8.3 Probability Applications of Counting Principles


 Quentin Hunt
 2 years ago
 Views:
Transcription
1 8. Probability Applications of Counting Principles In this section, we will see how we can apply the counting principles from the previous two sections in solving probability problems. Many of the probability problems involving dependent events that were solved earlier by using tree diagrams can also be solved by using permutations and combinations. Permutations and combinations are especially helpful when the numbers involved are large. To compare the method of using permutations and combinations with the method of tree diagrams used in Section 7.5, the first example repeats Example 8 from that section. (It is a good idea to review that example from section 7.5 before/after going through the following example.) Example. The Environmental Protection Agency is considering inspecting 6 plants for environmental compliance: in Chicago, in Los Angeles, and in New York. Due to a lack of inspectors, they decide to inspect two plants selected at random, one this month and one next month, with each plant equally likely to be selected, but no plant selected twice. What is the probability that Chicago plant and Los Angeles plant are selected? Solution. We note that, although the plants are selected one at a time, with one labeled as the first plant and the other as the second, the probability that Chicago plant and Los Angeles plant are selected should not depend upon the order in which the plants are selected (the outcomes, Chicago followed by Los Angeles, and Los Angeles followed by Chicago are both contained in the required event). So, we may use combinations. The number of ways to select Chicago plant out of Chicago plants and Los Angeles plant out of Los Angeles plants is ( ) ( ) = = 6. The number of ways to select any plants out of 6 is ( ) 6 = 6!!! = 6 5!! = 6 5 = 5. Thus, the probability that Chicago plant and Los Angeles plant are selected is This agrees with the answer found earlier. 6 5 = 5. Example. From a group of nurses, are to be selected to present a list of grievances to management. (a) In how many ways can this be done? Solution. nurses can be selected from a group of in combinations, since the group of is an unordered set.) ( ) =! 0 9 8! = = 8!! 8! There are 75 ways to choose nurses from. ( ) 0 9 ways. = 75. (We use Fall 00 Page Penn State University
2 (b) One of the nurses is Julie Davis. Find the probability that Davis will be among the selected. Solution. The probability that Davis will be selected is given by n(e) n(s), where E is the event that the chosen group includes Davis, and S is the sample space for the experiment of choosing a group of nurses. ( ) There is only = way to choose Davis. The number of ways that the other nurses can be chosen from the remaining nurses is ( ) =! 0 9 = = 0. 8!! Therefore, the probability that Davis will be one of the chosen is ( ) ( ) P (Davis is chosen) = P (E) = n(e) n(s) = ( ) = Notice that the two numbers in red in the numerator, and, add up to the number in red in the denominator,. This indicates that nurses have been split into two groups, one of size (Davis) and the other of size (the other nurses). Similarly, the green numbers indicate that the nurses chosen consist of two groups of size (Davis) and size (the other nurses chosen). (c) Find the probability that Davis will not be selected. Solution. The probability that Davis will not be chosen is P (E) = 0.88 = Remark. The problems in the previous two sections 8. and 8. of this chapter asked how many ways a certain operation can be done. The problems in this section ask what is the probability that a certain event occurs; the solution involves answering questions about how many ways the event and the operation can be done. If a problem asks how many ways something can be done, the answer must be a nonnegative integer. If a problem asks for a probability, the answer must be a number between 0 and. Example. When shipping diesel engines abroad, it is common to pack engines in one container that is then loaded on a rail car and sent to a port. Suppose that a company has received complaints from its customers that many of the engines arrive in nonworking condition. To help solve this problem, the company decides to make a spot check of containers after loading. The company will test engines from the container at random; if any of the are nonworking, the container will not be shipped until each engine in it is checked. Suppose that a given container has nonworking engines. Find the probability that container will not be shipped. Fall 00 Page Penn State University
3 Solution. The container will not be shipped if the sample of engines contains at least defective engine, that is, or defective engines (note that the container contains only defective engines). If P ( defective) represents that probability of exactly defective engine in the sample, then There are P (not shipping) = P ( defective) + P ( defective). ( ) ways to choose the engines for testing: ( ) ( ) =! 9!! = 0. There are ways of choosing defective engine from the in the container, and for each ( ) 0 of these ways there are ways of choosing good engines from among the 0 good engines in the container. By the multiplication principle, there are ( ) ( ) 0 =!!! 0! = 5 = 90 8!! ways to choose a sample of engines containing defective engine. Thus, ( ) There are ( ) 0 container, and P ( defective) = 90 0 = 9. ways of choosing defective engines from the defective engines in the there are ( ) ways of choosing good engine from among the 0 good engines. Thus ( ) 0 = 0 = 0 ways of choosing a sample of engines containing defective engines. So, Finally, P ( defective) = 0 0 =. P (not shipping) = P ( defective) + P ( defective) = ( ) ( ) 0 ( ) + ( ) ( ) 0 ) ( = 9 + = Fall 00 Page Penn State University
4 Remark. Observe that in Example, the complement of finding or defective engines is finding 0 defective engines. Then instead of finding the sum P ( defective) + P ( defective), the result in Example could be found as P (0 defective). P (not shipping) = P (0 defective) = = 0 0 ( ) ( ) 0 0 ( ) = 0 0 = Example. In a common form of the card game poker, a hand of 5 cards is dealt to each player from a deck of 5 cards. There are a total of ( ) 5 = 5! =, 598, ! 5! such hands possible. Find the probability of getting each of the following hands. (a) A hand containing only hearts, called a heart flush. Solution. There are hearts, and 5 = 9 other cards in a deck. Thus there are ( ) ( ) 9 =! 0 9 = = ! 5! 5 different hands containing only hearts. Hence, the probability of a heart flush is ( ) ( ) P (heart flush) = ( ) = 5, 598, (b) A flush of any suit (5 cards of the same suit). Solution. There are suits in a deck, so P (flush) = P (heart flush) = (c) A full house of aces and eights ( aces and eights). Fall 00 Page Penn State University
5 ( ) Solution. There are ways to choose aces from among the in the deck, and ( ) ways to choose eights out of the in the deck. Thus, ( ) ( ) ( ) 0 P ( aces, eights) = ( ) = 6 5, 598, (d) Any full house ( cards of one value, of another). Solution. There are values in a deck King, Queen, Jack, 0, 9, 8, 7, 6, 5,, (,, ) Ace, and cards of each value. Thus, there are choices for the first value, and ways to choose cards from among the cards that have that value. This leaves choices for the second value (order is important here, since a full house of aces, and eights is not the same as a full house of eights, ( and ) aces). From the cards that have the second value, cards can be chosen in ways. The probability of any full house is then ( ) ( ) P (full house) = 0.00., 598, 960 Example 5. A music teacher has violin pupils, Fred, Carl, and Helen. For a recital, the teacher selects a first violinist and a second violinist. The third pupil will play with the others, but not solo. If the teacher selects randomly, what is the probability that Helen is first violinist, Carl is second violinist, and Fred does not solo? Solution. We use permutations to find the number of arrangements in the sample space. P (, ) =! = 6 (We can think of this as filling the positions of the first violin, second violin, and no solo.) The 6 arrangements are equally likely, since the teacher will select randomly. Now there is only one arrangement where Helen is first violinist, Card is second violinist, and Fred does not solo. Thus, the required probability is 6. Example 6. Suppose a group of n people is in a room. Find the probability that at least of the people have the same birthday. Solution. Same birthday refers to the month and the day, not necessarily the same year. Also, we ignore leap years, and assume that each day in the year is equally likely as a birthday. To see how to proceed, we look at the case in which n = 5 and find the probability Fall 00 Page 5 Penn State University
6 that no people from among the 5 people have the same birthday. There are 65 different birthdays possible for the first of the 5 people, 6 for the second (so that the people have different birthdays), 6 for the third, and so on. The number of ways that 5 people can have different birthdays is thus the number of permutations of 65 days taken 5 at a time or P (65, 5) = The number of ways that 5 people can have the same or different birthdays is = (65) 5. Finally, the probability that none of the 5 people have the same birthday is P (65, 5) 65 5 = Thus, the probability that at least of the 5 people do have the same birthday is = Now this result can be extended to more than 5 people. Generalizing, the probability that no people among n people have the same birthday is P (65, n) 65 n. Therefore, the probability that at least of the n people do have the same birthday is P (65, n) 65 n. The following table shows this probability for various values of n. Number of People, Probability That Two n Have the Same Birthday > 65 The probability that people among have the same birthday is 0.507, a little over half. This is quite surprising! Fall 00 Page 6 Penn State University
7 Example 7. Ray and Nate are arranging a row of fruit at random on a table. They have 5 apples, 6 oranges, and 7 lemons. What is the probability that all fruit of the same kind are together? Solution. Method : Ray can t tell individual pieces of fruit of the same kind apart. All apples look the same to him, as do all oranges and all lemons. So, in the denominator of the probability, he calculates the number of distinguishable ways to arrange the = 8 pieces of fruit, given that all apples are indistinguishable, as are all oranges and all lemons. 8! 5! 6! 7! =, 70, 688 As for the numerator, the only choice is how to arrange the kinds of fruit, for which there are! = 6 ways. Thus P (all fruit of the same kind are together) = 6, 70, 688 = Method : Nate has better eyesight than Ray and can tell the individual pieces of fruit apart. So in the denominator of the probability, he calculates the number of ways to arrange the 8 pieces of fruit, which is 8! = For the numerator he must choose how to arrange the kinds of fruit, for which there are! ways. Then there are 5! ways to arrange the apples, 6! ways to arrange the oranges, and 7! ways to arrange the lemons, for a total number of possibilities of Therefore,! 5! 6! 7! =, 6, 76, 000. P (all fruit of the same kind are together) =, 6, 76, = The results for Method and Method are the same. The probability does not depend on whether a person can distinguish individual pieces of the same kind of fruit. Fall 00 Page 7 Penn State University
(b) You draw two balls from an urn and track the colors. When you start, it contains three blue balls and one red ball.
Examples for Chapter 3 Probability Math 10401 Section 3.1 1. Draw a tree diagram for each of the following situations. State the size of the sample space. (a) You flip a coin three times. (b) You draw
More informationJan 17 Homework Solutions Math 151, Winter 2012. Chapter 2 Problems (pages 5054)
Jan 17 Homework Solutions Math 11, Winter 01 Chapter Problems (pages 0 Problem In an experiment, a die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample
More informationIf we know that LeBron s next field goal attempt will be made in a game after 3 days or more rest, it would be natural to use the statistic
Section 7.4: Conditional Probability and Tree Diagrams Sometimes our computation of the probability of an event is changed by the knowledge that a related event has occurred (or is guaranteed to occur)
More information33 Probability: Some Basic Terms
33 Probability: Some Basic Terms In this and the coming sections we discuss the fundamental concepts of probability at a level at which no previous exposure to the topic is assumed. Probability has been
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 information2.5 Conditional Probabilities and 2Way Tables
2.5 Conditional Probabilities and 2Way Tables Learning Objectives Understand how to calculate conditional probabilities Understand how to calculate probabilities using a contingency or 2way table It
More informationChapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.
MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment.
More informationFind the indicated probability. 1) If a single fair die is rolled, find the probability of a 4 given that the number rolled is odd.
Math 0 Practice Test 3 Fall 2009 Covers 7.5, 8.8.3 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated probability. ) If a single
More informationWhat is the probability of throwing a fair die and receiving a six? Introduction to Probability. Basic Concepts
Basic Concepts Introduction to Probability A probability experiment is any experiment whose outcomes relies purely on chance (e.g. throwing a die). It has several possible outcomes, collectively called
More informationHoover High School Math League. Counting and Probability
Hoover High School Math League Counting and Probability Problems. At a sandwich shop there are 2 kinds of bread, 5 kinds of cold cuts, 3 kinds of cheese, and 2 kinds of dressing. How many different sandwiches
More informationSome special discrete probability distributions
University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Some special discrete probability distributions Bernoulli random variable: It is a variable that
More informationPERMUTATIONS AND COMBINATIONS
PERMUTATIONS AND COMBINATIONS Mathematics for Elementary Teachers: A Conceptual Approach New Material for the Eighth Edition Albert B. Bennett, Jr., Laurie J. Burton and L. Ted Nelson Math 212 Extra Credit
More informationSTAT 319 Probability and Statistics For Engineers PROBABILITY. Engineering College, Hail University, Saudi Arabia
STAT 319 robability and Statistics For Engineers LECTURE 03 ROAILITY Engineering College, Hail University, Saudi Arabia Overview robability is the study of random events. The probability, or chance, that
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 7, due Wedneday, March 14 Happy Pi Day! (If any errors are spotted, please email them to morrison at math dot berkeley dot edu..5.10 A croissant
More information9.2 The Multiplication Principle, Permutations, and Combinations
9.2 The Multiplication Principle, Permutations, and Combinations Counting plays a major role in probability. In this section we shall look at special types of counting problems and develop general formulas
More informationAn approach to Calculus of Probabilities through real situations
MaMaEuSch Management Mathematics for European Schools http://www.mathematik.unikl.de/ mamaeusch An approach to Calculus of Probabilities through real situations Paula Lagares Barreiro Federico Perea RojasMarcos
More informationExam. Name. How many distinguishable permutations of letters are possible in the word? 1) CRITICS
Exam Name How many distinguishable permutations of letters are possible in the word? 1) CRITICS 2) GIGGLE An order of award presentations has been devised for seven people: Jeff, Karen, Lyle, Maria, Norm,
More informationUSING PLAYING CARDS IN THE CLASSROOM. What s the Pattern Goal: To create and describe a pattern. Closest to 24!
USING PLAYING CARDS IN THE CLASSROOM You can purchase a deck of oversized playing cards at: www.oame.on.ca PRODUCT DESCRIPTION: Cards are 4 ½ by 7. Casino quality paperboard stock, plastic coated. The
More informationDetermine the empirical probability that a person selected at random from the 1000 surveyed uses Mastercard.
Math 120 Practice Exam II Name You must show work for credit. 1) A pair of fair dice is rolled 50 times and the sum of the dots on the faces is noted. Outcome 2 4 5 6 7 8 9 10 11 12 Frequency 6 8 8 1 5
More informationAP Stats  Probability Review
AP Stats  Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose
More informationCombinatorial Proofs
Combinatorial Proofs Two Counting Principles Some proofs concerning finite sets involve counting the number of elements of the sets, so we will look at the basics of counting. Addition Principle: If A
More informationREPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.
REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game
More informationSection 2.1. Tree Diagrams
Section 2.1 Tree Diagrams Example 2.1 Problem For the resistors of Example 1.16, we used A to denote the event that a randomly chosen resistor is within 50 Ω of the nominal value. This could mean acceptable.
More informationA (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes.
Chapter 7 Probability 7.1 Experiments, Sample Spaces, and Events A (random) experiment is an activity with observable results. The sample space S of an experiment is the set of all outcomes. Each outcome
More informationChapter 3: The basic concepts of probability
Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording
More informationSpring 2007 Math 510 Hints for practice problems
Spring 2007 Math 510 Hints for practice problems Section 1 Imagine a prison consisting of 4 cells arranged like the squares of an chessboard There are doors between all adjacent cells A prisoner in one
More information1 Combinations, Permutations, and Elementary Probability
1 Combinations, Permutations, and Elementary Probability Roughly speaking, Permutations are ways of grouping things where the order is important. Combinations are ways of grouping things where the order
More information94 Counting Solutions for Chapter 3. Section 3.2
94 Counting 3.11 Solutions for Chapter 3 Section 3.2 1. Consider lists made from the letters T, H, E, O, R, Y, with repetition allowed. (a How many length4 lists are there? Answer: 6 6 6 6 = 1296. (b
More informationTopic 1 Probability spaces
CSE 103: Probability and statistics Fall 2010 Topic 1 Probability spaces 1.1 Definition In order to properly understand a statement like the chance of getting a flush in fivecard poker is about 0.2%,
More informationProbability Theory, Part 4: Estimating Probabilities from Finite Universes
8 Resampling: The New Statistics CHAPTER 8 Probability Theory, Part 4: Estimating Probabilities from Finite Universes Introduction Some Building Block Programs Problems in Finite Universes Summary Introduction
More informationCombinations If 5 sprinters compete in a race, how many different ways can the medals for first, second and third place, be awarded
Combinations If 5 sprinters compete in a race, how many different ways can the medals for first, second and third place, be awarded If 5 sprinters compete in a race and the fastest 3 qualify for the relay
More informationDISCRETE RANDOM VARIABLES
DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES Documents prepared for use in course B01.1305, New York University, Stern School of Business Definitions page 3 Discrete random variables are introduced
More informationSection 6.4: Counting Subsets of a Set: Combinations
Section 6.4: Counting Subsets of a Set: Combinations In section 6.2, we learnt how to count the number of rpermutations from an nelement set (recall that an rpermutation is an ordered selection of r
More informationPROBABILITY 14.3. section. The Probability of an Event
4.3 Probability (43) 727 4.3 PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques
More informationProbability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.
1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event
More informationLecture 2 : Basics of Probability Theory
Lecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different
More informationGrade 7/8 Math Circles Fall 2012 Probability
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Grade 7/8 Math Circles Fall 2012 Probability Probability is one of the most prominent uses of mathematics
More informationLaws of probability. Information sheet. Mutually exclusive events
Laws of probability In this activity you will use the laws of probability to solve problems involving mutually exclusive and independent events. You will also use probability tree diagrams to help you
More informationExample: If we roll a dice and flip a coin, how many outcomes are possible?
12.5 Tree Diagrams Sample space Sample point Counting principle Example: If we roll a dice and flip a coin, how many outcomes are possible? TREE DIAGRAM EXAMPLE: Use a tree diagram to show all the possible
More informationHomework 6 (due November 4, 2009)
Homework 6 (due November 4, 2009 Problem 1. On average, how many independent games of poker are required until a preassigned player is dealt a straight? Here we define a straight to be cards of consecutive
More informationLecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett
Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.
More informationPigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
More informationProbability definitions
Probability definitions 1. Probability of an event = chance that the event will occur. 2. Experiment = any action or process that generates observations. In some contexts, we speak of a datagenerating
More informationPROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA
PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet
More informationContemporary Mathematics MAT 130. Probability. a) What is the probability of obtaining a number less than 4?
Contemporary Mathematics MAT 30 Solve the following problems:. A fair die is tossed. What is the probability of obtaining a number less than 4? What is the probability of obtaining a number less than
More informationCS 341 Software Design Homework 5 Identifying Classes, UML Diagrams Due: Oct. 22, 11:30 PM
CS 341 Software Design Homework 5 Identifying Classes, UML Diagrams Due: Oct. 22, 11:30 PM Objectives To gain experience doing objectoriented design To gain experience developing UML diagrams A Word about
More informationSTAT 201 INTRODUCTION TO BUSINESS STATISTICS PROBABILITY REVIEW QUESTIONS
STAT 201 INTRODUCTION TO BUSINESS STATISTICS PROBABILITY REVIEW QUESTIONS Question 1: Five standard sixsided dice are rolled. What is the probability of getting the same number on all five dice? The probability
More informationLesson 3 Chapter 2: Introduction to Probability
Lesson 3 Chapter 2: Introduction to Probability Department of Statistics The Pennsylvania State University 1 2 The Probability Mass Function and Probability Sampling Counting Techniques 3 4 The Law of
More informationPure Math 30: Explained! 334
www.puremath30.com 334 Lesson 2, Part One: Basic Combinations Basic combinations: In the previous lesson, when using the fundamental counting principal or permutations, the order of items to be arranged
More informationnumber of favorable outcomes total number of outcomes number of times event E occurred number of times the experiment was performed.
12 Probability 12.1 Basic Concepts Start with some Definitions: Experiment: Any observation of measurement of a random phenomenon is an experiment. Outcomes: Any result of an experiment is called an outcome.
More informationLecture 2: Probability
Lecture 2: Probability Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics İstanbul Kültür University Outline 1 Introduction 2 Sample Spaces 3 Event 4 The Probability
More informationto name the four suits of cards to sort the pack into suits the names ace, king, queen, jack the value of the cards ie ace is highest, 2 is lowest.
1 Lesson 1 Today we are learning: to name the four suits of cards to sort the pack into suits the names ace, king, queen, jack the value of the cards ie ace is highest, 2 is lowest. one pack of cards
More information4.3. Addition and Multiplication Laws of Probability. Introduction. Prerequisites. Learning Outcomes. Learning Style
Addition and Multiplication Laws of Probability 4.3 Introduction When we require the probability of two events occurring simultaneously or the probability of one or the other or both of two events occurring
More informationPossibilities and Probabilities
Possibilities and Probabilities Counting The Basic Principle of Counting: Suppose that two experiments are to be performed. Then if experiment 1 can result in any one of m possible outcomes and if, for
More informationZeroknowledge games. Christmas Lectures 2008
Security is very important on the internet. You often need to prove to another person that you know something but without letting them know what the information actually is (because they could just copy
More informationCounting principle, permutations, combinations, probabilities
Counting Methods Counting principle, permutations, combinations, probabilities Part 1: The Fundamental Counting Principle The Fundamental Counting Principle is the idea that if we have a ways of doing
More informationGrade 7 & 8 Math Circles. Mathematical Games  Solutions
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Grade 7 & 8 Math Circles November 19/20/21, 2013 Mathematical Games  Solutions 1. Classic Nim and Variations (a) If the Nim Game started with 40 matchsticks,
More informationLesson 1. Basics of Probability. Principles of Mathematics 12: Explained! www.math12.com 314
Lesson 1 Basics of Probability www.math12.com 314 Sample Spaces: Probability Lesson 1 Part I: Basic Elements of Probability Consider the following situation: A six sided die is rolled The sample space
More informationMath 117 Chapter 7 Sets and Probability
Math 117 Chapter 7 and Probability Flathead Valley Community College Page 1 of 15 1. A set is a welldefined collection of specific objects. Each item in the set is called an element or a member. Curly
More informationFor 2 coins, it is 2 possible outcomes for the first coin AND 2 possible outcomes for the second coin
Problem Set 1. 1. If you have 10 coins, how many possible combinations of heads and tails are there for all 10 coins? Hint: how many combinations for one coin; two coins; three coins? Here there are 2
More informationAll You Ever Wanted to Know About Probability Theory, but Were Afraid to Ask
All You Ever Wanted to Know About Probability Theory, but Were Afraid to Ask 1 Theoretical Exercises 1. Let p be a uniform probability on a sample space S. If S has n elements, what is the probability
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,
More informationDeal or No Deal Lesson Plan
Deal or No Deal Lesson Plan Grade Level: 7 (This lesson could be adapted for 6 th through 8 th grades) Materials: Deck of Playing Cards Fair Coin (coin with head and tail sides) for each pair of students
More information4. Binomial Expansions
4. Binomial Expansions 4.. Pascal's Triangle The expansion of (a + x) 2 is (a + x) 2 = a 2 + 2ax + x 2 Hence, (a + x) 3 = (a + x)(a + x) 2 = (a + x)(a 2 + 2ax + x 2 ) = a 3 + ( + 2)a 2 x + (2 + )ax 2 +
More informationProbabilistic Strategies: Solutions
Probability Victor Xu Probabilistic Strategies: Solutions Western PA ARML Practice April 3, 2016 1 Problems 1. You roll two 6sided dice. What s the probability of rolling at least one 6? There is a 1
More informationPERMUTATIONS and COMBINATIONS. If the order doesn't matter, it is a Combination. If the order does matter it is a Permutation.
Page 1 PERMUTATIONS and COMBINATIONS If the order doesn't matter, it is a Combination. If the order does matter it is a Permutation. PRACTICE! Determine whether each of the following situations is a Combination
More informationUnderstanding 31derful. Abstract
Understanding 31derful Emily Alfs Mathematics and Computer Science Doane College Crete, Nebraska 68333 emily.alfs@doane.edu Abstract Similar to Sudoku, 31derful is a game where the player tries to place
More informationDiscrete Mathematics & Mathematical Reasoning Chapter 6: Counting
Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting Colin Stirling Informatics Slides originally by Kousha Etessami Colin Stirling (Informatics) Discrete Mathematics (Chapter 6) Today 1 /
More informationSection 65 Sample Spaces and Probability
492 6 SEQUENCES, SERIES, AND PROBABILITY 52. How many committees of 4 people are possible from a group of 9 people if (A) There are no restrictions? (B) Both Juan and Mary must be on the committee? (C)
More information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationINTRODUCTION TO PROBABILITY AND STATISTICS
INTRODUCTION TO PROBABILITY AND STATISTICS Conditional probability and independent events.. A fair die is tossed twice. Find the probability of getting a 4, 5, or 6 on the first toss and a,,, or 4 on the
More informationMath 728 Lesson Plan
Math 728 Lesson Plan Tatsiana Maskalevich January 27, 2011 Topic: Probability involving sampling without replacement and dependent trials. Grade Level: 812 Objective: Compute the probability of winning
More informationMathematical Foundations of Computer Science Lecture Outline
Mathematical Foundations of Computer Science Lecture Outline September 21, 2016 Example. How many 8letter strings can be constructed by using the 26 letters of the alphabet if each string contains 3,4,
More informationCh5: Discrete Probability Distributions Section 51: Probability Distribution
Recall: Ch5: Discrete Probability Distributions Section 51: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.
More informationReady, Set, Go! Math Games for Serious Minds
Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 
More informationPoker. 10,Jack,Queen,King,Ace. 10, Jack, Queen, King, Ace of the same suit Five consecutive ranks of the same suit that is not a 5,6,7,8,9
Poker Poker is an ideal setting to study probabilities. Computing the probabilities of different will require a variety of approaches. We will not concern ourselves with betting strategies, however. Our
More informationProbability. Sample space: all the possible outcomes of a probability experiment, i.e., the population of outcomes
Probability Basic Concepts: Probability experiment: process that leads to welldefined results, called outcomes Outcome: result of a single trial of a probability experiment (a datum) Sample space: all
More informationTexas Hold em. From highest to lowest, the possible five card hands in poker are ranked as follows:
Texas Hold em Poker is one of the most popular card games, especially among betting games. While poker is played in a multitude of variations, Texas Hold em is the version played most often at casinos
More informationClock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system
CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical
More information4.5 Finding Probability Using Tree Diagrams and Outcome Tables
4.5 Finding Probability Using ree Diagrams and Outcome ables Games of chance often involve combinations of random events. hese might involve drawing one or more cards from a deck, rolling two dice, or
More informationFINAL EXAM, Econ 171, March, 2015, with answers
FINAL EXAM, Econ 171, March, 2015, with answers There are 9 questions. Answer any 8 of them. Good luck! Problem 1. (True or False) If a player has a dominant strategy in a simultaneousmove game, then
More informationPROBABILITY. Chapter Overview
Chapter 6 PROBABILITY 6. Overview Probability is defined as a quantitative measure of uncertainty a numerical value that conveys the strength of our belief in the occurrence of an event. The probability
More informationLECTURE 3. Probability Computations
LECTURE 3 Probability Computations Pg. 67, #42 is one of the hardest problems in the course. The answer is a simple fraction there should be a simple way to do it. I don t know a simple way I used the
More information24 Random Questions (to help with Exam 1)
4 Random Questions (to help with Exam ). State the binomial theorem: See p.. The probability that rain is followed by rain is 0.8, a sunny day is followed by rain is 0.6. Find the probability that one
More informationSTATISTICS 230 COURSE NOTES. Chris Springer, revised by Jerry Lawless and Don McLeish
STATISTICS 230 COURSE NOTES Chris Springer, revised by Jerry Lawless and Don McLeish JANUARY 2006 Contents 1. Introduction to Probability 1 2. Mathematical Probability Models 5 2.1 SampleSpacesandProbability...
More informationModule 6: Basic Counting
Module 6: Basic Counting Theme 1: Basic Counting Principle We start with two basic counting principles, namely, the sum rule and the multiplication rule. The Sum Rule: If there are n 1 different objects
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 informationChampion Poker Texas Hold em
Champion Poker Texas Hold em Procedures & Training For the State of Washington 4054 Dean Martin Drive, Las Vegas, Nevada 89103 1 Procedures & Training Guidelines for Champion Poker PLAYING THE GAME Champion
More informationMost of us would probably believe they are the same, it would not make a difference. But, in fact, they are different. Let s see how.
PROBABILITY If someone told you the odds of an event A occurring are 3 to 5 and the probability of another event B occurring was 3/5, which do you think is a better bet? Most of us would probably believe
More informationLecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University
Lecture 1 Introduction Properties of Probability Methods of Enumeration Asrat Temesgen Stockholm University 1 Chapter 1 Probability 1.1 Basic Concepts In the study of statistics, we consider experiments
More informationC.4 Tree Diagrams and Bayes Theorem
A26 APPENDIX C Probability and Probability Distributions C.4 Tree Diagrams and Bayes Theorem Find probabilities using tree diagrams. Find probabilities using Bayes Theorem. Tree Diagrams A type of diagram
More informationIf a tennis player was selected at random from the group, find the probability that the player is
Basic Probability. The table below shows the number of left and right handed tennis players in a sample of 0 males and females. Left handed Right handed Total Male 3 29 32 Female 2 6 8 Total 4 0 If a tennis
More informationEXAM. Exam #3. Math 1430, Spring 2002. April 21, 2001 ANSWERS
EXAM Exam #3 Math 1430, Spring 2002 April 21, 2001 ANSWERS i 60 pts. Problem 1. A city has two newspapers, the Gazette and the Journal. In a survey of 1, 200 residents, 500 read the Journal, 700 read the
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Ch.  Problems to look at 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 informationDiscrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22
CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette
More informationCurriculum Design for Mathematic Lesson Probability
Curriculum Design for Mathematic Lesson Probability This curriculum design is for the 8th grade students who are going to learn Probability and trying to show the easiest way for them to go into this class.
More informationMath 3C Homework 3 Solutions
Math 3C Homework 3 s Ilhwan Jo and Akemi Kashiwada ilhwanjo@math.ucla.edu, akashiwada@ucla.edu Assignment: Section 2.3 Problems 2, 7, 8, 9,, 3, 5, 8, 2, 22, 29, 3, 32 2. You draw three cards from a standard
More informationDefinition and Calculus of Probability
In experiments with multivariate outcome variable, knowledge of the value of one variable may help predict another. For now, the word prediction will mean update the probabilities of events regarding the
More informationChapter 5 A Survey of Probability Concepts
Chapter 5 A Survey of Probability Concepts True/False 1. Based on a classical approach, the probability of an event is defined as the number of favorable outcomes divided by the total number of possible
More informationMath 2020 Quizzes Winter 2009
Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile
More information