Finite Mathematics MAT 141: Chapter 8 Notes


 Francis Andrews
 4 years ago
 Views:
Transcription
1 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 principle as counting the branches (combinations) on a tree diagram.
2 Multiplication Principle Example: You have 2 pairs of shoes, 5 pairs of pants and 6 different tops. How many outfits can you make? Multiplication Principle Example: A housing builder provides the following options: Deck or patio. Hardwood, carpet or linoleum Granite, quartz, or compound Expensive landscaping, cheap landscaping or no landscaping. How many different combinations do they offer? Multiplication Principle Example: How many possible 0 digit phone numbers can there be if the first and fourth digit cannot be a zero? Multiplication Principle (Dependence) Example: For the Iowa Lottery one must choose 5 out of 59 numbers for the white balls; then choose out of 35 numbers for the Powerball. How many possible tickets are there? 2
3 Multiplication Principle (Dependence) Example: You have 6 books and want to place them on a shelf. In how many ways can you organize the books? Factorial! As you saw in the last few examples we often end up multiply successive terms until we reach one. There is a mathematical notation for this. Factorial! Example: 5 people on a committee need to be lined up for a photo. In how many ways can this be done? Factorial! Example: 42 cars are in a race. How many possible finishing orders can there be? 3
4 Permutations Permutations Let s consider the last example. What if we only cared about the possible top 5 finishing orders? Go to math and arrow over to PRB. Permutations Example: You have 6 books and want to place 4 of them on a shelf. In how many ways can you organize the books? Permutations Example: There are 5 people on a committee. How many ways can they select a president, vice president, and treasurer? 4
5 MAT 4  Chapter 8 Permutations Example 8..0: A television talk show will include 4 men and 5 women as panelists. (a) In how many ways can the panelists be seated? (b) In many ways can they be seated if they need to alternate? Repetition in Permutations Suppose you are asked to arrange some objects but the objects are indistinguishable. For example, arranging the numbers, 2, 2,,. You cannot tell the ones apart or the twos apart. If you have to arrange objects where of them repeat, of them repeat, of them repeat, Then you can calculate the number of arrangements as!!!! (c) In how many ways can they be seated if the men and women must be seated together? Permutations (Repetitions) Example: In how many ways can you arrange the letters of the word Mississippi? Permutations (Repetitions) Recall when were taking marbles out of the bag and I kept saying, in that order. Now we don t need to worry about the order. Example: A bag contains 5 blue, 4 black, 6 yellow, and 5 white marbles. If two blues and a black marble were drawn, how many ways can they be arranged? 5
6 Permutations (Repetitions) Example: You selected a full house with 3 tens and 2 aces. In how many ways could the cards have been drawn? License Plates Example: For many years, the State of California used 3 letters followed by 3 digits on its automobile license plates. (a) How many different license plates are possible with this arrangement? (b) When the state ran out of new numbers, the order was reversed to 3 digits followed by 3 letters. How many new license plate numbers were then possible? (c) Several years ago. the numbers described in b were also used up. The stale then issued plates with I letter followed by 3 digits and then 3 letters. How many new license plate numbers will this provide? Combination
7 Combination versus Permutation A permutation is when order matters. Lining people up. Selecting people from a group and assigning them positions. A combination is when order does not matter. Selecting cards Picking people for a team. vs. Permutations Example: For each problem, tell whether permutations or combinations should be used to solve the problem. (a) How many 4digit code numbers arc possible if no digits are repeated? (b) A sample of 3 light bulbs is randomly selected from a batch of 5. How many different samples are possible? (c) In a baseball conference with 8 teams, how many games must be played so that each team plays every other team exactly once? (d) In how many ways can 4 patients be assigned to 6 different hospital rooms so that each patient has a private room? Locker Combination If you have a locker combination does the order matter? Example: There are 5 people in a class. How many ways can 4 of them be selected to work on a project together? 7
8 Example: There are 52 cards in a standard deck of cards. In how many ways can you select 5 cards? Example: There are 20 people in Club West. Five students must be selected to work on an activity. (a) How many ways can this be done? (b) How many ways can this be done if Joe must be in the group? (c) What if the question was change to at most 3 people need to selected for the group? Example: In how many ways can you select a full house with tens over aces? Example: When selecting a five card hand: (a) How many possible hands are there with two hearts? (b) How many possible hands are there with all face cards? (c) How many possible hands are there where all 5 cards are of the same suite? 8
9 Example: How many possible full houses are there? Example: The senate contains 5 democrats, 47 republicans, and independent. How many ways can a committee be formed if it must contain 3 democrats and 2 republicans? Recall that probability can be calculated as = Applications of Counting Principles = () () For more complicated situations we can use the multiplication principle, combinations, and permutations to calculate probability. 9
10 Example: There are 52 cards in a standard deck of cards. What is the probability of getting four of a kind? : Example : What is the probability of being dealt a full house? : Using the Multiplication Principle: Using the Multiplication Principle: Example : There are 20 people in Club West, 2 of which are women. Five students must be selected to work on an activity. If the people were randomly chosen, (a) What is the probability of there being 3 women and 2 men? Example: The senate contains 5 democrats, 47 republicans, and independent. A committee of 0 must be formed. If the people were randomly chosen, (a) What is the probability of there being 3 democrats, 6 republicans, and independent? (b) What is the probability of there being all women? (b) What is the probability of there being 9 democrats, and republican? 0
11 Example: Find the probability of each. (a) A 5card hand with two hearts? We don t need no stinkn order! Remember when we did probability before and I kept saying, in that order. (b) A 5card hand with all face cards? (c) How many possible hands are there where all 5 cards are of the same suite? We don t need it any more. Example: A bag of marbles contains 3 red, 5 blue, 7 yellow, and 5 green marbles. If 3 are drawn, what is the probability of (a) Drawing 3 marbles and they are yellow, green, and blue? Example: The Environmental Protection Agency is considering inspecting 6 plants for environmental compliance: 3 in Chicago, 2 in Los Angeles, and in New York. Due to a lack of inspectors, they decide to inspect 2 plants selected at random, this month and next month, with each plant equally likely to be selected, but no plant is selected twice. What is the probability that Chicago plant and Los Angeles plant are selected? (b) Drawing or more green marbles?
12 Example: From a group of 50 accountants at a firm 5 are selected to write a review of management policies. (a) How many possible groups of 5 can be made? Example: During the 988 college football season, the Big Eight conference ended the season with a perfect progression. (a) How many games did the 8 teams play? (b) Assuming no ties, how many different outcomes are there for all the games together? (b) What is the probability that Jessica Smith, one of the accountants, is selected? (c) In how many ways could the eight teams end in a perfect progression? (d) Assuming each team had an equal chance of winning, what is the probability of a perfect progression. Definition Binomial Flipping a coin. Do a behavior, don t do a behavior. Faulty, not faulty. 2
13 MAT 4  Chapter 8 Binomial Formula Example: You flip a coin 0 times. What is the probability of getting 4 heads? You flip a coin 3 times. What is the probability that you get two heads?, Here are the possible outcomes. The probability could be thought of as , Example: Over a long period of time it has been observed that a given rifleman can hit a target on a single trial with probability equal to.8. Suppose that he has four shots at a target: Example: A bag contains 4 red, 5 green, 7 blue, and 4 white marbles. (a) What is the probability of drawing 0 marbles and three of them are blue? (a) What is the probability that he will hit it twice? (b) What is the probability that he will hit it at least twice? (b) What is the probability of drawing 0 marbles and at least four of them are red? 3
14 Example: A company produces light bulbs. It has found that out of every 0,000 is faulty. If a sample of 30 light bulbs is taken what it the probability that at least one is faulty? Distributions; Expected Value Making a Distribution A probability distribution represents the probabilities of all possible simple events. To make a probability distribution, do the following: Step : List all possible simple outcomes. A Distribution Example 8.5.: All possible outcomes and a probability distribution for the sum when two dice are rolled are shown below. Possible outcomes Step 2: Step 3: Identify outcomes that represent the same event and determine the probability of each event. Make a table listing each event and probability. 4
15 A Distribution Example: Create a probability distribution for the experiment of flipping 3 coins. Random Variable Event 3 Heads H, 2 T 2H, T 3 Tails Outcomes* *Technically a probability distribution would not include an outcomes row. We include it here to organize our work. Any experiment where we can assign numerical values. Rather than probability of 3 heads in 5 tosses we analyze the number of heads in 5 tosses. This could take on the values of 0,, 2, 3, 4, 5 The number of heads is a random variable. Looking at all the values of a random variable we can calculate the probability of each value and create a probability distribution. A Distribution Example 8.5.3: Create a probability distribution for the random variable of the number of heads in 5 tosses. # Heads 0 Heads Heads 2 Heads 3 Heads 4 Heads 5 Heads To calculate these use the binomial probability. 0 Head: 5 0 (.5) (.5) = Head: 5 4 (.5) (.5) = Distribution (Histogram) Distribution for Number 5 Tosses Head: 5 (.5) (.5) = Head: 5 2 (.5) (.5) = Head: 5 3 (.5) (.5) = Head: 5 5 (.5) (.5) = Number of Heads 5
16 MAT 4  Chapter 8 Distribution (Histogram) Here is the distribution and histogram if we had 2 tosses. Expected Value Consider two or more events, each with its own value and probability. = ( ) ( 2 ) The expected value can be thought of as the theoretical result if you averaged a HUGE number of trials of an event. Expected Value Book formula. Expected Value, Insurance Example: Suppose an automobile insurance company sells an insurance policy with an annual premium of $200. Based on data from past claims, the company has calculated the following probabilities: An average of in 50 policyholders will file a claim of $2,000. An average of in 20 policyholders will file a claim of $,000. An average of in 0 policyholders will file a claim of $500. Assuming that the policyholder could file any of the claims above, what is the expected value to the company for each policy sold? = $200 $2000 $000 $500 = $ *It is $200 to buy a policy and you must buy one, hence a probability of, if you are going to make a claim. Also, as this is money coming into the company it is positive. The other amounts are claims, which is money going out and therefore negatives. The result tells the company that they expect to make $60, on average, for every $200 policy that they sell. Slide
17 Expected Value, Insurance What if the company in the last example sold 200,000 such policies? Expected Value, Coin Example: Let s play a game. We flip a coin. If it is heads I pay you $2 and if it is tails you pay me $3. What is the expected value (from your perspective)? Expected Value, Deal or No Deal? Example: You are playing Deal or No Deal?. The following cases are left. The banker offers you $278,000. Should you take it? Expected Value (Raffle) Example: A group is raising money by selling raffle tickets for $5 each. The prizes are $2500 vacation package $500 local wine collection 4 $00 gift cards to a restaurant It is known that a 2,000 tickets were sold. Create a probability distribution. Prize $2495 $495 $95 $5 /2000 /2000 4/ /2000 7
18 Raffle Expected Value What is the expected value of one ticket? Example: Through the season we see that Sue is a 65% freethrow shooter. What is the expected value if she shot 8 freethrows in last nights game? 8
4.1 Sample Spaces and Events
4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an
More 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 informationFind the probability of an event by using the definition of probability
LESSON 101 Probability Lesson Objectives Find the probability of an event by using the definition of probability Vocabulary experiment (p. 522) trial (p. 522) outcome (p. 522) sample space (p. 522) event
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 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 informationFunctional Skills Mathematics
Functional Skills Mathematics Level Learning Resource Probability D/L. Contents Independent Events D/L. Page  Combined Events D/L. Page  9 West Nottinghamshire College D/L. Information Independent Events
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 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 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 informationSTATISTICAL COUNTING TECHNIQUES
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
More informationEx 1: A coin is flipped. Heads, you win $1. Tails, you lose $1. What is the expected value of this game?
AFM Unit 7 Day 5 Notes Expected Value and Fairness Name Date Expected Value: the weighted average of possible values of a random variable, with weights given by their respective theoretical probabilities.
More informationWEEK 11 REVIEW ( and )
Math 141 Review 1 (c) 2014 J.L. Epstein WEEK 11 REVIEW (7.5 7.6 and 8.1 8.2) Conditional Probability (7.5 7.6) P E F is the probability of event E occurring given that event F has occurred. Notation: (
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationUnit 5, Activity 1, The Counting Principle
Unit 5, Activity 1, The Counting Principle Directions: With a partner find the answer to the following problems. 1. A person buys 3 different shirts (Green, Blue, and Red) and two different pants (Khaki
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 informationThe next several lectures will be concerned with probability theory. We will aim to make sense of statements such as the following:
CS 70 Discrete Mathematics for CS Fall 2004 Rao Lecture 14 Introduction to Probability The next several lectures will be concerned with probability theory. We will aim to make sense of statements such
More informationWeek in Review #5 ( , 3.1)
Math 166 WeekinReview  S. Nite 10/6/2012 Page 1 of 5 Week in Review #5 (2.32.4, 3.1) n( E) In general, the probability of an event is P ( E) =. n( S) Distinguishable Permutations Given a set of n objects
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 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 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 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 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 informationCOMPOUND EVENTS. Judo Math Inc.
COMPOUND EVENTS Judo Math Inc. 7 th grade Statistics Discipline: Black Belt Training Order of Mastery: Compound Events 1. What are compound events? 2. Using organized Lists (7SP8) 3. Using tables (7SP8)
More informationMath 1070 Sample Exam 2
University of Connecticut Department of Mathematics Math 1070 Sample Exam 2 Exam 2 will cover sections 4.6, 4.7, 5.2, 5.3, 5.4, 6.1, 6.2, 6.3, 6.4, F.1, F.2, F.3 and F.4. This sample exam is intended to
More informationUnit Nine Precalculus Practice Test Probability & Statistics. Name: Period: Date: NONCALCULATOR SECTION
Name: Period: Date: NONCALCULATOR SECTION Vocabulary: Define each word and give an example. 1. discrete mathematics 2. dependent outcomes 3. series Short Answer: 4. Describe when to use a combination.
More informationMath 1342 Exam 2 Review
Math 1342 Exam 2 Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) If a sportscaster makes an educated guess as to how well a team will do this
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 informationUnit 14 Probability. Target 3 Calculate the probability of independent and dependent events (compound) AND/THEN statements
Target 1 Calculate the probability of an event Unit 14 Probability Target 2 Calculate a sample space 14.2a Tree Diagrams, Factorials, and Permutations 14.2b Combinations Target 3 Calculate the probability
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 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 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 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 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 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 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 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 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 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 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 informationMATH STUDENT BOOK. 8th Grade Unit 10
MATH STUDENT BOOK 8th Grade Unit 10 Math 810 Probability Introduction 3 1. Outcomes 5 Tree Diagrams and the Counting Principle 5 Permutations 12 Combinations 17 Mixed Review of Outcomes 22 SELF TEST 1:
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 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 informationSection 5.4 Permutations and Combinations
Section 5.4 Permutations and Combinations Definition: nfactorial For any natural number n, n! n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to
More 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 informationProbability Test Review Math 2. a. What is? b. What is? c. ( ) d. ( )
Probability Test Review Math 2 Name 1. Use the following venn diagram to answer the question: Event A: Odd Numbers Event B: Numbers greater than 10 a. What is? b. What is? c. ( ) d. ( ) 2. In Jason's homeroom
More 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 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 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 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 informationSection 5.4 Permutations and Combinations
Section 5.4 Permutations and Combinations Definition: nfactorial For any natural number n, n! = n( n 1)( n 2) 3 2 1. 0! = 1 A combination of a set is arranging the elements of the set without regard to
More 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 informationMath 1101 Combinations Handout #17
Math 1101 Combinations Handout #17 1. Compute the following: (a) C(8, 4) (b) C(17, 3) (c) C(20, 5) 2. In the lottery game Megabucks, it used to be that a person chose 6 out of 36 numbers. The order of
More information13 Probability CHAPTER. Chapter Outline. Chapter 13. Probability
Chapter 13 www.ck12.org Chapter 13. Probability CHAPTER 13 Probability Chapter Outline 13.1 INTRODUCTION TO PROBABILITY 13.2 PERMUTATIONS AND COMBINATIONS 13.3 THE FUNDAMENTAL COUNTING PRINCIPLE 13.4 THE
More informationWeek 1: Probability models and counting
Week 1: Probability models and counting Part 1: Probability model Probability theory is the mathematical toolbox to describe phenomena or experiments where randomness occur. To have a probability model
More informationMATH STUDENT BOOK. 7th Grade Unit 6
MATH STUDENT BOOK 7th Grade Unit 6 Unit 6 Probability and Graphing Math 706 Probability and Graphing Introduction 3 1. Probability 5 Theoretical Probability 5 Experimental Probability 13 Sample Space 20
More informationLAMC Junior Circle February 3, Oleg Gleizer. Warmup
LAMC Junior Circle February 3, 2013 Oleg Gleizer oleg1140@gmail.com Warmup Problem 1 Compute the following. 2 3 ( 4) + 6 2 Problem 2 Can the value of a fraction increase, if we add one to the numerator
More informationProbability of Independent Events. If A and B are independent events, then the probability that both A and B occur is: P(A and B) 5 P(A) p P(B)
10.5 a.1, a.5 TEKS Find Probabilities of Independent and Dependent Events Before You found probabilities of compound events. Now You will examine independent and dependent events. Why? So you can formulate
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 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 information1. The chance of getting a flush in a 5card poker hand is about 2 in 1000.
CS 70 Discrete Mathematics for CS Spring 2008 David Wagner Note 15 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette wheels. Today
More informationSection A Calculating Probabilities & Listing Outcomes Grade F D
Name: Teacher Assessment Section A Calculating Probabilities & Listing Outcomes Grade F D 1. A fair ordinary sixsided dice is thrown once. The boxes show some of the possible outcomes. Draw a line from
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 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 information, x {1, 2, k}, where k > 0. (a) Write down P(X = 2). (1) (b) Show that k = 3. (4) Find E(X). (2) (Total 7 marks)
1. The probability distribution of a discrete random variable X is given by 2 x P(X = x) = 14, x {1, 2, k}, where k > 0. Write down P(X = 2). (1) Show that k = 3. Find E(X). (Total 7 marks) 2. In a game
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 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. 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 informationUsing a table: regular fine micro. red. green. The number of pens possible is the number of cells in the table: 3 2.
Counting Methods: Example: A pen has tip options of regular tip, fine tip, or micro tip, and it has ink color options of red ink or green ink. How many different pens are possible? Using a table: regular
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 informationBell Work. WarmUp Exercises. Two sixsided dice are rolled. Find the probability of each sum or 7
WarmUp Exercises Two sixsided dice are rolled. Find the probability of each sum. 1. 7 Bell Work 2. 5 or 7 3. You toss a coin 3 times. What is the probability of getting 3 heads? WarmUp Notes Exercises
More information* Order Matters For Permutations * Section 4.6 Permutations MDM4U Jensen. Part 1: Factorial Investigation
Section 4.6 Permutations MDM4U Jensen Part 1: Factorial Investigation You are trying to put three children, represented by A, B, and C, in a line for a game. How many different orders are possible? a)
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 informationa) 2, 4, 8, 14, 22, b) 1, 5, 6, 10, 11, c) 3, 9, 21, 39, 63, d) 3, 0, 6, 15, 27, e) 3, 8, 13, 18, 23,
Prealculus Midterm Exam Review Name:. Which of the following is an arithmetic sequence?,, 8,,, b),, 6, 0,, c), 9,, 9, 6, d), 0, 6,, 7, e), 8,, 8,,. What is a rule for the nth term of the arithmetic sequence
More informationNovember 11, Chapter 8: Probability: The Mathematics of Chance
Chapter 8: Probability: The Mathematics of Chance November 11, 2013 Last Time Probability Models and Rules Discrete Probability Models Equally Likely Outcomes Probability Rules Probability Rules Rule 1.
More informationQ1) 6 boys and 6 girls are seated in a row. What is the probability that all the 6 gurls are together.
Required Probability = where Q1) 6 boys and 6 girls are seated in a row. What is the probability that all the 6 gurls are together. Solution: As girls are always together so they are considered as a group.
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 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 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 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 12 Academic Assignment 9: Probability Outcomes: B8, G1, G2, G3, G4, G7, G8
Math 12 Academic Assignment 9: Probability Outcomes: B8, G1, G2, G3, G4, G7, G8 Name: 45 1. A customer chooses 5 or 6 tapes from a bin of 40. What is the expression that gives the total number of possibilities?
More informationMaking Predictions with Theoretical Probability
? LESSON 6.3 Making Predictions with Theoretical Probability ESSENTIAL QUESTION Proportionality 7.6.H Solve problems using qualitative and quantitative predictions and comparisons from simple experiments.
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 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 information