Chapter 4. Probability and Counting Rules. McGrawHill, Bluman, 7 th ed, Chapter 4


 Jonas Boyd
 3 years ago
 Views:
Transcription
1 Chapter 4 Probability and Counting Rules McGrawHill, Bluman, 7 th ed, Chapter 4
2 Chapter 4 Overview Introduction 41 Sample Spaces and Probability 42 Addition Rules for Probability 43 Multiplication Rules & Conditional Probability 44 Counting Rules 45 Probability and Counting Rules
3 Chapter 4 Objectives 1. Determine sample spaces and find the probability of an event, using classical probability or empirical probability. 2. Find the probability of compound events, using the addition rules. 3. Find the probability of compound events, using the multiplication rules. 4. Find the conditional probability of an event.
4 Chapter 4 Objectives 5. Find total number of outcomes in a sequence of events, using the fundamental counting rule. 6. Find the number of ways that r objects can be selected from n objects, using the permutation rule. 7. Find the number of ways for r objects selected from n objects without regard to order, using the combination rule. 8. Find the probability of an event, using the counting rules.
5 Probability 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.
6 41 Sample Spaces and Probability 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. An event consists of outcomes.
7 Sample Spaces Experiment Toss a coin Sample Space Head, Tail Roll a die 1, 2, 3, 4, 5, 6 Answer a true/false True, False question Toss two coins HH, HT, TH, TT
8 Chapter 4 Probability and Counting Rules Section 41 Example 41 Page #184 8
9 Example 41: Rolling Dice Find the sample space for rolling two dice. 9
10 Chapter 4 Probability and Counting Rules Section 41 Example 43 Page #184 10
11 Example 43: Gender of Children Find the sample space for the gender of the children if a family has three children. Use B for boy and G for girl. BBB BBG BGB BGG GBB GBG GGB GGG 11
12 Chapter 4 Probability and Counting Rules Section 41 Example 44 Page #185 12
13 Example 44: Gender of Children Use a tree diagram to find the sample space for the gender of three children in a family. B B G B G G B G B G B G B G BBB BBG BGB BGG GBB GBG GGB GGG 13
14 Sample Spaces and Probability There are three basic interpretations of probability: Classical probability Empirical probability Subjective probability
15 Sample Spaces and Probability Classical probability uses sample spaces to determine the numerical probability that an event will happen and assumes that all outcomes in the sample space are equally likely to occur. P E ( ) n( E) = = n S ( ) # of desired outcomes Total # of possible outcomes
16 Sample Spaces and Probability Rounding Rule for Probabilities Probabilities should be expressed as reduced fractions or rounded to two or three decimal places. When the probability of an event is an extremely small decimal, it is permissible to round the decimal to the first nonzero digit after the decimal point.
17 Chapter 4 Probability and Counting Rules Section 41 Example 46 Page #187 17
18 Example 46: Gender of Children If a family has three children, find the probability that two of the three children are girls. Sample Space: BBB BBG BGB BGG GBB GBG GGB GGG Three outcomes (BGG, GBG, GGB) have two girls. The probability of having two of three children being girls is 3/8. 18
19 Chapter 4 Probability and Counting Rules Section 41 Exercise 413c Page #196 19
20 Exercise 413c: Rolling Dice If two dice are rolled one time, find the probability of getting a sum of 7 or 11. P sum of 7 or 11 ( ) = =
21 Sample Spaces and Probability The complement of an event E, denoted by E, is the set of outcomes in the sample space that are not included in the outcomes of event E. ( ) 1P( E) P E =
22 Chapter 4 Probability and Counting Rules Section 41 Example 410 Page #189 22
23 Example 410: Finding Complements Find the complement of each event. Event Complement of the Event Rolling a die and getting a 4 Getting a 1, 2, 3, 5, or 6 Selecting a letter of the alphabet and getting a vowel Selecting a month and getting a month that begins with a J Selecting a day of the week and getting a weekday Getting a consonant (assume y is a consonant) Getting February, March, April, May, August, September, October, November, or December Getting Saturday or Sunday 23
24 Chapter 4 Probability and Counting Rules Section 41 Example 411 Page #190 24
25 Example 411: Residence of People If the probability that a person lives in an 1 industrialized country of the world is 5, find the probability that a person does not live in an industrialized country. P Not living in industrialized country ( ) = 1 P 1 4 = 1 = 5 5 living in industrialized country ( ) 25
26 Sample Spaces and Probability There are three basic interpretations of probability: Classical probability Empirical probability Subjective probability
27 Sample Spaces and Probability Empirical probability relies on actual experience to determine the likelihood of outcomes. P E ( ) f = = n frequency of desired class Sum of all frequencies
28 Chapter 4 Probability and Counting Rules Section 41 Example 413 Page #192 28
29 Example 413: Blood Types In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities. a. A person has type O blood. Type Frequency A 22 B 5 AB 2 O 21 Total 50 P ( O) = = f n
30 Example 413: Blood Types In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities. b. A person has type A or type B blood. Type Frequency A 22 B 5 AB 2 O 21 Total P ( A or B) = = 50 30
31 Example 413: Blood Types In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities. c. A person has neither type A nor type O blood. Type Frequency A 22 B 5 AB 2 O 21 Total 50 P ( neither A nor O) 5 2 = = 50 31
32 Example 413: Blood Types In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities. d. A person does not have type AB blood. Type Frequency A 22 B 5 AB 2 O 21 Total 50 P not AB ( ) = 1 P AB ( ) = 1 = =
33 Sample Spaces and Probability There are three basic interpretations of probability: Classical probability Empirical probability Subjective probability
34 Sample Spaces and Probability Subjective probability uses a probability value based on an educated guess or estimate, employing opinions and inexact information. Examples: weather forecasting, predicting outcomes of sporting events
35 4.2 Addition Rules for Probability Two events are mutually exclusive events if they cannot occur at the same time (i.e., they have no outcomes in common) Addition Rules P Aor B = P A + P B Mutually Exclusive ( ) ( ) ( ) P Aor B = P A + P B P A and B Not M. E. ( ) ( ) ( ) ( )
36 Chapter 4 Probability and Counting Rules Section 42 Example 415 Page #200 36
37 Example 415: Rolling a Die Determine which events are mutually exclusive and which are not, when a single die is rolled. a. Getting an odd number and getting an even number Getting an odd number: 1, 3, or 5 Getting an even number: 2, 4, or 6 Mutually Exclusive 37
38 Example 415: Rolling a Die Determine which events are mutually exclusive and which are not, when a single die is rolled. b. Getting a 3 and getting an odd number Getting a 3: 3 Getting an odd number: 1, 3, or 5 Not Mutually Exclusive 38
39 Example 415: Rolling a Die Determine which events are mutually exclusive and which are not, when a single die is rolled. c. Getting an odd number and getting a number less than 4 Getting an odd number: 1, 3, or 5 Getting a number less than 4: 1, 2, or 3 Not Mutually Exclusive 39
40 Example 415: Rolling a Die Determine which events are mutually exclusive and which are not, when a single die is rolled. d. Getting a number greater than 4 and getting a number less than 4 Getting a number greater than 4: 5 or 6 Getting a number less than 4: 1, 2, or 3 Mutually Exclusive 40
41 Chapter 4 Probability and Counting Rules Section 42 Example 418 Page #201 41
42 Example 418: Political Affiliation At a political rally, there are 20 Republicans, 13 Democrats, and 6 Independents. If a person is selected at random, find the probability that he or she is either a Democrat or an Independent. Mutually Exclusive Events P Democrat or Republican ( ) = P Democrat + P Republican ( ) ( ) = + = =
43 Chapter 4 Probability and Counting Rules Section 42 Example 421 Page #202 43
44 Example 421: Medical Staff In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male. Staff Females Males Total Nurses Physicians Total P Nurse or Male = P Nurse + P Male P Male Nurse = + = ( ) ( ) ( ) ( ) 44
45 4.3 Multiplication Rules Two events A and B are independent events if the fact that A occurs does not affect the probability of B occurring. Multiplication Rules P Aand B = P A P B Independent ( ) ( ) ( ) P Aand B P A P B A Dependent ( ) = ( ) ( )
46 Chapter 4 Probability and Counting Rules Section 43 Example 423 Page #211 46
47 Example 423: Tossing a Coin A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die. Independent Events P Head and 4 = P Head P = = ( ) ( ) ( ) This problem could be solved using sample space. H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6 47
48 Chapter 4 Probability and Counting Rules Section 43 Example 426 Page #212 48
49 Example 426: Survey on Stress A Harris poll found that 46% of Americans say they suffer great stress at least once a week. If three people are selected at random, find the probability that all three will say that they suffer great stress at least once a week. Independent Events P S and S and S = P S P S P S ( ) ( ) ( ) ( ) = = ( 0.46)( 0.46)( 0.46)
50 Chapter 4 Probability and Counting Rules Section 43 Example 428 Page #214 50
51 Example 428: University Crime At a university in western Pennsylvania, there were 5 burglaries reported in 2003, 16 in 2004, and 32 in If a researcher wishes to select at random two burglaries to further investigate, find the probability that both will have occurred in Dependent Events P P P ( C and C ) = ( C ) ( C C ) = =
52 4.3 Conditional Probability Conditional probability is the probability that the second event B occurs given that the first event A has occurred. Conditional Probability ( ) P B A = P Aand B ( ) P A ( )
53 Chapter 4 Probability and Counting Rules Section 43 Example 433 Page #217 53
54 Example 433: Parking Tickets The probability that Sam parks in a noparking zone and gets a parking ticket is 0.06, and the probability that Sam cannot find a legal parking space and has to park in the noparking zone is On Tuesday, Sam arrives at school and has to park in a noparking zone. Find the probability that he will get a parking ticket. N= parking in a noparking zone, T= getting a ticket P( N and T) 0.06 P ( TN) = = = 0.30 P N 0.20 ( ) 54
55 Chapter 4 Probability and Counting Rules Section 43 Example 434 Page #217 55
56 Example 434: Women in the Military A recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. The results of the survey are shown. 56
57 Example 434: Women in the Military a. Find the probability that the respondent answered yes (Y), given that the respondent was a female (F). P ( YF) P( F and Y) = = P F ( ) = 8 50 =
58 Example 434: Women in the Military b. Find the probability that the respondent was a male (M), given that the respondent answered no (N). P ( MN) P( N and M) = = P N ( ) = =
59 Chapter 4 Probability and Counting Rules Section 43 Example 437 Page #219 59
60 Example 437: Bow Ties The Neckware Association of America reported that 3% of ties sold in the United States are bow ties (B). If 4 customers who purchased a tie are randomly selected, find the probability that at least 1 purchased a bow tie. P ( ) P( ) B = 0.03, B = = 0.97 P P P P P P ( no bow ties) = ( B) ( B) ( B) ( B) = = ( )( )( )( ) at least 1 bow tie = 1 P no bow ties ( ) ( ) = =
61 4.4 Counting Rules The fundamental counting rule is also called the multiplication of choices. 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
62 Chapter 4 Probability and Counting Rules Section 44 Example 439 Page #225 62
63 Example 439: Paint Colors A paint manufacturer wishes to manufacture several different paints. The categories include Color: red, blue, white, black, green, brown, yellow Type: latex, oil Texture: flat, semigloss, high gloss Use: outdoor, indoor How many different kinds of paint can be made if you can select one color, one type, one texture, and one use? ( # of )( # of )( # of )(# of ) colors types textures uses different kinds of paint 63
64 Counting Rules Factorial is the product of all the positive numbers from 1 to a number. n! = n n 1 n ( )( ) 0! = 1 Permutation is an arrangement of objects in a specific order. Order matters. n! npr = = n( n 1)( n 2) ( n r+ 1) ( n r)! r items
65 Counting Rules Combination is a grouping of objects. Order does not matter. n C r = = n! n r! r! ( ) n Pr r!
66 Chapter 4 Probability and Counting Rules Section 44 Example 442/443 Page #228 66
67 Example 442: Business Locations Suppose a business owner has a choice of 5 locations in which to establish her business. She decides to rank each location according to certain criteria, such as price of the store and parking facilities. How many different ways can she rank the 5 locations? ( first )( second)( third )( fourth )( fifth ) choice choice choice choice choice different ways to rank the locations Using factorials, 5! = 120. Using permutations, 5 P 5 =
68 Example 443: Business Locations Suppose the business owner in Example 4 42 wishes to rank only the top 3 of the 5 locations. How many different ways can she rank them? ( first )( second)( third ) choice choice choice different ways to rank the locations Using permutations, 5 P 3 =
69 Chapter 4 Probability and Counting Rules Section 44 Example 444 Page #229 69
70 Example 444: Television News Stories A television news director wishes to use 3 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 8 stories to choose from, how many possible ways can the program be set up? Since there is a lead, second, and closing story, we know that order matters. We will use permutations. 8! P = = 336 or 8P 3 = = 336 5!
71 Chapter 4 Probability and Counting Rules Section 44 Example 445 Page #229 71
72 Example 445: School Musical Plays A school musical director can select 2 musical plays to present next year. One will be presented in the fall, and one will be presented in the spring. If she has 9 to pick from, how many different possibilities are there? Order matters, so we will use permutations. 9! P = = 72 or 9P 2 = 9 8 = 72 7!
73 Chapter 4 Probability and Counting Rules Section 44 Example 448 Page #231 73
74 Example 448: School Musicals A newspaper editor has received 8 books to review. He decides that he can use 3 reviews in his newspaper. How many different ways can these 3 reviews be selected? The placement in the newspaper is not mentioned, so order does not matter. We will use combinations. 8! C = = 8!/ ( 5!3! ) = 56 5!3! or 8C3 = = P 8 3 or 8C 3 = = 56 3! 74
75 Chapter 4 Probability and Counting Rules Section 44 Example 449 Page #231 75
76 Example 449: Committee Selection In a club there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there? There are not separate roles listed for each committee member, so order does not matter. We will use combinations. 7! 5! Women: 7C3 = = 35, Men: 5C2 = = 10 4!3! 3!2! There are 35 10=350 different possibilities. 76
77 4.5 Probability and Counting Rules The counting rules can be combined with the probability rules in this chapter to solve many types of probability problems. By using the fundamental counting rule, the permutation rules, and the combination rule, you can compute the probability of outcomes of many experiments, such as getting a full house when 5 cards are dealt or selecting a committee of 3 women and 2 men from a club consisting of 10 women and 10 men.
78 Chapter 4 Probability and Counting Rules Section 45 Example 452 Page #238 78
79 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
80 Chapter 4 Probability and Counting Rules Section 45 Example 453 Page #239 80
81 Example 453: Combination Locks A combination lock consists of the 26 letters of the alphabet. If a 3letter combination is needed, find the probability that the combination will consist of the letters ABC in that order. The same letter can be used more than once. (Note: A combination lock is really a permutation lock.) There are = 17,576 possible combinations. The letters ABC in order create one combination. P ABC = ( ) 1 17,576 81
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
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 informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 4 Probability and Counting Rules 2 Objectives Determine sample spaces and find the probability of an event using classical probability or empirical
More 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 information4.4: 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 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 informationSec 4.4. Counting Rules. Bluman, Chapter 4
Sec 4.4 Counting Rules A Question to Ponder: A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, replaced and a second chip is selected. Display the sample space. Do you think
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 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 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 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 information, the of all of a probability experiment. consists of outcomes. (b) List the elements of the event consisting of a number that is greater than 4.
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 informationAn outcome is the result of a single trial of a probability experiment.
2 Sample Spaces and Probability The theory of probability grew out of the study of various games of chance using coins, dice, and cards. Since these devices lend themselves well to the application of concepts
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 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 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 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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Ch. 3 Probability 3.1 Basic Concepts of Probability and Counting 1 Find Probabilities 1) A coin is tossed. Find the probability that the result is heads. A) 0. B) 0.1 C) 0.9 D) 1 2) A single sixsided
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 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 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 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 informationChapter 13 From Randomness to Probability
Chapter 13 From Randomness to Probability 247 Chapter 13 From Randomness to Probability 1. Sample spaces. a) S = { HH, HT, TH, TT} All of the outcomes are equally likely to occur. b) S = { 0, 1, 2, 3}
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 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 informationSection 11.4: Tree Diagrams, Tables, and Sample Spaces
Section 11.4: Tree Diagrams, Tables, and Sample Spaces Diana Pell Exercise 1. Use a tree diagram to find the sample space for the genders of three children in a family. Exercise 2. (You Try!) A soda machine
More 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 informationStatistics By: Mr. Danilo J. Salmorin
Statistics By: Mr. Danilo J. Salmorin Ihr Logo COUNTING TECHNIQUES PROBABILITY Your Logo Continue COUNTING TECHNI QUES Tree diagram Multiplication Rule Permutation Combination TREE DIAGRAM a device used
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 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 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 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 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 informationMath141_Fall_2012 ( Business Mathematics 1) Week 7. Dr. Marco A. Roque Sol Department of Mathematics Texas A&M University
( Business Mathematics 1) Week 7 Dr. Marco A. Roque Department of Mathematics Texas A&M University In this sections we will consider two types of arrangements, namely, permutations and combinations 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 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 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 informationDiamond ( ) (Black coloured) (Black coloured) (Red coloured) ILLUSTRATIVE EXAMPLES
CHAPTER 15 PROBABILITY Points to Remember : 1. In the experimental approach to probability, we find the probability of the occurence of an event by actually performing the experiment a number of times
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 informationMTH 103 H Final Exam. 1. I study and I pass the course is an example of a. (a) conjunction (b) disjunction. (c) conditional (d) connective
MTH 103 H Final Exam Name: 1. I study and I pass the course is an example of a (a) conjunction (b) disjunction (c) conditional (d) connective 2. Which of the following is equivalent to (p q)? (a) p q (b)
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 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 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 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 informationCS 361: Probability & Statistics
January 31, 2018 CS 361: Probability & Statistics Probability Probability theory Probability Reasoning about uncertain situations with formal models Allows us to compute probabilities Experiments will
More informationM146  Chapter 5 Handouts. Chapter 5
Chapter 5 Objectives of chapter: Understand probability values. Know how to determine probability values. Use rules of counting. Section 51 Probability Rules What is probability? It s the of the occurrence
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 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 informationReview 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
More informationBasic Probability Concepts
6.1 Basic Probability Concepts How likely is rain tomorrow? What are the chances that you will pass your driving test on the first attempt? What are the odds that the flight will be on time when you go
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 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 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 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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
More 9.9.3 Practice Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. ) In how many ways can you answer the questions on
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 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 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 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 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 informationMath 147 Elementary Probability/Statistics I Additional Exercises on Chapter 4: Probability
Math 147 Elementary Probability/Statistics I Additional Exercises on Chapter 4: Probability Student Name: Find the indicated probability. 1) If you flip a coin three times, the possible outcomes are HHH
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 information7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count
7.1 Chance Surprises, 7.2 Predicting the Future in an Uncertain World, 7.4 Down for the Count Probability deals with predicting the outcome of future experiments in a quantitative way. The experiments
More informationName: Partners: Math Academy I. Review 6 Version A. 5. There are over a billion different possible orders for a line of 14 people.
Name: Partners: Math Academy I Date: Review 6 Version A [A] Circle whether each statement is true or false. 1. Odd and less than 4 are mutually exclusive. 2. The probability of a card being red given it
More informationTree Diagrams and the Multiplication Rule for Counting Tree Diagrams. tree diagram.
4 2 Tree Diagrams and the Multiplication Rule for Counting Tree Diagrams Objective 1. Determine the number of outcomes of a sequence of events using a tree diagram. Example 4 1 Many times one wishes to
More informationTest 2 Review Solutions
Test Review Solutions. A family has three children. Using b to stand for and g to stand for, and using ordered triples such as bbg, find the following. a. draw a tree diagram to determine the sample space
More informationFinite Mathematics MAT 141: Chapter 8 Notes
Finite Mathematics MAT 4: Chapter 8 Notes Counting Principles; More David J. Gisch The Multiplication Principle; Permutations Multiplication Principle Multiplication Principle You can think of the multiplication
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 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 informationCSC/MATA67 Tutorial, Week 12
CSC/MATA67 Tutorial, Week 12 November 23, 2017 1 More counting problems A class consists of 15 students of whom 5 are prefects. Q: How many committees of 8 can be formed if each consists of a) exactly
More informationHomework Set #1. 1. The Supreme Court (9 members) meet, and all the justices shake hands with each other. How many handshakes are there?
Homework Set # Part I: COMBINATORICS (follows Lecture ). The Supreme Court (9 members) meet, and all the justices shake hands with each other. How many handshakes are there? 2. A country has license plates
More informationChapter 3: Probability (Part 1)
Chapter 3: Probability (Part 1) 3.1: Basic Concepts of Probability and Counting Types of Probability There are at least three different types of probability Subjective Probability is found through people
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 informationExam Time. Final Exam Review. TR class Monday December 9 12:30 2:30. These review slides and earlier ones found linked to on BlackBoard
Final Exam Review These review slides and earlier ones found linked to on BlackBoard Bring a photo ID card: Rocket Card, Driver's License Exam Time TR class Monday December 9 12:30 2:30 Held in the regular
More informationDate. Probability. Chapter
Date Probability Contests, lotteries, and games offer the chance to win just about anything. You can win a cup of coffee. Even better, you can win cars, houses, vacations, or millions of dollars. Games
More informationProbability Essential Math 12 Mr. Morin
Probability Essential Math 12 Mr. Morin Name: Slot: Introduction Probability and Odds Single Event Probability and Odds Two and Multiple Event Experimental and Theoretical Probability Expected Value (Expected
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 informationProbability Models. Section 6.2
Probability Models Section 6.2 The Language of Probability What is random? Empirical means that it is based on observation rather than theorizing. Probability describes what happens in MANY trials. Example
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 1332 Review Test 4 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem by applying the Fundamental Counting Principle with two
More informationUse this information to answer the following questions.
1 Lisa drew a token out of the bag, recorded the result, and then put the token back into the bag. She did this 30 times and recorded the results in a bar graph. Use this information to answer the following
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 informationUnit on Permutations and Combinations (Counting Techniques)
Page 1 of 15 (Edit by Y.M. LIU) Page 2 of 15 (Edit by Y.M. LIU) Unit on Permutations and Combinations (Counting Techniques) e.g. How many different license plates can be made that consist of three digits
More informationNAME DATE PERIOD. Study Guide and Intervention
91 Section Title The probability of a simple event is a ratio that compares the number of favorable outcomes to the number of possible outcomes. Outcomes occur at random if each outcome occurs by chance.
More informationMath 102 Practice for Test 3
Math 102 Practice for Test 3 Name Show your work and write all fractions and ratios in simplest form for full credit. 1. If you draw a single card from a standard 52card deck what is P(King face card)?
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 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 informationBellwork Write each fraction as a percent Evaluate P P C C 6
Bellwork 21915 Write each fraction as a percent. 1. 2. 3. 4. Evaluate. 5. 6 P 3 6. 5 P 2 7. 7 C 4 8. 8 C 6 1 Objectives Find the theoretical probability of an event. Find the experimental probability
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 informationThis unit will help you work out probability and use experimental probability and frequency trees. Key points
Get started Probability This unit will help you work out probability and use experimental probability and frequency trees. AO Fluency check There are 0 marbles in a bag. 9 of the marbles are red, 7 are
More informationMultiplication and Probability
Problem Solving: Multiplication and Probability Problem Solving: Multiplication and Probability What is an efficient way to figure out probability? In the last lesson, we used a table to show the probability
More informationProbability, Continued
Probability, Continued 12 February 2014 Probability II 12 February 2014 1/21 Last time we conducted several probability experiments. We ll do one more before starting to look at how to compute theoretical
More informationMEP Practice Book ES5. 1. A coin is tossed, and a die is thrown. List all the possible outcomes.
5 Probability MEP Practice Book ES5 5. Outcome of Two Events 1. A coin is tossed, and a die is thrown. List all the possible outcomes. 2. A die is thrown twice. Copy the diagram below which shows all the
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 informationCH 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:
More informationTopic : ADDITION OF PROBABILITIES (MUTUALLY EXCLUSIVE EVENTS) TIME : 4 X 45 minutes
Worksheet 6 th Topic : ADDITION OF PROBABILITIES (MUTUALLY EXCLUSIVE EVENTS) TIME : 4 X 45 minutes STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of
More informationClassical vs. Empirical Probability Activity
Name: Date: Hour : Classical vs. Empirical Probability Activity (100 Formative Points) For this activity, you will be taking part in 5 different probability experiments: Rolling dice, drawing cards, drawing
More informationName: Spring P. Walston/A. Moore. Topic worksheet # assigned #completed Teacher s Signature Tree Diagrams FCP
Name: Spring 2016 P. Walston/A. Moore Topic worksheet # assigned #completed Teacher s Signature Tree Diagrams 10 13 FCP 11 16 Combinations/ Permutations Factorials 12 22 13 20 Intro to Probability
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 informationName: Class: Date: Probability/Counting Multiple Choice PreTest
Name: _ lass: _ ate: Probability/ounting Multiple hoice PreTest Multiple hoice Identify the choice that best completes the statement or answers the question. 1 The dartboard has 8 sections of equal area.
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