Ch5: Discrete Probability Distributions Section 51: Probability Distribution


 Oswald Booker
 4 years ago
 Views:
Transcription
1 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. X, Y, Z) are used to represent variables. A random variable is a variable whose values are determined by chance. Discrete variables are countable. Example: Roll a die and let X represent the outcome so X = {1,2,3,4,5,6} Ch5: Discrete Probability Distributions Santorico  Page 147
2 Discrete probability distribution  the values a random variable can assume and the corresponding probabilities of the values. The probabilities may be determined theoretically or by observation. They can be displayed by a graph or a table. How does this connect to our frequency distributions, tables and graphs from Chapter 2? Ch5: Discrete Probability Distributions Santorico  Page 148
3 Example: Create a probability distribution for the number of girls out of 3 children. We previously used a tree diagram to construct the sample space which consisted of 8 possible outcomes: BBB X=0 BBG, BGB, GBB X=1 BGG, GBG, GGB X=2 GGG X=3 The corresponding (discrete) probability distribution is: Number of Girls X Probability P(X) 1/8 3/8 3/8 1/8 Check of calculations in table: MUST SUM TO 1! Ch5: Discrete Probability Distributions Santorico  Page 149
4 Probability Graph the probability distribution above Number of Girls Ch5: Discrete Probability Distributions Santorico  Page 150
5 Example: The World Series played by Major League Baseball is a 4 to 7 game series won by the team winning four games. The data shown consists of the number of games played in the World Series from 1965 through The number of games played is represented by the variable X. Ch5: Discrete Probability Distributions Santorico  Page 151
6 Construct the corresponding discrete probability distribution and graph the probability distribution above. Ch5: Discrete Probability Distributions Santorico  Page 152
7 Two Requirements for a Probability Distribution 1. The sum of the probabilities of all the outcomes in the sample space must be 1; that is P(X) The probability of each outcome in the sample space must be between or equal to 0 and 1; that is 0 P(X) 1. These are good checks for you to use after you have computed a discrete probability distribution! The sums to 1 check will often find a calculation error! Ch5: Discrete Probability Distributions Santorico  Page 153
8 Example: Determine whether each distribution is a probability distribution. Explain. Ch5: Discrete Probability Distributions Santorico  Page 154
9 Section 52: Mean, Variance, Standard Deviation, and Expectation The mean, variance, and standard deviation for a probability distribution are computed differently from the mean, variance, and standard deviation for sample. Recall that a parameter is a numerical characteristic of a population. The mean of a probability distribution is denoted by the symbol,. The mean of a probability distribution for a discrete random variable is X P( X ) where the sum is taken over all possible values of X. Rounding Rule: Round to one more decimal place than the outcome X when finding the mean, variance, and standard deviation for variables of a probability distribution. Ch5: Discrete Probability Distributions Santorico  Page 155
10 Example: Find the mean number of girls in a family with two children using the probability distribution below X P( X ) X Ch5: Discrete Probability Distributions Santorico  Page 156
11 Example: Find the mean number of trips lasting five nights or longer that American adults take per year using the probability distribution below. X P(X) Ch5: Discrete Probability Distributions Santorico  Page 157
12 Variance and Standard Deviation The variance of a probability distribution, σ², for a discrete random variable is found by multiplying the square of each outcome, X, by its corresponding probability, summing those products, and subtracting the square of the mean. 2 X 2 P(X) 2 The standard deviation, σ, of a probability distribution is: 2 Ch5: Discrete Probability Distributions Santorico  Page 158
13 Example: Calculate the variance and standard deviation for the number of girls in the previous example: [ X P( X )] [ X P( X )] Ch5: Discrete Probability Distributions Santorico  Page 159
14 Example: Calculate the variance and standard deviation for the number of trips five nights or more in the previous example. 2 [X 2 P(X)] 2 2 Ch5: Discrete Probability Distributions Santorico  Page 160
15 Expectation Another concept closely related to the mean of a probability distribution is the concept of expectation. Expected value has wide uses in the insurance industry, gambling, and other areas such as decision theory. The expected value of a discrete random variable of a probability distribution is the theoretical average of the variable. Does this look familiar? E(X) X P(X) Ch5: Discrete Probability Distributions Santorico  Page 161
16 Example: Suppose one thousand tickets are sold at $10 each to win a used car valued at $5,000. What is the expected value of the gain if a person purchases one ticket? The person will either win or lose. If they win which will happen with probability 1/1000, they have gained $5000$10. If they lose, they have lost $10. Gain, X Probability, P(X) $4990 1/1000 $10 999/ E X $4990 ( $10) $ Ch5: Discrete Probability Distributions Santorico  Page 162
17 Example: Suppose one thousand tickets are sold at $1 each for 3 prizes of $150, $100, and $50. After each prize drawing, the winning ticket is then returned to the pool of tickets. What is the expected value if a person purchases 3 tickets? Gain, X Probability, P(X) E(X) = Ch5: Discrete Probability Distributions Santorico  Page 163
18 When gambling: If the expected value of the game is zero, the game is said to be fair. If the expected value of a game is positive, then the game is in the favor of the player. If the expected value of the game is negative, then the game is said to be in the favor of the house. o This means you lose should expect to lose money in the long run. o Every game in Las Vegas has a negative expected value!!! Ch5: Discrete Probability Distributions Santorico  Page 164
19 Section 53: The Binomial Distribution A binomial experiment is a probability experiment that satisfies the following four requirements: 1. Each of the n trials has two possible outcomes or can be reduced to two outcomes: success and failure. The outcome of interest is called a success and the other outcome is called a failure. 2. The outcomes of each trial must be independent of each other. 3. There must be a fixed number of trials. 4. Each trial has the same probability of success, denoted by p. Ch5: Discrete Probability Distributions Santorico  Page 165
20 The acronym BINS may help you remember the conditions: B Binary outcomes I Independent outcomes N number of trials is fixed S same probability of success Examples: Ch5: Discrete Probability Distributions Santorico  Page 166
21 Notation: P(S), probability of success P(F), probability of failure p, the numerical probability of success q, The numerical probability of failure P(S) p and P(F) 1 P(S) 1 p q n, the number of trials. X, the number of successes in n trials. NOTE: 0 X n and X 0,1,2,3,..., n Binomial distribution the outcomes of a binomial experiment along with the probabilities of these outcomes. Ch5: Discrete Probability Distributions Santorico  Page 167
22 Probabilities for a Binomial Distribution In a binomial experiment, the probability of exactly X successes in n trials is P(X) n! X!(n X)! px (1 p) n X. Note: x! stands for x factorial where x is a nonnegative integer. x! x( x 1)( x 2)...(2)(1) when x > 0 0! 1 You can use your calculator or Table C at the back of the book to solve binomial probabilities for selected values of n and p. Ch5: Discrete Probability Distributions Santorico  Page 168
23 Examples: 5! = 5*4*3*2*1 = 120 8! = 5! = 5*4=20 3! 25! = Ch5: Discrete Probability Distributions Santorico  Page 169
24 Example: Dionne Warwick claims to possess ESP. An experiment is conducted to test her. A person in one room picks one of the integers 1, 2, 3, 4, 5 at random. In another room, Dionne identifies the number she believes was picked. The experiment is done with eight trials. Dionne gets the correct answer four times. If Dionne does not actually have ESP and is actually guessing the number, what is the probability that she d make a correct guess in four of the eight trials? We have a Binomial experiment here since with each guess she will either be right (success) or wrong (failure). If she does not have ESP, then the probability of a correct guess is 1/5. Hence, we would like to know P(X=4) given we have a Binomial distribution with n=8 and p=1/ ! 1 4 PX ( 4) !8! 5 5 Ch5: Discrete Probability Distributions Santorico  Page 170
25 Example: Consider a family with six children and suppose there is a 25% chance that each child will be a carrier of a particular mutated gene, independent of the other children. What is the probability that exactly 2 of the children will carry the mutated gene? What is the probability that 2 or less children will carry the mutated gene? Ch5: Discrete Probability Distributions Santorico  Page 171
26 Binomial Mean and Standard Deviation The binomial probability distribution for n trials with probability p of success on each trial has mean, variance 2, and standard deviation given by: np 2 npq npq Ch5: Discrete Probability Distributions Santorico  Page 172
27 Example: You will take a 10 question multiplechoice test with 4 possible answers for each question. Find the mean, variance, and standard deviation if you simply guess the answer for each question. Each question represents a success/failure. We have: X=number of correct answers n=10 p=0.25 np npq npq Ch5: Discrete Probability Distributions Santorico  Page 173
28 Example: In the U.S., 85% of the population has Rh positive blood. Suppose we take an independent random sample of 10,000 persons and count the number with Rh positive blood. Find the mean, variance, and standard deviation for the number of Rh positive individuals in the sample. Ch5: Discrete Probability Distributions Santorico  Page 174
Construct and Interpret Binomial Distributions
CH 6.2 Distribution.notebook A random variable is a variable whose values are determined by the outcome of the experiment. 1 CH 6.2 Distribution.notebook A probability distribution is a function which
More informationSection 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 informationChapter 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 informationProbability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections 4.1 4.2
Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections 4.1 4.2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability
More informationMULTIPLE 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 informationMA 1125 Lecture 14  Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.
MA 5 Lecture 4  Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More informationChapter 4. Probability Distributions
Chapter 4 Probability Distributions Lesson 41/42 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive
More informationProbability Distributions
Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.
More informationLecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000
Lecture 4 Nancy Pfenning Stats 000 Chapter 7: Probability Last time we established some basic definitions and rules of probability: Rule : P (A C ) = P (A). Rule 2: In general, the probability of one event
More informationChapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution
Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 35, 36 Special discrete random variable distributions we will cover
More informationCh. 13.2: Mathematical Expectation
Ch. 13.2: Mathematical Expectation Random Variables Very often, we are interested in sample spaces in which the outcomes are distinct real numbers. For example, in the experiment of rolling two dice, we
More informationReview for Test 2. Chapters 4, 5 and 6
Review for Test 2 Chapters 4, 5 and 6 1. You roll a fair sixsided die. Find the probability of each event: a. Event A: rolling a 3 1/6 b. Event B: rolling a 7 0 c. Event C: rolling a number less than
More informationChapter 5  Practice Problems 1
Chapter 5  Practice Problems 1 Identify the given random variable as being discrete or continuous. 1) The number of oil spills occurring off the Alaskan coast 1) A) Continuous B) Discrete 2) The ph level
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
STATISTICS/GRACEY PRACTICE TEST/EXAM 2 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identify the given random variable as being discrete or continuous.
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a realvalued function defined on the sample space of some experiment. For instance,
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationChapter 5. Random variables
Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like
More informationSOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions
SOLUTIONS: 4.1 Probability Distributions and 4.2 Binomial Distributions 1. The following table contains a probability distribution for a random variable X. a. Find the expected value (mean) of X. x 1 2
More informationThe Binomial Distribution
The Binomial Distribution James H. Steiger November 10, 00 1 Topics for this Module 1. The Binomial Process. The Binomial Random Variable. The Binomial Distribution (a) Computing the Binomial pdf (b) Computing
More informationYou flip a fair coin four times, what is the probability that you obtain three heads.
Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.
More information3.4. The Binomial Probability Distribution. Copyright Cengage Learning. All rights reserved.
3.4 The Binomial Probability Distribution Copyright Cengage Learning. All rights reserved. The Binomial Probability Distribution There are many experiments that conform either exactly or approximately
More information16. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION
6. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION It is sometimes difficult to directly compute probabilities for a binomial (n, p) random variable, X. We need a different table for each value of
More informationChapter 5: Discrete Probability Distributions
Chapter 5: Discrete Probability Distributions Section 5.1: Basics of Probability Distributions As a reminder, a variable or what will be called the random variable from now on, is represented by the letter
More informationNormal Distribution as an Approximation to the Binomial Distribution
Chapter 1 Student Lecture Notes 11 Normal Distribution as an Approximation to the Binomial Distribution : Goals ONE TWO THREE 2 Review Binomial Probability Distribution applies to a discrete random variable
More informationThe 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 informationSample Questions for Mastery #5
Name: Class: Date: Sample Questions for Mastery #5 Multiple Choice Identify the choice that best completes the statement or answers the question.. For which of the following binomial experiments could
More informationSolution Let us regress percentage of games versus total payroll.
Assignment 3, MATH 2560, Due November 16th Question 1: all graphs and calculations have to be done using the computer The following table gives the 1999 payroll (rounded to the nearest million dolars)
More informationMath 151. Rumbos Spring 2014 1. Solutions to Assignment #22
Math 151. Rumbos Spring 2014 1 Solutions to Assignment #22 1. An experiment consists of rolling a die 81 times and computing the average of the numbers on the top face of the die. Estimate the probability
More information2. Discrete random variables
2. Discrete random variables Statistics and probability: 21 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationAn Introduction to Basic Statistics and Probability
An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random
More informationFeb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172179)
Feb 7 Homework Solutions Math 151, Winter 2012 Chapter Problems (pages 172179) Problem 3 Three dice are rolled. By assuming that each of the 6 3 216 possible outcomes is equally likely, find the probabilities
More informationDiscrete Probability Distributions
blu497_ch05.qxd /5/0 :0 PM Page 5 Confirming Pages C H A P T E R 5 Discrete Probability Distributions Objectives Outline After completing this chapter, you should be able to Introduction Construct a probability
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 informationX X AP Statistics Solutions to Packet 7 X Random Variables Discrete and Continuous Random Variables Means and Variances of Random Variables
AP Statistics Solutions to Packet 7 Random Variables Discrete and Continuous Random Variables Means and Variances of Random Variables HW #44, 3, 6 8, 3 7 7. THREE CHILDREN A couple plans to have three
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability
More informationMind on Statistics. Chapter 8
Mind on Statistics Chapter 8 Sections 8.18.2 Questions 1 to 4: For each situation, decide if the random variable described is a discrete random variable or a continuous random variable. 1. Random variable
More informationSTT315 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 informationAP Statistics 7!3! 6!
Lesson 64 Introduction to Binomial Distributions Factorials 3!= Definition: n! = n( n 1)( n 2)...(3)(2)(1), n 0 Note: 0! = 1 (by definition) Ex. #1 Evaluate: a) 5! b) 3!(4!) c) 7!3! 6! d) 22! 21! 20!
More informationWHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
More informationProbability Review Solutions
Probability Review Solutions. A family has three children. Using b to stand for and g to stand for, and using ordered triples such as bbg, find the following. a. draw a tree diagram to determine the sample
More informationReview #2. Statistics
Review #2 Statistics Find the mean of the given probability distribution. 1) x P(x) 0 0.19 1 0.37 2 0.16 3 0.26 4 0.02 A) 1.64 B) 1.45 C) 1.55 D) 1.74 2) The number of golf balls ordered by customers of
More informationDETERMINE whether the conditions for a binomial setting are met. COMPUTE and INTERPRET probabilities involving binomial random variables
1 Section 7.B Learning Objectives After this section, you should be able to DETERMINE whether the conditions for a binomial setting are met COMPUTE and INTERPRET probabilities involving binomial random
More informationRandom Variables and Probability
CHAPTER 9 Random Variables and Probability IN THIS CHAPTER Summary: We ve completed the basics of data analysis and we now begin the transition to inference. In order to do inference, we need to use the
More informationExample: Find the expected value of the random variable X. X 2 4 6 7 P(X) 0.3 0.2 0.1 0.4
MATH 110 Test Three Outline of Test Material EXPECTED VALUE (8.5) Super easy ones (when the PDF is already given to you as a table and all you need to do is multiply down the columns and add across) Example:
More informationWeek 5: Expected value and Betting systems
Week 5: Expected value and Betting systems Random variable A random variable represents a measurement in a random experiment. We usually denote random variable with capital letter X, Y,. If S is the sample
More information4.1 4.2 Probability Distribution for Discrete Random Variables
4.1 4.2 Probability Distribution for Discrete Random Variables Key concepts: discrete random variable, probability distribution, expected value, variance, and standard deviation of a discrete random variable.
More informationDecision Making Under Uncertainty. Professor Peter Cramton Economics 300
Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam 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 that the result
More informationChapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.
Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
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 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 informationThe normal approximation to the binomial
The normal approximation to the binomial The binomial probability function is not useful for calculating probabilities when the number of trials n is large, as it involves multiplying a potentially very
More information2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways.
Math 142 September 27, 2011 1. How many ways can 9 people be arranged in order? 9! = 362,880 ways 2. How many ways can the letters in PHOENIX be rearranged? 7! = 5,040 ways. 3. The letters in MATH are
More informationThe mathematical branch of probability has its
ACTIVITIES for students Matthew A. Carlton and Mary V. Mortlock Teaching Probability and Statistics through Game Shows The mathematical branch of probability has its origins in games and gambling. And
More informationDiscrete Math in Computer Science Homework 7 Solutions (Max Points: 80)
Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you
More informationRandom 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 informationProbability Distributions
CHAPTER 5 Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Probability Distribution 5.3 The Binomial Distribution
More informationNotes 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 informationTHE BINOMIAL DISTRIBUTION & PROBABILITY
REVISION SHEET STATISTICS 1 (MEI) THE BINOMIAL DISTRIBUTION & PROBABILITY The main ideas in this chapter are Probabilities based on selecting or arranging objects Probabilities based on the binomial distribution
More informationECE302 Spring 2006 HW4 Solutions February 6, 2006 1
ECE302 Spring 2006 HW4 Solutions February 6, 2006 1 Solutions to HW4 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
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 informationSession 8 Probability
Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome
More informationQuestion: What is the probability that a fivecard 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 informationECE302 Spring 2006 HW3 Solutions February 2, 2006 1
ECE302 Spring 2006 HW3 Solutions February 2, 2006 1 Solutions to HW3 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in
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 informationChapter 3 RANDOM VARIATE GENERATION
Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.
More informationProbability and Expected Value
Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are
More informationNUMBER SYSTEMS. William Stallings
NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html
More informationUniversity of California, Los Angeles Department of Statistics. Random variables
University of California, Los Angeles Department of Statistics Statistics Instructor: Nicolas Christou Random variables Discrete random variables. Continuous random variables. Discrete random variables.
More informationChapter 5. Decimals. Use the calculator.
Chapter 5. Decimals 5.1 An Introduction to the Decimals 5.2 Adding and Subtracting Decimals 5.3 Multiplying Decimals 5.4 Dividing Decimals 5.5 Fractions and Decimals 5.6 Square Roots 5.7 Solving Equations
More informationSTAT 3502. x 0 < x < 1
Solution  Assignment # STAT 350 Total mark=100 1. A large industrial firm purchases several new word processors at the end of each year, the exact number depending on the frequency of repairs in the previous
More information5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.
The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution
More informationImportant Probability Distributions OPRE 6301
Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in reallife applications that they have been given their own names.
More informationProbability Models.S1 Introduction to Probability
Probability Models.S1 Introduction to Probability Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard The stochastic chapters of this book involve random variability. Decisions are
More informationIntroduction to the Practice of Statistics Fifth Edition Moore, McCabe
Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 5.1 Homework Answers 5.7 In the proofreading setting if Exercise 5.3, what is the smallest number of misses m with P(X m)
More information6.4 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 information14.4. Expected Value Objectives. Expected Value
. Expected Value Objectives. Understand the meaning of expected value. 2. Calculate the expected value of lotteries and games of chance.. Use expected value to solve applied problems. Life and Health Insurers
More informationBinomial Probability Distribution
Binomial Probability Distribution In a binomial setting, we can compute probabilities of certain outcomes. This used to be done with tables, but with graphing calculator technology, these problems are
More informationChapter 2: Systems of Linear Equations and Matrices:
At the end of the lesson, you should be able to: Chapter 2: Systems of Linear Equations and Matrices: 2.1: Solutions of Linear Systems by the Echelon Method Define linear systems, unique solution, inconsistent,
More informationProbability Generating Functions
page 39 Chapter 3 Probability Generating Functions 3 Preamble: Generating Functions Generating functions are widely used in mathematics, and play an important role in probability theory Consider a sequence
More informationGaming the Law of Large Numbers
Gaming the Law of Large Numbers Thomas Hoffman and Bart Snapp July 3, 2012 Many of us view mathematics as a rich and wonderfully elaborate game. In turn, games can be used to illustrate mathematical ideas.
More informationRandom variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.
Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationExam 3 Review/WIR 9 These problems will be started in class on April 7 and continued on April 8 at the WIR.
Exam 3 Review/WIR 9 These problems will be started in class on April 7 and continued on April 8 at the WIR. 1. Urn A contains 6 white marbles and 4 red marbles. Urn B contains 3 red marbles and two white
More informationLecture 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 informationMath 431 An Introduction to Probability. Final Exam Solutions
Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <
More informationPractice 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 multiplechoice
More informationSection 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 informationStatistics Class 10 2/29/2012
Statistics Class 10 2/29/2012 Quiz 8 When playing roulette at the Bellagio casino in Las Vegas, a gambler is trying to decide whether to bet $5 on the number 13 or to bet $5 that the outcome is any one
More informationMAT 155. Key Concept. September 22, 2010. 155S5.3_3 Binomial Probability Distributions. Chapter 5 Probability Distributions
MAT 155 Dr. Claude Moore Cape Fear Community College Chapter 5 Probability Distributions 5 1 Review and Preview 5 2 Random Variables 5 3 Binomial Probability Distributions 5 4 Mean, Variance, and Standard
More informationChapter 5: Normal Probability Distributions  Solutions
Chapter 5: Normal Probability Distributions  Solutions Note: All areas and zscores are approximate. Your answers may vary slightly. 5.2 Normal Distributions: Finding Probabilities If you are given that
More informationNormal and Binomial. Distributions
Normal and Binomial Distributions Library, Teaching and Learning 14 By now, you know about averages means in particular and are familiar with words like data, standard deviation, variance, probability,
More informationSECTION 105 Multiplication Principle, Permutations, and Combinations
105 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 informationV. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE
V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay $ to play. A penny and a nickel are flipped. You win $ if either
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 0.4987 B) 0.9987 C) 0.0010 D) 0.
Ch. 5 Normal Probability Distributions 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 1 Find Areas Under the Standard Normal Curve 1) Find the area under the standard normal
More informationStat 20: Intro to Probability and Statistics
Stat 20: Intro to Probability and Statistics Lecture 16: More Box Models Tessa L. ChildersDay UC Berkeley 22 July 2014 By the end of this lecture... You will be able to: Determine what we expect the sum
More informationChapter 6. 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? Ans: 4/52.
Chapter 6 1. What is the probability that a card chosen from an ordinary deck of 52 cards is an ace? 4/52. 2. What is the probability that a randomly selected integer chosen from the first 100 positive
More information