STATISTICAL COUNTING TECHNIQUES


 Gloria Fisher
 4 years ago
 Views:
Transcription
1 STATISTICAL COUNTING TECHNIQUES I. Counting Principle The counting principle states that if there are n 1 ways of performing the first experiment, n 2 ways of performing the second experiment, n 3 ways of performing the third experiment, and so on, then the total number of ways of performing the experiments in order is found by multiplying n 1 x n 2 x n 3. Note: Repetition is allowed. Example 1: How many outcomes are possible when rolling a pair of fair dice? There are 6 outcomes from each die: 1, 2, 3, 4, 5, 6. Thus, there are 6 x 6 = 36 possible outcomes. Example 2: A package to be mailed is charged according to volume (length, width, and height), and weight. There are 5 different volume prices and 8 different weight prices. How many different prices are possible? There are two distinct categories: volume and weight. Thus, by the counting principle, there are 5 x 8 = 40 possible prices. Example 3: How many different 5letter codes (repetition is allowed) are possible? ABSAT is one such code. There are 5 positions, each one with 26 possibilities (26 letters in the alphabet). Thus, there are 26 x 26 x 26 x 26 x 26 = 11,881,376 possible codes. Example 4: Burger World sells hamburgers with or without each of the following: cheese, lettuce, tomatoes, mustard, mayonnaise, and ketchup. Also, you can have it on a white, wheat, or sourdough bun. How many different hamburgers does Burger World sell? You can have with or without cheese (2 ways), with or without lettuce (2 ways), with or without tomatoes (2 ways), with or without mustard (2 ways), with or without mayonnaise (2 ways), with or without ketchup (2 ways), and on white, wheat, or sourdough bun (3 ways). Therefore, by the counting principle, there are 2 x 2 x 2 x 2 x 2 x 2 x 3 =192 different hamburgers. 1
2 II. Factorial A factorial computes the number of ways to arrange n items in order with repetition not allowed. It is written as n! (read as n factorial ), and is found by multiplying n times (n1) times (n2) times times (1). n! = n (n1) (n2) (n3) (2) (1) For example: 9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 4! = 4 x 3 x 2 x 1 = 24 2! = 2 x 1 = 2 1! = 1 0! = 1 (by definition) Note: To solve (93)!, you must work inside parenthesis first and then take the factorial of the result. So, (93)! = 6! = 720. To solve 9! 3!, take factorial of each term, and then subtract. So, 9! 3! = 362,880 6 = 362,874. Find 7! Using the TI83/84 Enter 7, MATH, PRB, 4:!, ENTER ANSWER: 5040 Example 5: How many ways can you arrange 3 different books on a shelf? This is done using factorial. There are 3! = 3 x 2 x 1= 6 ways. To understand this, note that any of the 3 books can be put in the first position on the shelf, then either of the remaining 2 books can be placed in the second position, then there is 1 book left to be put in the last position. Example 6: How many ways can 8 ducks sit on a fence? Using factorial, there are 8! =8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 ways. Find 8! Using the TI83/84 Enter 8, MATH, PRB, 4:!, ENTER ANSWER:
3 III. Permutations A permutation is an ordered arrangement of objects where repetition is not allowed. Use the following when order is important: Permutation Formula: n P r = for ordered arrangements of n items taken r at a time. Example 7: How many ways can a president and vicepresident be selected from a group of 8 candidates? First, we determine that order of arrangement i& important. For example, the outcome (John, Bill) differs from outcome (Bill, John) since the first outcome has John as the president and Bill as the vicepresident, and the second viceversa. 8 P 2 = = = 56 Find 8 P 2 using the TI83/84 Enter 8, MATH, PRB, 2: n P r, 2, ENTER ANSWER: 56 Example 8: How many possible 1st, 2nd, and 3rd place finishes can there be in a race of 7 runners? Order is important in a race, so use the permutation formula. 7P 3 = = 210 Find 7 P 3 using the TI83/84 Enter 7, MATH, PRB, 2: n P r, 3, ENTER ANSWER: 210 3
4 IV. Combinations A combination is an unordered grouping of objects where repetition is not allowed. Use when order is not important: Combination Formula: n P r = for ordered arrangements of n items taken r at a time. Example 10: How many sundaes can be made using 2 of 5 ice cream flavors? Order of choosing the flavor is not important. For example, the outcome "vanilla, chocolate is the same as chocolate, vanilla. 5C 2 = Find 5 C 2 using the TI83/84 Enter 5, MATH, PRB, 3: n C r, 2, ENTER ANSWER: 10 Example 11: How many threemember committees can be chosen from a group of 8 people? The order picked is not important since (Mary, John, and Ann) is the same committee as (John, Ann, and Mary). 8C 3 = Find 8 C 3 using the TI83/84 Enter 8, MATH, PRB, 3: n C r, 3, ENTER ANSWER: 56 4
5 Example 12: How many 9p1ayer baseball teams can be made from a group of 25 players? 25C 9 = Find 25 C 9 using the TI83/84 Enter 25, MATH, PRB, 3: n C r, 9, ENTER ANSWER: 2,042,975 When choosing from different subsets of a group with order not important, use the Combination Formula for each subgroup. The following are examples involving the Counting Principle and the Combinations Formula. Example 13: How many committees of 2 men and 3 women can be chosen from a group of 6 men and 4 women? We need 2 of 6 men and 3 of 4 women: men women 6C 2 4 C 3 = 15 4 = 60 Find 6 C 2 * 4 C 3 using the TI83/84 Enter 6, MATH, PRB, 3: n C r, 2, ENTER * Enter 4, MATH, PRB, 3: n C r, 3, ENTER ANSWER: 60 Example 14: Suppose there are 11 batteries on a shelf, 4 defective and 7 nondefective. If 3 are chosen at random; how many ways can a) any 3 be chosen, b) no defectives be chosen, c) 1 defective and 2 nondefectives? a) 11 C 3 = 165 b) We need 0 of 4 defectives and 3 of 7 nondefectives def non 4C 0 7C 3 = 1 35 = 35 c) We need 1 of 4 defectives and 2 of 7 nondefectives def non 4C 1 7C 2 = 4 21 = 84 5
6 V. Probability Involving the Counting Principle Sometimes, the number of total possible outcomes is found using the Counting Principle. Recall that the probability of event E is Example 15: If a balanced coin is tossed 5 times, what is the probability of obtaining a) all heads and b) at least one head? Each of the 5 coins has 2 possibilities, heads or tails, so there are 2*2*2*2*2 = 32 possible outcomes. a) There is one outcome with all heads: {HHHHH} P(all heads) = 1/32 =.031 b) We need all outcomes with at least one head. This means outcomes with 1, 2, 3, 4, or 5 heads. This requires much work, so use the Complementation Rule. P( at least 1 head) = 1 P(not at least 1 head) = 1 P(O heads) = 1 P({TTTTT}) = 11/32 = 31/32 =.969 Example 16: If three dice are tossed, what is the probability of obtaining a sum of a) 3, b) 4, c) 5 or more? Since each die has 6 possibilities, the total number of outcomes is 6*6*6 = 216. a) There is one outcome with sum 3: {1,1,1} P (sum =3) = 1/216 =.005 b) 3 outcomes with sum = 4: {(2, 1, 1), (1, 2, 1), (1, 1, 2)} P (sum = 4) = 3/216 =.014 c) Use Complementation Rule: P (sum = 5 or more) = 1P (sum is not 5 or more) = 1P (sum is less than 5) = 1P (sum=3 or 4) = 14/216 (from a) and b)) = 212/216 =.981 6
7 VI. Probability Involving Combinations Example 17: Winning the lottery consists of picking the 6 winning numbers, in any order, from the numbers 1 through 49. What is the probability of winning this lottery? Example 18: What is the probability of picking 3 men and 2 women to sit on a fiveperson committee from a group of 7 men and 8 women? =.326 Example 19: A bin contains 5 faulty and 7 good parts. If 4 parts are picked at random from the bin, what is the probability of picking a) all good parts and b) 2 of each? =.071 =.424 7
8 VII. Exercises 1. How many outcomes are there when a fair die is rolled 4 times? 2. How many license plates (3 letters followed by 3 digits are possible with a) repetition allowed (so AAB155 is possible) and b) repetition not allowed (so AABl15 is not allowed)? 3. How many ways can 10 people stand in a line? 4. In how many orders can you watch 7 different videos? 5. How many ways can a president, vicepresident, and secretary be chosen from a group of 9 people? 6. How many ways can 8 people finish 1st, 2nd, 3rd, and 4th in a race? 7. How many 5member basketball teams can be made from a group of 15 players? 8. How many ways can 2 positions be filled by a group of 12 applicants? 9. If 4 coins are flipped, what is the probability of observing 4 tails? 10. If a die is tossed 5 times, what is the probability of obtaining a sum of 6? Hint: one possible outcome is (2,1,1,1,1}. 11. If a box contains 20 light bulbs, 14 good and 6 bad. If 5 are picked at random, what is the probability of picking no bad ones? 12. What is the probability of choosing a panel of 4 men and 2 women from a group of 6 men and 5 women? 8
9 VIII. Solutions 1. Each roll has 6 possible outcomes, so =1296 outcomes. 2. A license plate is letterietterietterdigitdigitdigit. There are 26 letters and 10 digits (0 through 9). a) = 17,576,000 b) = 11,232, Order is important, so 10! = =3,628, Order was asked for, so 7! = = 5, Order is important, so 9 P 3 = Order is important, so 8 P 4 = 1, Order is not important, so 15 C 5 = 3, Order is not important, so 12 C 2 = P(4 tails) = (# of outcomes with 4 tails)/(total outcomes) = (1 outcome: {TTTT})/ ( ) = 1/16 = outcomes total 6: {2,1,1,1,1}, {1,2,1,1,1}, {1,1,2,1,1}, {1,1,1,2,1}, {1,1,1,1,2} Total outcomes = = 7,776 P (sum 6) =5/7776 = P(0 bad & 5 good) = 6 C 0 * 14 C 5 / 20 C 5 =1 * 2002/15504 = P(4 of 6 men & 2 of 5 women) = = 6 C 4 * 5 C 2 / 11 C 6 = 15 * 10 / 462 =.325 9
7.1 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 informationProbability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37
Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete
More 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 information1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?
1. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building? 2. A particular brand of shirt comes in 12 colors, has a male version and a female version,
More 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 information6. In how many different ways can you answer 10 multiplechoice questions if each question has five choices?
PreCalculus Section 4.1 Multiplication, Addition, and Complement 1. Evaluate each of the following: a. 5! b. 6! c. 7! d. 0! 2. Evaluate each of the following: a. 10! b. 20! 9! 18! 3. In how many different
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 informationMATH 215 DISCRETE MATHEMATICS INSTRUCTOR: P. WENG
MATH DISCRETE MATHEMATICS INSTRUCTOR: P. WENG Counting and Probability Suggested Problems Basic Counting Skills, InclusionExclusion, and Complement. (a An office building contains 7 floors and has 7 offices
More informationWEEK 7 REVIEW. Multiplication Principle (6.3) Combinations and Permutations (6.4) Experiments, Sample Spaces and Events (7.1)
WEEK 7 REVIEW Multiplication Principle (6.3) Combinations and Permutations (6.4) Experiments, Sample Spaces and Events (7.) Definition of Probability (7.2) WEEK 87.3, 7.4 and Test Review THE MULTIPLICATION
More informationMath 1116 Probability Lecture Monday Wednesday 10:10 11:30
Math 1116 Probability Lecture Monday Wednesday 10:10 11:30 Course Web Page http://www.math.ohio state.edu/~maharry/ Chapter 15 Chances, Probabilities and Odds Objectives To describe an appropriate sample
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 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 informationSection 6.1 #16. Question: What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit?
Section 6.1 #16 What is the probability that a fivecard poker hand contains a flush, that is, five cards of the same suit? page 1 Section 6.1 #38 Two events E 1 and E 2 are called independent if p(e 1
More informationName: 1. Match the word with the definition (1 point each  no partial credit!)
Chapter 12 Exam Name: Answer the questions in the spaces provided. If you run out of room, show your work on a separate paper clearly numbered and attached to this exam. SHOW ALL YOUR WORK!!! Remember
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 informationMAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability. Preliminary Concepts, Formulas, and Terminology
MAT104: Fundamentals of Mathematics II Summary of Counting Techniques and Probability Preliminary Concepts, Formulas, and Terminology Meanings of Basic Arithmetic Operations in Mathematics Addition: Generally
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 informationCourse Learning Outcomes for Unit V
UNIT V STUDY GUIDE Counting Reading Assignment See information below. Key Terms 1. Combination 2. Fundamental counting principle 3. Listing 4. Permutation 5. Tree diagrams Course Learning Outcomes for
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 informationSTAT 430/510 Probability Lecture 1: Counting1
STAT 430/510 Probability Lecture 1: Counting1 Pengyuan (Penelope) Wang May 22, 2011 Introduction In the early days, probability was associated with games of chance, such as gambling. Probability is describing
More informationSTAT 430/510 Probability
STAT 430/510 Probability Hui Nie Lecture 1 May 26th, 2009 Introduction Probability is the study of randomness and uncertainty. In the early days, probability was associated with games of chance, such as
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 informationW = {Carrie (U)nderwood, Kelly (C)larkson, Chris (D)aughtry, Fantasia (B)arrino, and Clay (A)iken}
UNIT V STUDY GUIDE Counting Course Learning Outcomes for Unit V Upon completion of this unit, students should be able to: 1. Apply mathematical principles used in realworld situations. 1.1 Draw tree diagrams
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 informationProbability, Permutations, & Combinations LESSON 11.1
Probability, Permutations, & Combinations LESSON 11.1 Objective Define probability Use the counting principle Know the difference between combination and permutation Find probability Probability PROBABILITY:
More informationExercises Exercises. 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}?
Exercises Exercises 1. List all the permutations of {a, b, c}. 2. How many different permutations are there of the set {a, b, c, d, e, f, g}? 3. How many permutations of {a, b, c, d, e, f, g} end with
More informationName Class Date. Introducing Probability Distributions
Name Class Date Binomial Distributions Extension: Distributions Essential question: What is a probability distribution and how is it displayed? 86 CC.9 2.S.MD.5(+) ENGAGE Introducing Distributions Video
More informationIndependent Events. If we were to flip a coin, each time we flip that coin the chance of it landing on heads or tails will always remain the same.
Independent Events Independent events are events that you can do repeated trials and each trial doesn t have an effect on the outcome of the next trial. If we were to flip a coin, each time we flip that
More informationChapter 2. Permutations and Combinations
2. Permutations and Combinations Chapter 2. Permutations and Combinations In this chapter, we define sets and count the objects in them. Example Let S be the set of students in this classroom today. Find
More informationName. Is the game fair or not? Prove your answer with math. If the game is fair, play it 36 times and record the results.
Homework 5.1C You must complete table. Use math to decide if the game is fair or not. If Period the game is not fair, change the point system to make it fair. Game 1 Circle one: Fair or Not 2 six sided
More informationApril 10, ex) Draw a tree diagram of this situation.
April 10, 2014 121 Fundamental Counting Principle & Multiplying Probabilities 1. Outcome  the result of a single trial. 2. Sample Space  the set of all possible outcomes 3. Independent Events  when
More informationEECS 203 Spring 2016 Lecture 15 Page 1 of 6
EECS 203 Spring 2016 Lecture 15 Page 1 of 6 Counting We ve been working on counting for the last two lectures. We re going to continue on counting and probability for about 1.5 more lectures (including
More informationSection The Multiplication Principle and Permutations
Section 2.1  The Multiplication Principle and Permutations Example 1: A yogurt shop has 4 flavors (chocolate, vanilla, strawberry, and blueberry) and three sizes (small, medium, and large). How many different
More informationCounting and Probability Math 2320
Counting and Probability Math 2320 For a finite set A, the number of elements of A is denoted by A. We have two important rules for counting. 1. Union rule: Let A and B be two finite sets. Then A B = A
More informationAlgebra II Probability and Statistics
Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 20160115 www.njctl.org Slide 3 / 241 Table of Contents click on the topic to go to that section Sets Independence and Conditional Probability
More informationCSC/MTH 231 Discrete Structures II Spring, Homework 5
CSC/MTH 231 Discrete Structures II Spring, 2010 Homework 5 Name 1. A six sided die D (with sides numbered 1, 2, 3, 4, 5, 6) is thrown once. a. What is the probability that a 3 is thrown? b. What is the
More informationAlgebra II. Sets. Slide 1 / 241 Slide 2 / 241. Slide 4 / 241. Slide 3 / 241. Slide 6 / 241. Slide 5 / 241. Probability and Statistics
Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 20160115 www.njctl.org Slide 3 / 241 Slide 4 / 241 Table of Contents click on the topic to go to that section Sets Independence and Conditional
More informationPrecalc Unit 10 Review
Precalc Unit 10 Review Name: Use binomial expansion to expand. 1. 2. 3.. Use binomial expansion to find the term you are asked for. 4. 5 th term of (4x3y) 8 5. 3 rd term of 6. 4 th term of 7. 2 nd term
More informationAlgebra II. Slide 1 / 241. Slide 2 / 241. Slide 3 / 241. Probability and Statistics. Table of Contents click on the topic to go to that section
Slide 1 / 241 Slide 2 / 241 Algebra II Probability and Statistics 20160115 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 241 Sets Independence and Conditional Probability
More informationChapter 4: Introduction to Probability
MTH 243 Chapter 4: Introduction to Probability Suppose that we found that one of our pieces of data was unusual. For example suppose our pack of M&M s only had 30 and that was 3.1 standard deviations below
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 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 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 informationFundamental Counting Principle
Lesson 88 Probability with Combinatorics HL2 Math  Santowski Fundamental Counting Principle Fundamental Counting Principle can be used determine the number of possible outcomes when there are two or more
More informationName: Exam 1. September 14, 2017
Department of Mathematics University of Notre Dame Math 10120 Finite Math Fall 2017 Name: Instructors: Basit & Migliore Exam 1 September 14, 2017 This exam is in two parts on 9 pages and contains 14 problems
More informationAdvanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY
Advanced Intermediate Algebra Chapter 12 Summary INTRO TO PROBABILITY 1. Jack and Jill do not like washing dishes. They decide to use a random method to select whose turn it is. They put some red and blue
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 informationMost of the time we deal with theoretical probability. Experimental probability uses actual data that has been collected.
AFM Unit 7 Day 3 Notes Theoretical vs. Experimental Probability Name Date Definitions: Experiment: process that gives a definite result Outcomes: results Sample space: set of all possible outcomes Event:
More informationPermutations. and. Combinations
Permutations and Combinations Fundamental Counting Principle Fundamental Counting Principle states that if an event has m possible outcomes and another independent event has n possible outcomes, then there
More informationBlock 1  Sets and Basic Combinatorics. Main Topics in Block 1:
Block 1  Sets and Basic Combinatorics Main Topics in Block 1: A short revision of some set theory Sets and subsets. Venn diagrams to represent sets. Describing sets using rules of inclusion. Set operations.
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 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 informationPark Forest Math Team. Meet #5. Selfstudy Packet
Park Forest Math Team Meet #5 Selfstudy Packet Problem Categories for this Meet: 1. Mystery: Problem solving 2. Geometry: Angle measures in plane figures including supplements and complements 3. Number
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 informationNwheatleyschaller s The Next Step...Conditional Probability
CK12 FOUNDATION Nwheatleyschaller s The Next Step...Conditional Probability Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) Meery To access a customizable version of
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 informationCISC 1400 Discrete Structures
CISC 1400 Discrete Structures Chapter 6 Counting CISC1400 Yanjun Li 1 1 New York Lottery New York Megamillion Jackpot Pick 5 numbers from 1 56, plus a mega ball number from 1 46, you could win biggest
More informationACTIVITY 6.7 Selecting and Rearranging Things
ACTIVITY 6.7 SELECTING AND REARRANGING THINGS 757 OBJECTIVES ACTIVITY 6.7 Selecting and Rearranging Things 1. Determine the number of permutations. 2. Determine the number of combinations. 3. Recognize
More information1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 100 calculators is tested.
1. A factory makes calculators. Over a long period, 2 % of them are found to be faulty. A random sample of 0 calculators is tested. Write down the expected number of faulty calculators in the sample. Find
More 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 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 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 information\\\v?i. EXERCISES Activity a. Determine the complement of event A in the rolladie experiment.
ACTIVITY 6.2 CHOICES 719 11. a. Determine the complement of event A in the rolladie experiment. b. Describe what portion of the Venn diagram above represents the complement of A. SUMMARY Activity 6.2
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 informationMathematics 3201 Test (Unit 3) Probability FORMULAES
Mathematics 3201 Test (Unit 3) robability Name: FORMULAES ( ) A B A A B A B ( A) ( B) ( A B) ( A and B) ( A) ( B) art A : lace the letter corresponding to the correct answer to each of the following in
More 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 informationIntroduction. Firstly however we must look at the Fundamental Principle of Counting (sometimes referred to as the multiplication rule) which states:
Worksheet 4.11 Counting Section 1 Introduction When looking at situations involving counting it is often not practical to count things individually. Instead techniques have been developed to help us count
More informationPermutations and Combinations
Permutations and Combinations In statistics, there are two ways to count or group items. For both permutations and combinations, there are certain requirements that must be met: there can be no repetitions
More informationObjectives: Permutations. Fundamental Counting Principle. Fundamental Counting Principle. Fundamental Counting Principle
and Objectives:! apply fundamental counting principle! compute permutations! compute combinations HL2 Math  Santowski! distinguish permutations vs combinations can be used determine the number of possible
More informationThe Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you $5 that if you give me $10, I ll give you $20.)
The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you $ that if you give me $, I ll give you $2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If
More informationUniversity of Connecticut Department of Mathematics
University of Connecticut Department of Mathematics Math 070Q Exam A Fall 07 Name: TA Name: Discussion: Read This First! This is a closed notes, closed book exam. You cannot receive aid on this exam from
More informationFoundations of Computing Discrete Mathematics Solutions to exercises for week 12
Foundations of Computing Discrete Mathematics Solutions to exercises for week 12 Agata Murawska (agmu@itu.dk) November 13, 2013 Exercise (6.1.2). A multiplechoice test contains 10 questions. There are
More information3.6 Theoretical and Experimental Coin Tosses
wwwck12org Chapter 3 Introduction to Discrete Random Variables 36 Theoretical and Experimental Coin Tosses Here you ll simulate coin tosses using technology to calculate experimental probability Then you
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 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 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 informationCIS 2033 Lecture 6, Spring 2017
CIS 2033 Lecture 6, Spring 2017 Instructor: David Dobor February 2, 2017 In this lecture, we introduce the basic principle of counting, use it to count subsets, permutations, combinations, and partitions,
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 informationTEST A CHAPTER 11, PROBABILITY
TEST A CHAPTER 11, PROBABILITY 1. Two fair dice are rolled. Find the probability that the sum turning up is 9, given that the first die turns up an even number. 2. Two fair dice are rolled. Find the probability
More information101. Combinations. Vocabulary. Lesson. Mental Math. able to compute the number of subsets of size r.
Chapter 10 Lesson 101 Combinations BIG IDEA With a set of n elements, it is often useful to be able to compute the number of subsets of size r Vocabulary combination number of combinations of n things
More information5 Elementary Probability Theory
5 Elementary Probability Theory 5.1 What is Probability? The Basics We begin by defining some terms. Random Experiment: any activity with a random (unpredictable) result that can be measured. Trial: one
More information2. Combinatorics: the systematic study of counting. The Basic Principle of Counting (BPC)
2. Combinatorics: the systematic study of counting The Basic Principle of Counting (BPC) Suppose r experiments will be performed. The 1st has n 1 possible outcomes, for each of these outcomes there are
More informationMath 14 Lecture Notes Ch. 3.6
Math Lecture Notes h... ounting Rules xample : Suppose a lottery game designer wants to list all possible outcomes of the following sequences of events: a. tossing a coin once and rolling a sided die
More informationExam 2 Review (Sections Covered: 3.1, 3.3, , 7.1) 1. Write a system of linear inequalities that describes the shaded region.
Exam 2 Review (Sections Covered: 3.1, 3.3, 6.16.4, 7.1) 1. Write a system of linear inequalities that describes the shaded region. 5x + 2y 30 x + 2y 12 x 0 y 0 2. Write a system of linear inequalities
More informationProbability of Independent and Dependent Events. CCM2 Unit 6: Probability
Probability of Independent and Dependent Events CCM2 Unit 6: Probability Independent and Dependent Events Independent Events: two events are said to be independent when one event has no affect on the probability
More informationFinite Math B, Chapter 8 Test Review Name
Finite Math B, Chapter 8 Test Review Name Evaluate the factorial. 1) 6! A) 720 B) 120 C) 360 D) 1440 Evaluate the permutation. 2) P( 10, 5) A) 10 B) 30,240 C) 1 D) 720 3) P( 12, 8) A) 19,958,400 B) C)
More informationINDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2
INDEPENDENT AND DEPENDENT EVENTS UNIT 6: PROBABILITY DAY 2 WARM UP Students in a mathematics class pick a card from a standard deck of 52 cards, record the suit, and return the card to the deck. The results
More informationThe topic for the third and final major portion of the course is Probability. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Spring 2006 Vazirani Lecture 17 Introduction to Probability The topic for the third and final major portion of the course is Probability. We will aim to make sense of
More informationCombinations and Permutations LongTerm Memory Review Review 1
Review 1 1. A is an arrangement of a set of objects in which order IS important. 2. A is an arrangement of a set of objects in which order IS NOT important.. How do you read?. 4. How do your read C or.
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 informationAlgebra II Chapter 12 Test Review
Sections: Counting Principle Permutations Combinations Probability Name Choose the letter of the term that best matches each statement or phrase. 1. An illustration used to show the total number of A.
More informationGrade 7/8 Math Circles February 25/26, Probability
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Probability Grade 7/8 Math Circles February 25/26, 2014 Probability Centre for Education in Mathematics and Computing Probability is the study of how likely
More informationMATH 1324 (Finite Mathematics or Business Math I) Lecture Notes Author / Copyright: Kevin Pinegar
MATH 1324 Module 4 Notes: Sets, Counting and Probability 4.2 Basic Counting Techniques: Addition and Multiplication Principles What is probability? In layman s terms it is the act of assigning numerical
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 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 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 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 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 informationCSI 23 LECTURE NOTES (Ojakian) Topics 5 and 6: Probability Theory
CSI 23 LECTURE NOTES (Ojakian) Topics 5 and 6: Probability Theory 1. Probability Theory OUTLINE (References: 5.1, 5.2, 6.1, 6.2, 6.3) 2. Compound Events (using Complement, And, Or) 3. Conditional Probability
More informationThe study of probability is concerned with the likelihood of events occurring. Many situations can be analyzed using a simplified model of probability
The study of probability is concerned with the likelihood of events occurring Like combinatorics, the origins of probability theory can be traced back to the study of gambling games Still a popular branch
More information