Online Materials: Counting Rules The Poisson Distribution

Size: px
Start display at page:

Download "Online Materials: Counting Rules The Poisson Distribution"

Transcription

1 Chapter 6 The Poisson Distribution Counting Rules When the outcomes of a chance experiment are equally likely, one way to determine the probability of some event E is to calculate number of outcomes for which E occurs P(E ) total number of possible outcomes For example, if a chance experiment consists of rolling two dice, there are 36 possible outcomes, shown in the accompanying table. Possible Outcomes for Rolling Two Dice 1,1 1,2 1,3 1,4 1, 1,6 2,1 2,2 2,3 2,4 2, 2,6 3,1 3,2 3,3 3,4 3, 3,6 4,1 4,2 4,3 4,4 4, 4,6,1,2,3,4,,6 6,1 6,2 6,3 6,4 6, 6,6 If the dice are fair, each of these 36 outcomes is equally likely. Because 6 of these outcomes result in a total of 7 (the highlighted outcomes in the table), P (total on two dice is 7 ) 6 36 Listing the possible outcomes and then counting the total number of outcomes and the number of outcomes favorable to the event of interest is one way to determine the two counts needed to calculate the desired probability. However, if the number of possible outcomes is large, this can be tedious. In many situations, a quicker way to count is to use one of the following approaches: 1. Fundamental Counting Rule 2. Permutations 3. Combinations Fundamental Counting Rule When an outcome can be thought of as being generated by following a sequence of steps and the number of different ways each step can be completed is known, the fundamental counting rule specifies the total number of possible outcomes. Fundamental Counting Rule Suppose an outcome can be generated by following a sequence of k steps. If Step 1 can be completed in n 1 ways, Step 2 can be completed in n 2 ways, and so on, then total number of outcomes (n 1 )(n 2 ) (n k )

2 CHAPTER 6 3 For example, suppose that you have purchased a new combination lock. The lock has three wheels, each of which can be set to one of ten positions labeled, 1,,. How many different ways are there to set the combination for the lock? In this case, the answer should be obvious because each combination is a three-digit number, and there are 1, different three-digit numbers (, 1,, ). But the fundamental counting rule could have also been used. Setting a combination can be viewed as consisting of three steps: Step 1: Selecting a digit for the first wheel Step 2: Selecting a digit for the second wheel Step 3: Selecting a digit for the third wheel Step 1 can be completed in ten different ways, Step 2 can be completed in ten different ways, and Step 3 can be completed in ten different ways. Using the fundamental counting rule, total number of possible combinations (1 )(1 )(1 ) (1,) Example 6.34 Selecting a Password Suppose that the password for a particular debit card must consist of four characters. It must begin and end with a letter, and the two middle characters must be digits. How many different passwords are possible? You can view selecting a password as consisting of four steps, with the first step being selection of the first password character, and so on. Because there are 26 letters and ten digits, using the fundamental counting rule results in Permutations and Combinations In some situations, outcomes can be thought of as making selections from a fixed set of items. For example, an instructor who assigned ten homework problems may want to select three of these problems to put on a quiz. There are a large number of different possible quizzes that might result. If the homework problems are labeled H 1, H 2,, and H 1, one possible quiz might consist of H 1, H, and H 6 and another quiz might consist of H 2, H 7, and H 1. Each subset of three problems selected from the ten homework problems represents a different possible quiz. Before looking at a way to determine how many different quizzes are possible, you need to think about one more thing does order make a difference? That is, would you consider a quiz consisting of problems H 1, H, and H 6 in that order a different quiz than a quiz that had the same three problems, but in a different order (such as H 1, H 6, and H )? If different orders are considered as different outcomes, you want to count the number of different ordered subsets, which are called permutations. If different orders of the same items are considered as the same outcome, you want to count the number of unordered subsets, which are called combinations. Definition total number of passwords (26 )(1 )(1 )(26 ) 67,6 A permutation is an ordered subset of k distinct items selected from a set of n distinct items. A combination is an unordered subset of k distinct items selected from a set of n distinct items. Before considering formulas that will allow you to calculate the number of different permutations or the number of different combinations, let s look at a simple example.

3 4 CHAPTER 6 Example 6.3 Selecting Club Officers Suppose that five members of a service club (Allan, Betty, Cindy, David, and Eric) have volunteered to serve as officers for the upcoming year. There are three offices: president, vice-president, and treasurer. How many different ways are there to fill the three offices? Here, order is important, because the outcome A, B, C (where Allan is president, Betty is vice-president, and Cindy is treasurer) is a different outcome than C, B, A. To evaluate the number of different outcomes, you want to know how many different permutations of three members selected from the five volunteers are possible. Think of generating an outcome by first selecting a president ( choices), then selecting a vice-president (4 choices remaining), and then selecting a treasurer (3 choices). The fundamental counting rule can be used to determine that There are formulas that will allow you to calculate the number of permutations or the number of combinations directly, without having to go through the work illustrated in Example 6.3. These formulas use factorial notation. Recall that for any positive integer k, k factorial is denoted by k! and is defined as and! is defined to be 1. Permutations number of permutation ( )(4 )(3 ) 6 What if the club was managed by a three-person executive board and that these individuals did not hold specific offices, such as president? In this situation, order doesn t matter you would consider the executive board consisting of A, B, and C to be the same as the board consisting of B, A, and C. In this situation, you want to calculate the number of different combinations of three members selected from the five volunteers. Consider the executive board consisting of A, B, and C. These three members can be listed in six different orders: ABC, ACB, BAC, BCA, CAB, CBA. Each of these six ordered arrangements was counted as different when you determined that there were 6 permutations. The count of permutations over-counted combinations by a factor of six. This means that number of combinations k! (k)(k 2 1)(k 2 2) (1) The number of ordered subsets of k items selected from a set of n items, denoted by n P k, is calculated using the following formula: P n! n k (n 2 k )! Example 6.36 ipod Shuffle Suppose that Chris has ten songs in a playlist on his ipod. A shuffle is a random ordering of songs from a playlist. How many different shuffles of four songs can be created from songs in this playlist? Each possible shuffle is a permutation of four songs chosen from the ten in the playlist. The number of different shuffles is P 1! 1 4 (1 2 4)! 1! 6! 3,628,8,4 72 Most graphing calculators and statistics software packages can calculate the number of permutations (or the number of combinations) given n and k.

4 CHAPTER 6 Combinations The number of unordered subsets of k items selected from a set of n items, denoted by n C k, is calculated using the following formula: C n! n k k!(n 2 k )! Example 6.37 ipod Revisited Suppose you want to create a playlist of 1 songs on an ipod. A playlist is a collection of songs chosen from all the songs on an ipod. How many different playlists are possible if you have 3 songs on your ipod? Because a playlist is an unordered collection of songs, each playlist is a combination of 1 songs chosen from the 3 songs on the ipod. The number of different playlists is C 3! 3 1 1!(3 2 1)! 3! 1! 2! This can be evaluated using a graphing calculator or software to obtain C 3,4,1 3 1 That is over 3 million different playlists! And with more songs on the ipod, this number would be even larger. Example 6.38 Figure Skating Competitions In figure skating competitions, skaters perform both a short program and a long program. For the short program, skaters are divided at random into groups of 6 skaters. The skating order within each group is assigned at random. Suppose that 24 skaters have entered a competition. How many different possibilities are possible for the program listing for the first group of skaters to perform the short program? It is often the case with counting problems that there are multiple ways to solve them. Let s consider two different ways to answer this question. Solution 1 You can think of determining the program listing for the first group as consisting of two steps: Step 1: Select the six skaters who will be in the first group. Step 2: Determine the skating order for these six skaters. Step 1 can be completed in 24 C 6 ways, and then for each of these possible groups of skaters, there are 6 P 6 ways to order the skaters. The total number of different possible program listings is (using the fundamental counting rule) ( 24 C 6 ) ( 6 P 6 ) ( 24! 6!18! )( 6!! ) (134,6)(72) 6,,12 Solution 2 Another way to think about this problem is to recognize that even though the description of how the first group and skating order are determined was described as a two-step process (selecting the group, then determining the order), this is equivalent to just asking how many different ordered arrangements of six can be formed from the set of 24 skaters a permutation. So, total number of program listings 24 P 6 6,,12

5 6 CHAPTER 6 Exercises 6.11 How many different passwords are possible if a password must begin with a letter and must consist of eight characters that must be letters or numbers? California automobile license plate numbers begin with a non-zero digit, followed by three letters, followed by three digits (which can be zero). How many different license plate numbers are possible using this format? Jeanie has a list of seven errands she needs to complete, but she only has time to do three of them today. How many ways are possible for Jeanie to select the three errands she will complete today? A professor plans to ask 3 different students in her math class to participate in a class demonstration. How many different ways to select these three students are possible if there are students in the class? 6.11 A professor assigned ten homework problems. He plans to select four of these problems to be graded. a. Suppose you only have time to complete six of these problems. How many different ways are there to choose which six problems you will do? b. How many different ways are there for the professor to select four problems to be graded? c. How many different ways are there to select four problems from the six that you completed? d. If the professor selects the four problems to be graded at random, what is the probability that all four problems selected for grading are from the six that you completed? 6.12 How many different sequences of three different characters can be formed using the letters NEWYORK? A baseball team has 1 players. The coach must submit a starting batting order, which consists of players and the order in which they will bat. How many different batting orders are possible? The Poisson Distribution In Section 6.7 you learned about two discrete probability distributions the binomial distribution and the geometric distribution. In this section, you will see another widely used discrete probability distribution called the Poisson distribution. A Poisson distribution is used to describe the behavior of a random variable that counts the number of occurrences of some random event during a fixed time interval or over a fixed space. For example, you might be interested in the number of calls to a customer service center during a three-hour period or the number of accidents occurring along a particular stretch of highway. If the event you are interested in occurs at random, but with a known overall rate, and the properties in the following box are satisfied, then the random variable x number of occurrences in a fixed interval has a Poisson distribution. If x = number of occurrences in a fixed interval and the following conditions are met, x has a Poisson distribution. Conditions for a Poisson Distribution 1. The probability of more than one occurrence in a very small interval is. 2. The probabilities of an occurrence in each of two intervals of the same length are equal. 3. The number of occurrences in any particular interval is independent of the number of occurrences prior to that interval. These conditions are satisfied when occurrences are random but with a fixed overall rate. For example, consider calls coming in to a customer service center, and define x to be x number of calls received during a 1-hour period From past experience, you might know that calls come in at random times but with an overall rate of about two per minute. It is then reasonable to think that

6 CHAPTER The chance of more than one call in any very small time interval (for example,.1 seconds) is equal to. 2. The chances of a call coming in during each of two different intervals of the same length are equal. 3. The number of calls during any particular interval is independent of the number of calls that were received prior to this interval. These observations are consistent with calls coming in at random times, but with an overall rate of two per minute. Because the conditions are met, the behavior of the random variable x can be described by a Poisson distribution. The Poisson Distribution Let the overall rate of occurrence of the random event in 1 unit of time or space (such as 1 minute or 1 mile) t a fixed interval of time or space (such as 2 minutes or miles) Then if the conditions for a Poisson distribution are met x number of occurrences in an interval of length t has a Poisson distribution. The Poisson distribution is defined by the following formula: P(x) (x occurences in an interval of length t) ( t ) x e 2 t x, 1, 2, x! (e in this formula is a mathematical constant that is approximately equal to ) Example 6.3 Paint Bubbles Sometimes bubbles form in paint as it dries, creating flaws in the paint surface. Suppose that for a particular type of paint, bubbles form at random, with an overall rate of.3 per square foot. What is the probability of observing five bubbles in a 1-square-foot wall that has been painted with this paint? In this example, the occurrence you are interested in is a paint bubble. Bubbles occur at random with a known rate of.3 per square foot. This mean that.3, the unit of space for this rate is 1 square foot, and the interval of interest is t 1 square feet. The desired probability is then p() ( (.3)(1) ) e 2(.3)(1)! 3 (2.718) 23 (234)(.48) 11.64! Graphing calculators and statistical software can also be used to compute Poisson probabilities. For example, Minitab was used to compute p (x # ) p() 1 p(1) 1 p(2) 1 p(3) 1 p(4) 1 p() The difference in the value of p() from the value in the hand calculations (.181 versus.7) is a result of Minitab using more decimal accuracy in the computations and in the value of e. Example 6.4 Calls to -1-1 Suppose that in a particular county, calls come in to the -1-1 emergency number at random with an overall rate of three per hour. What is the probability of observing eight calls in a two-hour period? To answer this question, you would need to evaluate

7 8 CHAPTER 6 ( t) x e 2 t p(x) x! where x 8, 3, and t 2. This results in p (8) (6) 8 (2.718) 26 (1,67,616)(.248) 4, ! 4, Exercises An automobile club reports that requests for roadside assistance come in at random times with an overall rate of four per hour. a. What is the probability that exactly ten requests are received in a particular two-hour time period? b. What is the probability that there are no calls in a particular one-hour time period? A cookie manufacturer makes chocolate chip cookies. When the cookie dough is prepared, chocolate chips are added to the dough in a way that distributes them at random throughout the dough. Because about 12, chips are added to a batch of dough that makes 1, cookies, the company knows that, on average, there are 12 chocolate chips per cookie. a. What is the probability that a randomly selected cookie will have four or fewer chocolate chips? b. What is the probability that the total number of chocolate chips in three randomly selected cookies will be exactly 36? Suppose that the random variable x has a Poisson distribution with rate 6 per hour. For a particular one-hour time interval, is it more likely that x or that x 7? Support your answer with the appropriate probability calculations Suppose that the random variable x has a Poisson distribution with rate. per hour and the random variable y has a Poisson distribution with rate 2 per hour. a. Calculate the probabilities needed to complete the following tables. Round your answers to four decimal places. x p(x) b. Possible values for a random variable that has a Poison distribution are, 1, 2, 3, Even though possible values are all nonnegative integers, what do you notice about p(x) for t 1 as the value of x gets large relative to? Draw a probability histogram for each of the Poisson distributions for which you calculated probabilities in the previous exercise (., t 1 and 2, t 1 ) using possible values to on the horizontal axis. How do the two probability distributions compare with respect to shape, center, and variability? y p(y)

8 CHAPTER 6 Counting Rules Answers for Selected Exercises Counting Rules (26) (36) 6 6,6,34, () (26) 3 (1) 3 18,184, C C 3 1, (a) 1 C 6 21 (b) 1 C 4 21 (c) 6 C 4 1 (d) P P 1,816,214,4 The Poisson Distribution (a) 4, t 2, p(1).3 (b) 4, t 1, p() (a) 12, t 1 p() 1 p(1) 1 p(2) 1 p(3) 1 p(4) (b) 12, t 3, p(36) , t 1, p().16623, p(7) , p() is greater than p(7) (a) x p(x) y p(y) p(x) p(y) x 4 y Both distributions are positively skewed. The distribution of x is centered at about., whereas the distribution of y is centered at about 2. There is more variability in the y distribution (b) As the value of x gets large compared to the value of, p(x) gets close to.

Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution

Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.

More information

1 Introduction. 2 Basic Principles. 2.1 Multiplication Rule. [Ch 9] Counting Methods. 400 lecture note #9

1 Introduction. 2 Basic Principles. 2.1 Multiplication Rule. [Ch 9] Counting Methods. 400 lecture note #9 400 lecture note #9 [Ch 9] Counting Methods 1 Introduction In many discrete problems, we are confronted with the problem of counting. Here we develop tools which help us counting. Examples: o [9.1.2 (p.

More information

35 Permutations, Combinations and Probability

35 Permutations, Combinations and Probability 35 Permutations, Combinations and Probability Thus far we have been able to list the elements of a sample space by drawing a tree diagram. For large sample spaces tree diagrams become very complex to construct.

More information

Discrete Mathematics & Mathematical Reasoning Chapter 6: Counting

Discrete 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 information

Chapter 4. Probability Distributions

Chapter 4. Probability Distributions Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive

More information

Section 6-5 Sample Spaces and Probability

Section 6-5 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 information

Random variables, probability distributions, binomial random variable

Random variables, probability distributions, binomial random variable Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

More information

Stats Review Chapters 5-6

Stats Review Chapters 5-6 Stats Review Chapters 5-6 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

More information

Random Variable: A function that assigns numerical values to all the outcomes in the sample space.

Random Variable: A function that assigns numerical values to all the outcomes in the sample space. STAT 509 Section 3.2: Discrete Random Variables Random Variable: A function that assigns numerical values to all the outcomes in the sample space. Notation: Capital letters (like Y) denote a random variable.

More information

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions... MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

More information

Section 2.1. Tree Diagrams

Section 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 information

Introductory Problems

Introductory Problems Introductory Problems Today we will solve problems that involve counting and probability. Below are problems which introduce some of the concepts we will discuss.. At one of George Washington s parties,

More information

9.2 The Multiplication Principle, Permutations, and Combinations

9.2 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 information

Solutions for Review Problems for Exam 2 Math 1040 1 1. You roll two fair dice. (a) Draw a tree diagram for this experiment.

Solutions for Review Problems for Exam 2 Math 1040 1 1. You roll two fair dice. (a) Draw a tree diagram for this experiment. Solutions for Review Problems for Exam 2 Math 1040 1 1. You roll two fair dice. (a) Draw a tree diagram for this experiment. 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2

More information

6.1. Construct and Interpret Binomial Distributions. p Study probability distributions. Goal VOCABULARY. Your Notes.

6.1. Construct and Interpret Binomial Distributions. p Study probability distributions. Goal VOCABULARY. Your Notes. 6.1 Georgia Performance Standard(s) MM3D1 Your Notes Construct and Interpret Binomial Distributions Goal p Study probability distributions. VOCABULARY Random variable Discrete random variable Continuous

More information

Fundamentals of Probability

Fundamentals 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 information

37.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes

37.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes The Poisson distribution 37.3 Introduction In this block we introduce a probability model which can be used when the outcome of an experiment is a random variable taking on positive integer values and

More information

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 3-7, 3-8 The remaining discrete random

More information

Counting principle, permutations, combinations, probabilities

Counting 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 information

, for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0

, for x = 0, 1, 2, 3,... (4.1) (1 + 1/n) n = 2.71828... b x /x! = e b, x=0 Chapter 4 The Poisson Distribution 4.1 The Fish Distribution? The Poisson distribution is named after Simeon-Denis Poisson (1781 1840). In addition, poisson is French for fish. In this chapter we will

More information

NOTES ON COUNTING KARL PETERSEN

NOTES ON COUNTING KARL PETERSEN NOTES ON COUNTING KARL PETERSEN It is important to be able to count exactly the number of elements in any finite set. We will see many applications of counting as we proceed (number of Enigma plugboard

More information

Math Common Core Sampler Test

Math Common Core Sampler Test High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests

More information

Lesson 1: Experimental and Theoretical Probability

Lesson 1: Experimental and Theoretical Probability Lesson 1: Experimental and Theoretical Probability Probability is the study of randomness. For instance, weather is random. In probability, the goal is to determine the chances of certain events happening.

More information

Worksheet A2 : Fundamental Counting Principle, Factorials, Permutations Intro

Worksheet A2 : Fundamental Counting Principle, Factorials, Permutations Intro Worksheet A2 : Fundamental Counting Principle, Factorials, Permutations Intro 1. A restaurant offers four sizes of pizza, two types of crust, and eight toppings. How many possible combinations of pizza

More information

Random Variables and Their Expected Values

Random Variables and Their Expected Values Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution Function Discrete and Continuous Random Variables The Probability Mass Function The (Cumulative) Distribution

More information

Lecture 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 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 information

Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

More information

4. Joint Distributions

4. Joint Distributions Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose

More information

Chapter 6 Random Variables

Chapter 6 Random Variables Chapter 6 Random Variables Day 1: 6.1 Discrete Random Variables Read 340-344 What is a random variable? Give some examples. A numerical variable that describes the outcomes of a chance process. Examples:

More information

The Utah Basic Skills Competency Test Framework Mathematics Content and Sample Questions

The Utah Basic Skills Competency Test Framework Mathematics Content and Sample Questions The Utah Basic Skills Competency Test Framework Mathematics Content and Questions Utah law (53A-1-611) requires that all high school students pass The Utah Basic Skills Competency Test in order to receive

More information

Section 6.1 Discrete Random variables Probability Distribution

Section 6.1 Discrete Random variables Probability Distribution Section 6.1 Discrete Random variables Probability Distribution Definitions a) Random variable is a variable whose values are determined by chance. b) Discrete Probability distribution consists of the values

More information

Models for Discrete Variables

Models for Discrete Variables Probability Models for Discrete Variables Our study of probability begins much as any data analysis does: What is the distribution of the data? Histograms, boxplots, percentiles, means, standard deviations

More information

Quadratic Equations and Inequalities

Quadratic Equations and Inequalities MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

More information

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

More information

A1. Basic Reviews PERMUTATIONS and COMBINATIONS... or HOW TO COUNT

A1. Basic Reviews PERMUTATIONS and COMBINATIONS... or HOW TO COUNT A1. Basic Reviews Appendix / A1. Basic Reviews / Perms & Combos-1 PERMUTATIONS and COMBINATIONS... or HOW TO COUNT Question 1: Suppose we wish to arrange n 5 people {a, b, c, d, e}, standing side by side,

More information

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

More information

Math Review Large Print (18 point) Edition Chapter 4: Data Analysis

Math Review Large Print (18 point) Edition Chapter 4: Data Analysis GRADUATE RECORD EXAMINATIONS Math Review Large Print (18 point) Edition Chapter 4: Data Analysis Copyright 2010 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS,

More information

The Math. P (x) = 5! = 1 2 3 4 5 = 120.

The Math. P (x) = 5! = 1 2 3 4 5 = 120. The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct

More information

4 BASICS OF PROBABILITY. Experiment is a process of observation that leads to a single outcome that cannot be predicted with certainty.

4 BASICS OF PROBABILITY. Experiment is a process of observation that leads to a single outcome that cannot be predicted with certainty. 4 BASICS OF PROBABILITY Experiment is a process of observation that leads to a single outcome that cannot be predicted with certainty. Examples: 1. Pull a card from a deck 2. Toss a coin 3. Response time.

More information

34 Probability and Counting Techniques

34 Probability and Counting Techniques 34 Probability and Counting Techniques If you recall that the classical probability of an event E S is given by P (E) = n(e) n(s) where n(e) and n(s) denote the number of elements of E and S respectively.

More information

Lecture 5 : The Poisson Distribution. Jonathan Marchini

Lecture 5 : The Poisson Distribution. Jonathan Marchini Lecture 5 : The Poisson Distribution Jonathan Marchini Random events in time and space Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,

More information

Unit 21: Binomial Distributions

Unit 21: Binomial Distributions Unit 21: Binomial Distributions Summary of Video In Unit 20, we learned that in the world of random phenomena, probability models provide us with a list of all possible outcomes and probabilities for how

More information

The basics of probability theory. Distribution of variables, some important distributions

The basics of probability theory. Distribution of variables, some important distributions The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a

More information

Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty.

Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty. Chapter 1 Probability Spaces 11 What is Probability? Probability is concerned with quantifying the likelihoods of various events in situations involving elements of randomness or uncertainty Example 111

More information

Definition of an nth Root

Definition of an nth Root Radicals and Complex Numbers 7 7. Definition of an nth Root 7.2 Rational Exponents 7.3 Simplifying Radical Expressions 7.4 Addition and Subtraction of Radicals 7.5 Multiplication of Radicals 7.6 Rationalization

More information

STAT 360 Probability and Statistics. Fall 2012

STAT 360 Probability and Statistics. Fall 2012 STAT 360 Probability and Statistics Fall 2012 1) General information: Crosslisted course offered as STAT 360, MATH 360 Semester: Fall 2012, Aug 20--Dec 07 Course name: Probability and Statistics Number

More information

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED 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 information

Against all odds. Byron Schmuland Department of Mathematical Sciences University of Alberta, Edmonton

Against all odds. Byron Schmuland Department of Mathematical Sciences University of Alberta, Edmonton Against all odds Byron Schmuland Department of Mathematical Sciences University of Alberta, Edmonton schmu@stat.ualberta.ca In 1975, when I was a high school student, the Canadian government held its first

More information

MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS THE BINOMIAL THEOREM. (p + q) 0 =1,

MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS THE BINOMIAL THEOREM. (p + q) 0 =1, THE BINOMIAL THEOREM Pascal s Triangle and the Binomial Expansion Consider the following binomial expansions: (p + q) 0 1, (p+q) 1 p+q, (p + q) p +pq + q, (p + q) 3 p 3 +3p q+3pq + q 3, (p + q) 4 p 4 +4p

More information

Chapter 5 - Probability

Chapter 5 - Probability Chapter 5 - Probability 5.1 Basic Ideas An experiment is a process that, when performed, results in exactly one of many observations. These observations are called the outcomes of the experiment. The set

More information

5.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes. Learning Style

5.3. The Poisson distribution. Introduction. Prerequisites. Learning Outcomes. Learning Style The Poisson distribution 5.3 Introduction In this block we introduce a probability model which can be used when the outcome of an experiment is a random variable taking on positive integer values and where

More information

4.4 Other Discrete Distribution: Poisson and Hypergeometric S

4.4 Other Discrete Distribution: Poisson and Hypergeometric S 4.4 Other Discrete Distribution: Poisson and Hypergeometric S S time, area, volume, length Characteristics of a Poisson Random Variable 1. The experiment consists of counting the number of times x that

More information

7.5: Conditional Probability

7.5: Conditional Probability 7.5: Conditional Probability Example 1: A survey is done of people making purchases at a gas station: buy drink (D) no drink (Dc) Total Buy drink(d) No drink(d c ) Total Buy Gas (G) 20 15 35 No Gas (G

More information

Chapter 6: Random Variables

Chapter 6: Random Variables Chapter : Random Variables Section.3 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter Random Variables.1 Discrete and Continuous Random Variables.2 Transforming and Combining

More information

Discrete Random Variables and their Probability Distributions

Discrete Random Variables and their Probability Distributions CHAPTER 5 Discrete Random Variables and their Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Discrete Random Variable

More information

Mathematical Foundations of Computer Science Lecture Outline

Mathematical Foundations of Computer Science Lecture Outline Mathematical Foundations of Computer Science Lecture Outline September 21, 2016 Example. How many 8-letter strings can be constructed by using the 26 letters of the alphabet if each string contains 3,4,

More information

MA 485-1E, Probability (Dr Chernov) Final Exam Wed, Dec 10, 2003 Student s name Be sure to show all your work. Full credit is given for 100 points.

MA 485-1E, Probability (Dr Chernov) Final Exam Wed, Dec 10, 2003 Student s name Be sure to show all your work. Full credit is given for 100 points. MA 485-1E, Probability (Dr Chernov) Final Exam Wed, Dec 10, 2003 Student s name Be sure to show all your work. Full credit is given for 100 points. 1. (10 pts) Assume that accidents on a 600 miles long

More information

What is the probability of throwing a fair die and receiving a six? Introduction to Probability. Basic Concepts

What 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 information

Math/Stat 360-1: Probability and Statistics, Washington State University

Math/Stat 360-1: Probability and Statistics, Washington State University Math/Stat 360-1: Probability and Statistics, Washington State University Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2 Haijun Li Math/Stat 360-1: Probability and

More information

P(X = x k ) = 1 = k=1

P(X = x k ) = 1 = k=1 74 CHAPTER 6. IMPORTANT DISTRIBUTIONS AND DENSITIES 6.2 Problems 5.1.1 Which are modeled with a unifm distribution? (a Yes, P(X k 1/6 f k 1,...,6. (b No, this has a binomial distribution. (c Yes, P(X k

More information

Lecture 2 Binomial and Poisson Probability Distributions

Lecture 2 Binomial and Poisson Probability Distributions Lecture 2 Binomial and Poisson Probability Distributions Binomial Probability Distribution l Consider a situation where there are only two possible outcomes (a Bernoulli trial) H Example: u flipping a

More information

6.4 Normal Distribution

6.4 Normal Distribution Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

More information

SECTION 10-5 Multiplication Principle, Permutations, and Combinations

SECTION 10-5 Multiplication Principle, Permutations, and Combinations 10-5 Multiplication Principle, Permutations, and Combinations 761 54. Can you guess what the next two rows in Pascal s triangle, shown at right, are? Compare the numbers in the triangle with the binomial

More information

Discrete probability and the laws of chance

Discrete probability and the laws of chance Chapter 8 Discrete probability and the laws of chance 8.1 Introduction In this chapter we lay the groundwork for calculations and rules governing simple discrete probabilities. These steps will be essential

More information

Ch. 2.3, 2.4 Quiz/Test Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Ch. 2.3, 2.4 Quiz/Test Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Ch. 2.3, 2.4 Quiz/Test Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 1) Find the mean, median, and mode of the following

More information

STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

More information

Section 5 Part 2. Probability Distributions for Discrete Random Variables

Section 5 Part 2. Probability Distributions for Discrete Random Variables Section 5 Part 2 Probability Distributions for Discrete Random Variables Review and Overview So far we ve covered the following probability and probability distribution topics Probability rules Probability

More information

Applied Reliability ------------------------------------------------------------------------------------------------------------ Applied Reliability

Applied Reliability ------------------------------------------------------------------------------------------------------------ Applied Reliability Applied Reliability Techniques for Reliability Analysis with Applied Reliability Tools (ART) (an EXCEL Add-In) and JMP Software AM216 Class 6 Notes Santa Clara University Copyright David C. Trindade, Ph.

More information

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

More information

Probability Models for Continuous Random Variables

Probability Models for Continuous Random Variables Density Probability Models for Continuous Random Variables At right you see a histogram of female length of life. (Births and deaths are recorded to the nearest minute. The data are essentially continuous.)

More information

Worksheet 2 nd. STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving.

Worksheet 2 nd. STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving. Worksheet 2 nd Topic : PERMUTATIONS TIME : 4 X 45 minutes STANDARD COMPETENCY : 1. To use the statistics rules, the rules of counting, and the characteristic of probability in problem solving. BASIC COMPETENCY:

More information

Chapter 5. Discrete Probability Distributions

Chapter 5. Discrete Probability Distributions Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable

More information

Sets and set operations

Sets and set operations CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used

More information

MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables

MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables MATH 10: Elementary Statistics and Probability Chapter 4: Discrete Random Variables Tony Pourmohamad Department of Mathematics De Anza College Spring 2015 Objectives By the end of this set of slides, you

More information

Probability distributions

Probability distributions Probability distributions (Notes are heavily adapted from Harnett, Ch. 3; Hayes, sections 2.14-2.19; see also Hayes, Appendix B.) I. Random variables (in general) A. So far we have focused on single events,

More information

RELATIONS AND FUNCTIONS

RELATIONS AND FUNCTIONS 008 RELATIONS AND FUNCTIONS Concept 9: Graphs of some functions Graphs of constant function: Let k be a fixed real number. Then a function f(x) given by f ( x) = k for all x R is called a constant function.

More information

Math 202-0 Quizzes Winter 2009

Math 202-0 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

Grade 7/8 Math Circles Fall 2012 Probability

Grade 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 information

Notes on Continuous Random Variables

Notes on Continuous Random Variables Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

More information

In this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events.

In this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events. Lecture#4 Chapter 4: Probability In this chapter, we use sample data to make conclusions about the population. Many of these conclusions are based on probabilities of the events. 4-2 Fundamentals Definitions:

More information

Hoover High School Math League. Counting and Probability

Hoover 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 information

Practice Problems #4

Practice Problems #4 Practice Problems #4 PRACTICE PROBLEMS FOR HOMEWORK 4 (1) Read section 2.5 of the text. (2) Solve the practice problems below. (3) Open Homework Assignment #4, solve the problems, and submit multiple-choice

More information

CHAPTER 4: DISCRETE RANDOM VARIABLE

CHAPTER 4: DISCRETE RANDOM VARIABLE CHAPTER 4: DISCRETE RANDOM VARIABLE Exercise 1. A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have established the following

More information

3. Conditional probability & independence

3. Conditional probability & independence 3. Conditional probability & independence Conditional Probabilities Question: How should we modify P(E) if we learn that event F has occurred? Derivation: Suppose we repeat the experiment n times. Let

More information

Math 421: Probability and Statistics I Note Set 2

Math 421: Probability and Statistics I Note Set 2 Math 421: Probability and Statistics I Note Set 2 Marcus Pendergrass September 13, 2013 4 Discrete Probability Discrete probability is concerned with situations in which you can essentially list all the

More information

Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014

Chapter 3: Discrete Random Variable and Probability Distribution. January 28, 2014 STAT511 Spring 2014 Lecture Notes 1 Chapter 3: Discrete Random Variable and Probability Distribution January 28, 2014 3 Discrete Random Variables Chapter Overview Random Variable (r.v. Definition Discrete

More information

Math 55: Discrete Mathematics

Math 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 information

COURSE OUTLINE. Course Number Course Title Credits MAT201 Probability and Statistics for Science and Engineering 4. Co- or Pre-requisite

COURSE OUTLINE. Course Number Course Title Credits MAT201 Probability and Statistics for Science and Engineering 4. Co- or Pre-requisite COURSE OUTLINE Course Number Course Title Credits MAT201 Probability and Statistics for Science and Engineering 4 Hours: Lecture/Lab/Other 4 Lecture Co- or Pre-requisite MAT151 or MAT149 with a minimum

More information

Section 1-5 Functions: Graphs and Transformations

Section 1-5 Functions: Graphs and Transformations 1-5 Functions: Graphs and Transformations 61 (D) Write a brief description of the relationship between tire pressure and mileage, using the terms increasing, decreasing, local maximum, and local minimum

More information

Homework 5 Solutions

Homework 5 Solutions Math 130 Assignment Chapter 18: 6, 10, 38 Chapter 19: 4, 6, 8, 10, 14, 16, 40 Chapter 20: 2, 4, 9 Chapter 18 Homework 5 Solutions 18.6] M&M s. The candy company claims that 10% of the M&M s it produces

More information

Discrete Mathematics: Homework 6 Due:

Discrete Mathematics: Homework 6 Due: Discrete Mathematics: Homework 6 Due: 2011.05.20 1. (3%) How many bit strings are there of length six or less? We use the sum rule, adding the number of bit strings of each length to 6. If we include the

More information

Individual 5 th Grade

Individual 5 th Grade Individual 5 th Grade Instructions: Problems 1 10 are multiple choice and count towards your team score. Bubble in the letter on your answer sheet. Be sure to erase all mistakes completely. 1. What is

More information

Administrative - Master Syllabus COVER SHEET

Administrative - Master Syllabus COVER SHEET Administrative - Master Syllabus COVER SHEET Purpose: It is the intention of this to provide a general description of the course, outline the required elements of the course and to lay the foundation for

More information

Topic 1 Probability spaces

Topic 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 five-card poker is about 0.2%,

More information

Find the theoretical probability of an event. Find the experimental probability of an event. Nov 13 3:37 PM

Find the theoretical probability of an event. Find the experimental probability of an event. Nov 13 3:37 PM Objectives Find the theoretical probability of an event. Find the experimental probability of an event. Probability is the measure of how likely an event is to occur. Each possible result of a probability

More information

Math 55: Discrete Mathematics

Math 55: Discrete Mathematics Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Ch. 4 Discrete Probability Distributions 4.1 Probability Distributions 1 Decide if a Random Variable is Discrete or Continuous 1) State whether the variable is discrete or continuous. The number of cups

More information

Lecture 5 : The Poisson Distribution

Lecture 5 : The Poisson Distribution Lecture 5 : The Poisson Distribution Jonathan Marchini November 10, 2008 1 Introduction Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,

More information

Sets and functions. {x R : x > 0}.

Sets and functions. {x R : x > 0}. Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

More information

CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS

CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS TRUE/FALSE 235. The Poisson probability distribution is a continuous probability distribution. F 236. In a Poisson distribution,

More information