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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, inclusively

The sum of the probabilities of all mutually exclusive and collectively exhaustive events is 1

General Addition Rule

P(A or B) = P(A) + P(B) - P(A and B)

If A and B are mutually exclusive, then

P(A and B) = 0, so the rule can be simplified:

P(A or B) = P(A) + P(B)

For mutually exclusive events A and B

Computing Conditional Probabilities

A conditional…...

...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, 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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...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 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......

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...Odd-Numbered End-of-Chapter Exercises * Chapter 2 Review of Probability 2.1. (a) Probability distribution function for Y Outcome (number of heads) | Y 0 | Y 1 | Y 2 | Probability | 0.25 | 0.50 | 0.25 | (b) Cumulative probability distribution function for Y Outcome (number of heads) | Y 0 | 0 Y 1 | 1 Y 2 | Y 2 | Probability | 0 | 0.25 | 0.75 | 1.0 | (c) . Using Key Concept 2.3: and so that 2.3. For the two new random variables and we have: (a) (b) (c) 2.5. Let X denote temperature in F and Y denote temperature in C. Recall that Y 0 when X 32 and Y 100 when X 212; this implies Using Key Concept 2.3, X 70oF implies that and X 7oF implies 2.7. Using obvious notation, thus and This implies (a) per year. (b) , so that Thus where the units are squared thousands of dollars per year. (c) so that and thousand dollars per year. (d) First you need to look up the current Euro/dollar exchange rate in the Wall Street Journal, the Federal Reserve web page, or other financial data outlet. Suppose that this exchange rate is e (say e 0.80 Euros per dollar); each 1 dollar is therefore with e Euros. The mean is therefore e C (in units of thousands of Euros per year), and the standard deviation is e C (in units of thousands of Euros per year). The correlation is unit-free, and is unchanged. 2.9. | | Value of Y | Probability Distribution of X | | | 14 | 22 | 30 | 40 | 65 | | | Value......

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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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...concerning the market potential. the product management team asks you for an analysis of the stock-out probabilites for various order quantities, an estimate of the profit potential, and to help make an order quantity recommendation. Specialty (the company name) expects to sell Weather Teddy (the product) for $24 based on a cost of $16 per unit. If inventory remains after the holiday season, Specialty will sell all surplus inventory for $5 per unit. After reviewing the sales history of similiar products, Specialty's senior sales forecaster predicted an expected demand of 20,000 units with a 0.95 probability that demand would be between 10,000 units and 30,000 units. 1. Use the sales forecaster's prediction to describe a normal probability distribution that can be used to approximate the demand distribution. Sketch the distribution and show its mean and standard deviation. 2. Compute the probability of a stock-out for the order quantities suggested by members of the management team. 3. Compute the projected profit for the rder quantities suggested by the management team under three scenarios: worst case in which sales = 10,000 units, most likely case in which sales = 20,000 units, and best case in which sales = 30,000 units. 4. One of Specialty's managers felt that the profit potential was so great that the order quantity should have a 70% chance of meeting demand and only a 30% chance of any stock-outs. What quantity would be ordered under this policy, and what is......

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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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...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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...Probability Question 1 The comparison between the bar chart and histogram are bar graphs are normally used to represent the frequency of discrete items. They can be things, like colours, or things with no particular order. But the main thing about it is the items are not grouped, and they are not continuous. Where else for the histogram is mainly used to represent the frequency of a continuous variable like height or weight and anything that has a decimal placing and would not be exact in other words a whole number. An example of both the graphs:- Bar Graph Histogram These 2 graphs both look similar but however, in a histogram the bars must be touching. This is because the data used are number that are grouped and in a continuous range from left to right. But as for the bar graph the x axis would have its individual data like colours shown in the above. Question 2 a) i) The probability of females who enjoys shopping for clothing are 224/ 500 = 0.448. ii) The probability of males who enjoys shopping for clothing are 136/500 = 0.272. iii) The probability of females who wouldn’t enjoy shopping for clothing are 36/500 = 0.072. iv) The probability of males who wouldn’t enjoy shopping for clothing are 104/500 = 0.208. b) P (AᴗB) = P(A)+P(B)-P(AᴖB) P (A|B) = P(AᴖB)P(B) > 0 P (B|A) = PAᴖBPA PAᴖBPB = PAᴖBPA PAPB = 1 P(A) = P(B) P(AᴗB) = 1 P(A)+P(B) = 1 P(B) > 0.25 Question 3 1. Frequency Distribution of Burberry Clothing...

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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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