 presents from 10 presents, how many different combinations are there? 61
62 Example 4 : Your family vacation involves a crosscountry air flight, a rental car, and a hotel stay in Boston. If you can choose from four major air carriers, five car rental agencies, and three major hotel chains, how many options are available for your vacation accommodations? Multiplication Rule = 60 ways 62
63 Section 4.5 Probability and Counting Rules I. Basic Concepts By using the fundamental counting rule, the permutation rules, and the combination rule, the probability of outcomes of many experiments can be computed. 63
64 Example 1 : A combination lock consists of 26 letters of the alphabet. If a threeletter combination is needed, find the probability that the combination will consist of the first two letters AB in that order. The same letter can be used more than once. 64
65 Example 2 : Five cards are selected from a 52card deck for a poker hand. (A poker hand consists of 5 cards dealt in any order.) (a) How many outcomes are in the sample space? (b) A royal flush is a hand that contains that A, K, Q, J, 10, all in the same suit. How many ways are there to get a royal flush? A, K, Q, J, 10  A, K, Q, J, 10  A, K, Q, J, 10  A, K, Q, J, 10  n(a) = 4 combinations (c) What is the probability of being dealt a royal flush? 65
66 Example 452: Committee Selection A store has 6 TV Graphic magazines and 8 Newstime magazines on the counter. If two customers purchased a magazine, find the probability that one of each magazine was purchased. TV Graphic: One magazine of the 6 magazines Newstime: One magazine of the 8 magazines Total: Two magazines of the 14 magazines C C C Bluman, Chapter 4 66
67 Summary The three types of probability are classical, empirical, and subjective. Classical probability uses sample spaces. Empirical probability uses frequency distributions and is based on observations. In subjective probability, the researcher makes an educated guess about the chance of an event occurring. 67
68 Summary (cont) An event consists of one or more outcomes of a probability experiment. Two events are said to be mutually exclusive if they cannot occur at the same time. Events can also be classified as independent or dependent. If events are independent, whether or not the first event occurs does not affect the probability of the next event occurring. 68
69 Summary (cont.) If the probability of the second event occurring is changed by the occurrence of the first event, then the events are dependent. The complement of an event is the set of outcomes in the sample space that are not included in the outcomes of the event itself. Complementary events are mutually exclusive. 69
70 Summary The number of ways a sequence of n events can occur; if the first event can occur in k 1 ways, the second event can occur in k 2 ways, etc. (Multiplication rule) k 1 k 2 k 3 k n The arrangement of n objects in a specific order using r objects at a time (Permutation rule) n! n Pr ( n r)! The number of combinations of r objects selected from n objects (order is not important) (Combination rule) n C r n! ( n r)! r! 70
71 Conclusions Probability can be defined as the chance of an event occurring. It can be used to quantify what the odds are that a specific event will occur. Some examples of how probability is used everyday would be weather forecasting, 75% chance of snow or for setting insurance rates. 71
72 Conclusions A tree diagram can be used when a list of all possible outcomes is necessary. When only the total number of outcomes is needed, the multiplication rule, the permutation rule, and the combination rule can be used. 72
, 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.
41 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 informationChapter 4. Probability and Counting Rules. McGrawHill, Bluman, 7 th ed, Chapter 4
Chapter 4 Probability and Counting Rules McGrawHill, Bluman, 7 th ed, Chapter 4 Chapter 4 Overview Introduction 41 Sample Spaces and Probability 42 Addition Rules for Probability 43 Multiplication
More information1. 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
More informationProbability  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
More information4.1 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 informationMath 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 information4.3 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:
More informationProbability. 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 informationProbability 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
More informationSection Introduction to Sets
Section 1.1  Introduction to Sets Definition: A set is a welldefined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase
More informationFundamentals 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 informationEmpirical (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 31 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
More informationProbability Concepts and Counting Rules
Probability Concepts and Counting Rules Chapter 4 McGrawHill/Irwin Dr. Ateq Ahmed AlGhamedi Department of Statistics P O Box 80203 King Abdulaziz University Jeddah 21589, Saudi Arabia ateq@kau.edu.sa
More information7.1 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 informationPROBABILITY. 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 informationChapter 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 informationECON 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 informationChapter 3: Elements of Chance: Probability Methods
Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 34 20142015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,
More informationChapter 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
More informationCHAPTER 9  COUNTING PRINCIPLES AND PROBABILITY
CHAPTER 9  COUNTING PRINCIPLES AND PROBABILITY Probability is the Probability is used in many realworld fields, such as insurance, medical research, law enforcement, and political science. Objectives:
More informationChapter 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
More informationBusiness Statistics. Chapter 4 Using Probability and Probability Distributions QMIS 120. Dr. Mohammad Zainal
Department of Quantitative Methods & Information Systems Business Statistics Chapter 4 Using Probability and Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter,
More informationMULTIPLE 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 informationNovember 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 informationChapter 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 informationRaise 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
More informationChapter 1: Sets and Probability
Chapter 1: Sets and Probability Section 1.31.5 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 informationIntermediate 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 informationProbability 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
More informationProbability. 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
More informationProbability 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
More informationMULTIPLE 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
More informationHere are other examples of independent events:
5 The Multiplication Rules and Conditional Probability The Multiplication Rules Objective. Find the probability of compound events using the multiplication rules. The previous section showed that the addition
More informationSection : Combinations and Permutations
Section 11.111.2: 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
More informationSection 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 informationIf 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 informationChapter 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
More informationI. 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
More informationExam 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
More informationChapter 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 informationExamples: Experiment Sample space
Intro to Probability: A cynical person once said, The only two sure things are death and taxes. This philosophy no doubt arose because so much in people s lives is affected by chance. From the time a person
More information4.1 What is Probability?
4.1 What is Probability? between 0 and 1 to indicate the likelihood of an event. We use event is to occur. 1 use three major methods: 1) Intuition 3) Equally Likely Outcomes Intuition  prediction based
More informationSuch 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 informationProbability 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 informationMath 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 informationMath 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 informationContents 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
More informationAP Statistics Ch InClass Practice (Probability)
AP Statistics Ch 1415 InClass Practice (Probability) #1a) A batter who had failed to get a hit in seven consecutive times at bat then hits a gamewinning home run. When talking to reporters afterward,
More informationChapter 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 twoletter words (including nonsense words) can be formed when
More informationKey Concepts. Theoretical Probability. Terminology. Lesson 111
Key Concepts Theoretical Probability Lesson  Objective Teach students the terminology used in probability theory, and how to make calculations pertaining to experiments where all outcomes are equally
More informationProbability. 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!
More informationChapter 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 informationWeek 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
More informationImportant 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 informationMathematics 'A' level Module MS1: Statistics 1. Probability. The aims of this lesson are to enable you to. calculate and understand probability
Mathematics 'A' level Module MS1: Statistics 1 Lesson Three Aims The aims of this lesson are to enable you to calculate and understand probability apply the laws of probability in a variety of situations
More informationMath 7 Notes  Unit 11 Probability
Math 7 Notes  Unit 11 Probability Probability Syllabus Objective: (7.2)The student will determine the theoretical probability of an event. Syllabus Objective: (7.4)The student will compare theoretical
More informationProbability. Probabilty Impossibe Unlikely Equally Likely Likely Certain
PROBABILITY Probability The likelihood or chance of an event occurring If an event is IMPOSSIBLE its probability is ZERO If an event is CERTAIN its probability is ONE So all probabilities lie between 0
More informationIndependent 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
More informationProbability. 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
More informationProbability 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
More informationApplications of Probability
Applications of Probability CK12 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
More informationUnit 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,
More information6) 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 information05 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 informationClass 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:
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Chapter 3: Practice SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. ) A study of 000 randomly selected flights of a major
More informationProbability Review 41
Probability Review 41 For the following problems, give the probability to four decimals, or give a fraction, or if necessary, use scientific notation. Use P(A) = 1  P(not A) 1) A coin is tossed 6 times.
More informationMath 141 Exam 3 Review with Key. 1. P(E)=0.5, P(F)=0.6 P(E F)=0.9 Find ) b) P( E F ) c) P( E F )
Math 141 Exam 3 Review with Key 1. P(E)=0.5, P(F)=0.6 P(E F)=0.9 Find C C C a) P( E F) ) b) P( E F ) c) P( E F ) 2. A fair coin is tossed times and the sequence of heads and tails is recorded. Find a)
More informationProbability. March 06, J. Boulton MDM 4U1. P(A) = n(a) n(s) Introductory Probability
Most people think they understand odds and probability. Do you? Decision 1: Pick a card Decision 2: Switch or don't Outcomes: Make a tree diagram Do you think you understand probability? Probability Write
More informationSection 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 informationChapter 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
More information[Independent Probability, Conditional Probability, Tree Diagrams]
Name: Year 1 Review 119 Topic: Probability Day 2 Use your formula booklet! Page 5 Lesson 118: Probability Day 1 [Independent Probability, Conditional Probability, Tree Diagrams] Read and Highlight Station
More informationCounting Methods and Probability
CHAPTER Counting Methods and Probability Many good basketball players can make 90% of their free throws. However, the likelihood of a player making several free throws in a row will be less than 90%. You
More information4.4: The Counting Rules
4.4: The Counting Rules The counting rules can be used to discover the number of possible for a sequence of events. Fundamental Counting Rule In a sequence of n events in which the first one has k 1 possibilities
More informationMULTIPLE 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 informationUnit 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 informationDef: 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 information3 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
More informationMAT104: Fundamentals of Mathematics II Counting Techniques Class Exercises Solutions
MAT104: Fundamentals of Mathematics II Counting Techniques Class Exercises Solutions 1. Appetizers: Salads: Entrées: Desserts: 2. Letters: (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U,
More informationMULTIPLE 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 informationThe point value of each problem is in the lefthand margin. You must show your work to receive any credit, except on problems 1 & 2. Work neatly.
Introduction to Statistics Math 1040 Sample Exam II Chapters 57 4 Problem Pages 4 Formula/Table Pages Time Limit: 90 Minutes 1 No Scratch Paper Calculator Allowed: Scientific Name: The point value of
More informationAnswer each of the following problems. Make sure to show your work.
Answer each of the following problems. Make sure to show your work. 1. A board game requires each player to roll a die. The player with the highest number wins. If a player wants to calculate his or her
More informationNormal Distribution Lecture Notes Continued
Normal Distribution Lecture Notes Continued 1. Two Outcome Situations Situation: Two outcomes (for against; heads tails; yes no) p = percent in favor q = percent opposed Written as decimals p + q = 1 Why?
More informationConditional 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 informationProbability: 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 informationAlgebra 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 informationLC OL Probability. ARNMaths.weebly.com. As part of Leaving Certificate Ordinary Level Math you should be able to complete the following.
A Ryan LC OL Probability ARNMaths.weebly.com Learning Outcomes As part of Leaving Certificate Ordinary Level Math you should be able to complete the following. Counting List outcomes of an experiment Apply
More informationn(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 informationName: 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
More informationCHAPTER 7 Probability
CHAPTER 7 Probability 7.1. Sets A set is a welldefined 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 informationCHAPTER 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
More informationA Probability Work Sheet
A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair sixsided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we
More informationInstructions: 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 information3 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,
More informationQuiz 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 information2.5 Sample Spaces Having Equally Likely Outcomes
Sample Spaces Having Equally Likely Outcomes 3 Sample Spaces Having Equally Likely Outcomes Recall that we had a simple example (fair dice) before on equallylikely sample spaces Since they will appear
More informationBayes stuff Red Cross and Blood Example
Bayes stuff Red Cross and Blood Example 42% of the workers at Motor Works are female, while 67% of the workers at City Bank are female. If one of these companies is selected at random (assume a 5050 chance
More informationTextbook: pp Chapter 2: Probability Concepts and Applications
1 Textbook: pp. 3980 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.
More informationRANDOM EXPERIMENTS AND EVENTS
Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In daytoday 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
More informationMore 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