... 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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...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 theories...
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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], which...
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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 bounds...
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...Probability and Distributions Abstract This paper will discuss the trends and data values and how they relate to statistical terms. Also will describe the probability of different actions to the same group of data. The data will be broke down accordingly to qualitative and quantitative data, and will be grouped and manipulated to show how the data in each group can prove to be useful in the workplace. Memo To: Head of American Intellectual Union From: Abby Price Date: 3/05/2014 Subject: Data analysis from within the union’s surveys Dear Dr. Common: I will be analyzing data given to me which was taken from a survey within the union from 186 employees. I will discuss probability and how its information is important in the workplace. Overview of the Data Set The data group I was given to analyze has 9 categories: gender, age, department, position, tenure, job satisfaction, intrinsic, extrinsic, and benefits. The employees were asked to rate on a scale of 1-7 on how satisfied they were with the company. Gender, age, department, position and tenure are all qualitative data. This data is acknowledged by a code on the given data but cannot measured unlike the quantitative data: job satisfaction, intrinsic, extrinsic, and benefits. Use of Statistics and Probability in the Real World Statistics are just about everywhere in the business world, from the upper management to the lower line of employees, statistics are very useful and are a huge part of our...
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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 trial...
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...Unit 2 – Probability and Distributions Kimberly Reed American InterContinental University Abstract This week’s paper focuses on an email that will be written to AUI the email will contain information from the data set key and explain why this information is important to the company. Memo To: HR Department From: Senior Manager Date: 20 Sept, 2011 Subject: Data Set Dear Department Heads: The following memo will contain information that contains vital and confidential information. This information will need to be studied by all department heads. Overview of the data set This data set of information contains information on the breakdown of the survey that was conducted on the company Use of statistics and probability in the real world Companies use statistics in the real world to get and have an advantage. They can be used for things such as knowing the latest stats on a sports figure or what items a consumer will likely buy from the local hardware store Distributions Distribution table contains the information that gives the breakdown of how the study was conducted and who the participants were in the study. This information is important to AIU for the company will be able to better prepare for the future when they know how to better manage their work force Then complete the following distribution tables. Please pay attention to whether you should present the results in terms of percentages or simple counts. Gender |Gender |Percentage...
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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 x2...
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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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...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 978-0-321-62911-1...
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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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...PROBABILITY 1. ACCORDING TO STATISTICAL DEFINITION OF PROBABILITY P(A) = lim FA/n WHERE FA IS THE NUMBER OF TIMES EVENT A OCCUR AND n IS THE NUMBER OF TIMES THE EXPERIMANT IS REPEATED. 2. IF P(A) = 0, A IS KNOWN TO BE AN IMPOSSIBLE EVENT AND IS P(A) = 1, A IS KNOWN TO BE A SURE EVENT. 3. BINOMIAL DISTRIBUTIONS IS BIPARAMETRIC DISTRIBUTION, WHERE AS POISSION DISTRIBUTION IS UNIPARAMETRIC ONE. 4. THE CONDITIONS FOR THE POISSION MODEL ARE : • THE PROBABILIY OF SUCCESS IN A VERY SMALL INTERAVAL IS CONSTANT. • THE PROBABILITY OF HAVING MORE THAN ONE SUCCESS IN THE ABOVE REFERRED SMALL TIME INTERVAL IS VERY LOW. • THE PROBABILITY OF SUCCESS IS INDEPENDENT OF t FOR THE TIME INTERVAL(t ,t+dt) . 5. Expected Value or Mathematical Expectation of a random variable may be defined as the sum of the products of the different values taken by the random variable and the corresponding probabilities. Hence if a random variable X takes n values X1, X2,………… Xn with corresponding probabilities p1, p2, p3, ………. pn, then expected value of X is given by µ = E (x) = Σ pi xi . Expected value of X2 is given by E ( X2 ) = Σ pi xi2 Variance of x, is given by σ2 = E(x- µ)2 = E(x2)- µ2 Expectation of a constant k is k i.e. E(k) = k fo any constant k. Expectation of sum of two random variables is the sum of their expectations i.e. E(x +y) = E(x) + E(y) for any two...
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...Probability, Mean and Median In the last section, we considered (probability) density functions. We went on to discuss their relationship with cumulative distribution functions. The goal of this section is to take a closer look at densities, introduce some common distributions and discuss the mean and median. Recall, we define probabilities as follows: Proportion of population for Area under the graph of p ( x ) between a and b which x is between a and b p( x)dx a b The cumulative distribution function gives the proportion of the population that has values below t. That is, t P (t ) p( x)dx Proportion of population having values of x below t When answering some questions involving probabilities, both the density function and the cumulative distribution can be used, as the next example illustrates. Example 1: Consider the graph of the function p(x). p x 0.2 0.1 2 4 6 8 10 x Figure 1: The graph of the function p(x) a. Explain why the function is a probability density function. b. Use the graph to find P(X < 3) c. Use the graph to find P(3 § X § 8) 1 Solution: a. Recall, a function is a probability density function if the area under the curve is equal to 1 and all of the values of p(x) are non-negative. It is immediately clear that the values of p(x) are non-negative. To verify that the area under the curve is equal to 1, we recognize that the graph above can be viewed as a triangle. Its...
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...Review Chapter 18 discussed Theoretical Probability and Statistical Inference. Jakob Bernoulli, wanted to be able to quantify probabilities by looking at the results observed in many similar instances. It seemed reasonably obvious to Bernoulli that the more observations one made of a given situation, the better one would be able to predict future occurrences. Bernoulli presented this scientific proof in his theorem, the “Law of Large Numbers”, where the same experiment is performed a large number of times. De Moivre also made contributions in the area of Theoretical Probability. His major mathematical work, “The Doctrine of Chances,” began with a precise definition of probability. His definition stated that, “The Probability of an Event is greater or less, according to the number of Chances by which it may either happen or fail.” De Moivre used his definition in solving problems, such as the dice problem of de Mere. Another concept discussed in this chapter is Statistical Inference. Statistical Inference is the process of drawing conclusions from data that are subject to random variation. Thomas Bayes and Pierre Laplace were the first to attempt a direct answer to the question of how to determine probability from observed frequencies. Bayes develop a theorem that states if X represents the number of times the event has happened in n trials, x the probability of its happening in a single trial, and r and s the two given probabilities, Baye’s aim was to calculate P(r<x<s|X)...
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...DICE AND PROBABILITY LAB 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...
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