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Submitted By josephbickers

Words 883

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Words 883

Pages 4

Learning outcome:

Upon completion, students will be able to… * Compute experimental and theoretical probabilities using basic laws of probability.

Scoring/Grading Rubric: * Part 1: 5 points * Part 2: 5 points * Part 3: 22 points (2 per sum of 2-12) * Part 4: 5 points * Part 5: 5 points * Part 6: 38 points (4 per sum of 4-12, 2 per sum of 3) * Part 7: 10 points * Part 8: 10 points

Introduction:

While it is fairly simple to understand the outcomes of a single die roll, the outcomes when rolling two dice are a little more complicated. The goal of this lab is to get a better understanding of these outcomes and the probabilities that go with them. We will examine and compare the experimental and theoretical probabilities for rolling two dice and obtaining a certain sum.

Directions:

1. (5 pts) You are going to roll a pair of dice 108 times and record the sum of each roll. Before beginning, make a prediction about how you think the sums will be distributed. (Each sum will occur equally often, there will be more 12s than any other sum, there will be more 5s than any other sum, etc.) Record your prediction here: The more combinations available, the more possibility that the dice will roll that number. For example- there is only one way you can get 2, with rolling the pair of dice with 1 on each. Now with for example 8, you can roll a 3 and a 5 or a 2 and a 6 or a 4 and a 4 which means there is more possibility you will get the sum of 8 over 1.

2. (5 pts) Roll the dice 108 times and record the sum of each roll in the table provided below. 10 | 7 | 10 | 8 | 11 | 6 | 3 | 8 | 3 | 2 | 7 | 7 | 7 | 11 | 9 | 8 | 7 | 12 | 10 | 9 | 4 | 7 | 8 | 7 | 8 | 10 | 7 | 8 | 7 | 7 | 6 | 8 | 7 | 6 | 6 | 5 | 9 | 5 | 5 | 3 | 4 | 8 | 3 | 8 | 2 | 9 | 7 | 5 | 7 | 3 | 6 | 4 | 7 | 10 | 3 | 2 | 7 | 7…...

... Probability – the chance that an uncertain event will occur (always between 0 and 1) Impossible Event – an event that has no chance of occurring (probability = 0) Certain Event – an event that is sure to occur (probability = 1) Assessing Probability probability of occurrence= probability of occurrence based on a combination of an individual’s past experience, personal opinion, and analysis of a particular situation Events Simple event An event described by a single characteristic Joint event An event described by two or more characteristics Complement of an event A , All events that are not part of event A The Sample Space is the collection of all possible events Simple Probability refers to the probability of a simple event. Joint Probability refers to the probability of an occurrence of two or more events. ex. P(Jan. and Wed.) Mutually exclusive events is the Events that cannot occur simultaneously Example: Randomly choosing a day from 2010 A = day in January; B = day in February Events A and B are mutually exclusive Collectively exhaustive events One of the events must occur the set of events covers the entire sample space Computing Joint and Marginal Probabilities The probability of a joint event, A and B: Computing a marginal (or simple) probability: Probability is the numerical measure of the likelihood that an event will occur The probability of any event must be between 0 and 1,...

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...Statistics 100A Homework 5 Solutions Ryan Rosario Chapter 5 1. Let X be a random variable with probability density function c(1 − x2 ) −1 < x < 1 0 otherwise ∞ f (x) = (a) What is the value of c? We know that for f (x) to be a probability distribution −∞ f (x)dx = 1. We integrate f (x) with respect to x, set the result equal to 1 and solve for c. 1 1 = −1 c(1 − x2 )dx cx − c x3 3 1 −1 = = = = c = Thus, c = 3 4 c c − −c + c− 3 3 2c −2c − 3 3 4c 3 3 4 . (b) What is the cumulative distribution function of X? We want to ﬁnd F (x). To do that, integrate f (x) from the lower bound of the domain on which f (x) = 0 to x so we will get an expression in terms of x. x F (x) = −1 c(1 − x2 )dx cx − cx3 3 x −1 = But recall that c = 3 . 4 3 1 3 1 = x− x + 4 4 2 = 3 4 x− x3 3 + 2 3 −1 < x < 1 elsewhere 0 1 4. The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by, 10 x2 f (x) = (a) Find P (X > 20). 0 x > 10 x ≤ 10 There are two ways to solve this problem, and other problems like it. We note that the area we are interested in is bounded below by 20 and unbounded above. Thus, ∞ P (X > c) = c f (x)dx Unlike in the discrete case, there is not really an advantage to using the complement, but you can of course do so. We could consider P (X > c) = 1 − P (X < c), c P (X > c) = 1 − P (X < c) = 1 − −∞ f (x)dx P (X > 20) = 10 dx......

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...Probability & Statistics for Engineers & Scientists This page intentionally left blank Probability & Statistics for Engineers & Scientists NINTH EDITION Ronald E. Walpole Roanoke College Raymond H. Myers Virginia Tech Sharon L. Myers Radford University Keying Ye University of Texas at San Antonio Prentice Hall Editor in Chief: Deirdre Lynch Acquisitions Editor: Christopher Cummings Executive Content Editor: Christine O’Brien Associate Editor: Christina Lepre Senior Managing Editor: Karen Wernholm Senior Production Project Manager: Tracy Patruno Design Manager: Andrea Nix Cover Designer: Heather Scott Digital Assets Manager: Marianne Groth Associate Media Producer: Vicki Dreyfus Marketing Manager: Alex Gay Marketing Assistant: Kathleen DeChavez Senior Author Support/Technology Specialist: Joe Vetere Rights and Permissions Advisor: Michael Joyce Senior Manufacturing Buyer: Carol Melville Production Coordination: Liﬂand et al. Bookmakers Composition: Keying Ye Cover photo: Marjory Dressler/Dressler Photo-Graphics Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Pearson was aware of a trademark claim, the designations have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publication Data Probability & statistics for engineers & scientists/Ronald E. Walpole . . . [et al.] — 9th ed. p. cm. ISBN......

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...Probability: Introduction to Basic Concept Uncertainty pervades all aspects of human endeavor. Probability is one of our most important conceptual tools because we use it to assess degrees of uncertainty and thereby to reduce risk. Whether or not one has had formal instruction in this topic, s/he is already familiar with the concept of probability since it pervades almost all aspects of our lives. With out consciously realizing it many of our decisions are based on probability. For example, when you study for an examination, you concentrate more on areas that you feel are likely to be covered on the test. You may cancel or postpone an out door activity if you believe the likelihood of rain is high. In business, probability plays a key role in decision-making. The owner of a retail shoe store, for example, orders heavily in those sizes that s/he believes likely to sell fast. The owner of a movie theatre schedules matinees only during holiday seasons because the chances of filling the theatre are greater at that time. The two companies decide to merge when they believe the probability of success is greater for the consolidated company than for either independently. Some important Definitions: Experiment: Experiment is an act that can be repeated under given conditions. Usually, the exact result of the experiment cannot be predicted with certainly. Unit experiment is known as trial. This means that trial is a special case of experiment. Experiment may be a......

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...What are the two basic laws of probability? What are the differences between a discrete probability distribution and a continuous probability distribution? Provide at least one example of each type of probability distribution. The two basic laws of probability are adding mutually exclusive events and addition for events that are not mutually exclusive (Render, Stair & Hanna, 2008). The probability must be between zero and one for any event just as the sum must equal one of all the events. Therefore, according to Render, Stair& Hanna (2008), zero would represent an event that is never expected to occur while one represents an event that is always expected to occur. Probability distributions are represented by two types, discrete and continuous which is dependent upon probability association in regards to the random variables (Render, Stair & Hanna, 2008). The random variable is discrete only if it can assume finite or limited set of values. Probability values are assigned to each event in a discrete random variable which is between zero and one and must add up to one. The probability distribution is often described by its mean and variance. The continuous variable has to have an infinite or an unlimited set of values. Continuous probability distribution is a continuous mathematical function that describes the probability distribution which can be graphed to reflect the area under the curve as the probability as it is associated with the range of interest (Render,......

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...Problem 1: Question 1. The probability of a case being appealed for each judge in Common Pleas Court. p(a) | 0.04511031 | 0.03529063 | 0.03497615 | 0.03070624 | 0.04047164 | 0.04019435 | 0.03990765 | 0.04427171 | 0.03883194 | 0.04085893 | 0.04033333 | 0.04344897 | 0.04524181 | 0.06282723 | 0.04043298 | 0.02848818 | Question 2. The probability of a case being reversed for each judge in Common Pleas Court. P® | 0.00395127 | 0.0029656 | 0.0063593 | 0.0035824 | 0.00223072 | 0.00795053 | 0.00725594 | 0.00675904 | 0.00434918 | 0.00477185 | 0.002 | 0.00404176 | 0.00561622 | 0.0104712 | 0.00413881 | 0.00194238 | Question 3. The probability of reversal given an appeal for each judge in Common Pleas Court. p(R/A) | 0.08759124 | 0.08403361 | 0.18181818 | 0.11666667 | 0.05511811 | 0.1978022 | 0.18181818 | 0.15267176 | 0.112 | 0.11678832 | 0.04958678 | 0.09302326 | 0.12413793 | 0.16666667 | 0.1023622 | 0.06818182 | Question 4. The probability of cases being appealed in Common Pleas Court. Probability of cases being appealed in common pleas court | 0.0400956 | Question 5. Identify the best judges in Common Pleas Court according to the three criteria in Questions 1-3: 1) The best judge in Common Pleas Court with the smallest probability in Question 1; Ralph Winkler 2) The best judge in Common Pleas Court with the smallest probability in Question 2; and ......

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...Probability Discrete Event Simulation: Tutorial on Probability and Random Variables 1. A card is drawn from an ordinary deck of 52 playing cards. Find the probability that it is (a) an Ace, (b) a jack of hearts, (c) a three of clubs or a six of diamonds, (d) a heart, (e) any suit except hearts, (f) a ten of spade, (g) neither a four nor a club (1/3, 1/52, 1/26, 1/4, 3/4, 4/13, 9/13) 2. A ball is drawn at random from a box containing six red balls, 4 white balls and 5 blue balls. Determine the probability that it is (a) red, (b) white, (c) blue, (d) not red, (e) red or white (2/5, 4/15, 1/3, 3/5, 2/3) 3. Three balls are drawn successively from the box in the previous problem. Find the probability that they are drawn in the order red, white and blue if each ball is (a) replaced (b) not replaced (8/225, 4/91) 4. A fair die is tossed twice. Find the probability of getting a 4, 5 or 6 on the ﬁrst toss and a 1, 2, 3, or 4 on the second toss. (1/3) 5. Find the probability of not getting a 7 or 11 total on either of two tosses of a pair of fair dice. (49/81) 6. Two cards are drawn from a well-shuﬄed ordinary deck of 52 cards. Find the probability that they are both aces if the ﬁrst card is (a) replaced, (b) not replaced. (1/169, 1/221) 7. Box I contains 3 red and 2 blue marbles while box II contains 2 red marbles and 8 blue marbles. A fair coin is tossed. If the coin turns up heads a marble is chosen from box I and if it turns up tails, a marble......

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...PROBABILITY ASSIGNMENT 1. The National Highway Traffic Safety Administration (NHTSA) conducted a survey to learn about how drivers throughout the US are using their seat belts. Sample data consistent with the NHTSA survey are as follows. (Data as on May, 2015) Driver using Seat Belt? | Region | Yes | No | Northeast | 148 | 52 | Midwest | 162 | 54 | South | 296 | 74 | West | 252 | 48 | Total | 858 | 228 | a. For the U.S., what is the probability that the driver is using a seat belt? b. The seat belt usage probability for a U.S. driver a year earlier was .75. NHTSA Chief had hoped for a 0.78 probability in 2015. Would he have been pleased with the 2003 survey results? c. What is the probability of seat belt usage by region of the Country? What region has the highest seat belt usage? d. What proportion of the drivers in the sample came from each region of the country? What region had the most drivers selected? 2. A company that manufactures toothpaste is studying five different package designs. Assuming that one design is just as likely to be selected by a consumer as any other design, what selection probability would you assign to each of the package designs? In an actual experiment, 100 consumers were asked to pick the design they preferred. The following data were obtained. Do the data confirm the belief that one design is just as likely to be selected as other? Explain. Design | Number of Times Preferred | 1 | 5 | 2 |......

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...Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #07 Random Variables So, far we were discussing the laws of probability so, in the laws of the probability we have a random experiment, as a consequence of that we have a sample space, we consider a subset of the, we consider a class of subsets of the sample space which we call our event space or the events and then we define a probability function on that. Now, we consider various types of problems for example, calculating the probability of occurrence of a certain number in throwing of a die, probability of occurrence of certain card in a drain probability of various kinds of events. However, in most of the practical situations we may not be interested in the full physical description of the sample space or the events; rather we may be interested in certain numerical characteristic of the event, consider suppose I have ten instruments and they are operating for a certain amount of time, now after amount after working for a certain amount of time, we may like to know that, how many of them are actually working in a proper way and how many of them are not working properly. Now, if there are ten instruments, it may happen that seven of them are working properly and three of them are not working properly, at this stage we may not be interested in knowing the positions, suppose we are saying one instrument, two instruments and so, on tenth...

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... Title: The Probability that the Sum of two dice when thrown is equal to seven Purpose of Project * To carry out simple experiments to determine the probability that the sum of two dice when thrown is equal to seven. Variables * Independent- sum * Dependent- number of throws * Controlled- Cloth covered table top. Method of data collection 1. Two ordinary six-faced gaming dice was thrown 100 times using three different method which can be shown below. i. The dice was held in the palm of the hand and shaken around a few times before it was thrown onto a cloth covered table top. ii. The dice was placed into a Styrofoam cup and shaken around few times before it was thrown on a cloth covered table top. iii. The dice was placed into a glass and shaken around a few times before it was thrown onto a cloth covered table top. 2. All result was recoded and tabulated. 3. A probability tree was drawn. Presentation of Data Throw by hand Sum of two dice | Frequency | 23456789101112 | 4485161516121172 | Throw by Styrofoam cup Sum of two dice | Frequency | 23456789101112 | 2513112081481072 | Throw by Glass Sum of two dice | Frequency | 23456789101112 | 18910121214121174 | Sum oftwo dice | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | Total | Experiment1 | 4 | 4 | 8 | 5 | 16 | 15 | 16 | 12 | 11 | 7 | 2 | 100 | Experiment2 | 2 | 5 | 13 | 11 | 20 | 8 | 14 | 8 | 10 | 7 | 2 | 100 | Experiment3 | 1 | 8 | 9 | 10 | 12 | 12 | 13 | 12 |......

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...CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11 Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According to clinical trials, the test has the following properties: 1. When applied to an affected person, the test comes up positive in 90% of cases, and negative in 10% (these are called “false negatives”). 2. When applied to a healthy person, the test comes up negative in 80% of cases, and positive in 20% (these are called “false positives”). Suppose that the incidence of the condition in the US population is 5%. When a random person is tested and the test comes up positive, what is the probability that the person actually has the condition? (Note that this is presumably not the same as the simple probability that a random person has the condition, which is 1 just 20 .) This is an example of a conditional probability: we are interested in the probability that a person has the condition (event A) given that he/she tests positive (event B). Let’s write this as Pr[A|B]. How should we deﬁne Pr[A|B]? Well, since event B is guaranteed to happen, we should look not at the whole sample space Ω , but at the smaller sample space consisting only of the sample points in B. What should the conditional probabilities of these sample points be? If they all simply inherit their probabilities from Ω , then the sum of these probabilities will be ∑ω ∈B Pr[ω ] = Pr[B],......

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...Shamara MAT 143 03/02/16 Module 6 Project 1. (5 pts) You are going to roll a pair of dice 108 times and record the sum of each roll. Before beginning, make a prediction about how you think the sums will be distributed. (Each sum will occur equally often, there will be more 12s than any other sum, there will be more 5s than any other sum, etc.) Record your prediction here: I think in the beginning the sums will vary but then eventually repeat, and there will be more 7’s then any other sum. 2. (5 pts) Roll the dice 108 times and record the sum of each roll in the table provided below. 7 3 4 8 11 3 6 4 2 10 5 9 7 7 12 10 10 3 5 2 2 6 4 8 5 9 2 7 9 7 7 12 12 4 6 3 9 8 2 3 12 6 9 7 7 7 12 5 6 8 2 4 7 7 8 10 3 5 5 4 7 7 12 4 6 8 3 12 7 7 8 3 10 3 5 8 7 9 6 6 12 4 7 9 9 11 4 5 2 5 2 3 5 3 10 2 2 4 3 6 3 10 7 7 9 8 5 12 3. (22 pts) Find the experimental probability of rolling each sum. Fill out the following table: Sum of the dice Number of times each sum occurred Probability of occurrence for each sum out of your 108 total rolls (record your probabilities to three decimal places) 2 10 .092 3 13 .120 4 10 .092 5 11 .101 6 9 .083 7 19 .175 8 9 .083 9 9 .083 10 7 .064 11 2 .018 12 9 .083 4. (5 pts) Compare your outcomes to your prediction. Was your prediction correct? Why do you think this happened? Yes, my predictions were correct and I think it just happened coincidently. 5. (5 pts) What number(s)......

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...Probability XXXXXXXX MAT300 Professor XXXXXX Date Probability Probability is commonly applied to indicate an outlook of the mind with respect to some hypothesis whose facts are not yet sure. The scheme of concern is mainly of the frame “would a given incident happen?” the outlook of the mind is of the type “how sure is it that the incident would happen?” The surety we applied may be illustrated in form of numerical standards and this value ranges between 0 and 1; this is referred to as probability. The greater the probability of an incident, the greater the surety that the incident will take place. Therefore, probability in a used perspective is a measure of the likeliness, which a random incident takes place (Olofsson, 2005). The idea has been presented as a theoretical mathematical derivation within the probability theory that is applied in a given fields of study like statistics, mathematics, gambling, philosophy, finance, science, and artificial machine/intelligence learning. For instance, draw deductions concerning the likeliness of incidents. Probability is applied to show the underlying technicalities and regularities of intricate systems. Nevertheless, the term probability does not have any one straight definition for experimental application. Moreover, there are a number of wide classifications of probability whose supporters have varied or even conflicting observations concerning the vital state of probability. Just as other......

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...PROBABILITY SEDA YILDIRIM 2009421051 DOKUZ EYLUL UNIVERSITY MARITIME BUSINESS ADMINISTRATION CONTENTS Rules of Probability 1 Rule of Multiplication 3 Rule of Addition 3 Classical theory of probability 5 Continuous Probability Distributions 9 Discrete vs. Continuous Variables 11 Binomial Distribution 11 Binomial Probability 12 Poisson Distribution 13 PROBABILITY Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions. In common usage, the word "probability" is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%. The analysis of events governed by probability is called statistics. There are several competing interpretations of the actual "meaning" of probabilities. Frequentists view probability simply as a measure of the frequency of outcomes (the more conventional interpretation), while Bayesians treat probability more subjectively as a statistical procedure that endeavors to estimate parameters of an underlying distribution based on the observed distribution. The conditional probability of an event A assuming that B has occurred, denoted ,equals The two faces of probability introduces a central ambiguity which has been around for 350 years and still leads to disagreements about...

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...Massachusetts Institute of Technology 6.042J/18.062J, Fall ’02: Mathematics for Computer Science Professor Albert Meyer and Dr. Radhika Nagpal Course Notes 10 November 4 revised November 6, 2002, 572 minutes Introduction to Probability 1 Probability Probability will be the topic for the rest of the term. Probability is one of the most important subjects in Mathematics and Computer Science. Most upper level Computer Science courses require probability in some form, especially in analysis of algorithms and data structures, but also in information theory, cryptography, control and systems theory, network design, artiﬁcial intelligence, and game theory. Probability also plays a key role in ﬁelds such as Physics, Biology, Economics and Medicine. There is a close relationship between Counting/Combinatorics and Probability. In many cases, the probability of an event is simply the fraction of possible outcomes that make up the event. So many of the rules we developed for ﬁnding the cardinality of ﬁnite sets carry over to Probability Theory. For example, we’ll apply an Inclusion-Exclusion principle for probabilities in some examples below. In principle, probability boils down to a few simple rules, but it remains a tricky subject because these rules often lead unintuitive conclusions. Using “common sense” reasoning about probabilistic questions is notoriously unreliable, as we’ll illustrate with many real-life examples. This reading is longer than usual . To keep things in......

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