...Probability & Mathematical Statistics | “The frequency concept of Probability” | [Type the author name] | What is probability & Mathematical Statistics? It is the mathematical machinery necessary to answer questions about uncertain events. Where scientists, engineers and so forth need to make results and findings to these uncertain events precise... Random experiment “A random experiment is an experiment, trial, or observation that can be repeated numerous times under the same conditions... It must in no way be affected by any previous outcome and cannot be predicted with certainty.” i.e. it is uncertain (we don’t know ahead of time what the answer will be) and repeatable (ideally).The sample space is the set containing all possible outcomes from a random experiment. Often called S. (In set theory this is usually called U, but it’s the same thing) Discrete probability Finite Probability This is where there are only finitely many possible outcomes. Moreover, many of these outcomes will mostly be where all the outcomes are equally likely, that is, uniform finite probability. An example of such a thing is where a fair cubical die is tossed. It will come up with one of the six outcomes 1, 2, 3, 4, 5, or 6, and each with the same probability. Another example is where a fair coin is flipped. It will come up with one of the two outcomes H or T. Terminology and notation. We’ll call the tossing of a die a trial or an experiment. Where we...
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...are unknown. The latter part of the aforementioned statement is also known as decisions of uncertainty. To make these choices with more confidence, we will explore concepts that will formulate these judgments. We also have to include appropriate probability concepts that will help limit uncertainty in certain decisions. This paper will disclose the decision to reside in the tri-state area with the probability of destructive hurricanes occurring. Next this paper will reveal concepts and the outcome from the statistical analysis that was used to determine the final decision and, the tradeoffs between accuracy and precision required by various probability concepts. As a final point, this paper will demonstrate the effects the decision had on the data provided and the decision that was ultimately made. Probability Concepts and Application Hurricane Sandy hit Atlantic City, New Jersey on October 29, 2012. A 900-mile wide storm, Sandy affected the entire northeastern United States with devastating winds, rain and floods. New Jersey and New York suffered the worst from the super storm leaving thousands of people without power for days. People living in this area were not prepared or expected the storm to devastate the area as it did leaving homes and personal belongings destroyed. So, what is the probability of New Jersey and New York becoming destroyed again by treacherous hurricanes such as Sandy? First we need to look at how often hurricanes of this nature occur in...
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...departments causing delay in care. As possible solution to this problem for Saint Francis Medical Center in Peoria, Illinois is the development of a Pediatric Afterhours Clinic. The Pediatric Afterhours Clinic would be available Monday through Friday 5pm to 10pm, and Saturday, Sunday, and holidays 2pm to 10pm. The decision needs to be made by administration whether or not to opening the clinic would be beneficial to the emergency department as well as financially viable. In this paper I will be applying appropriate probability concepts to find resulting data to limit the uncertainty in this decision with rationale, identifying each discrete outcome from the statistical analysis, identifying tradeoffs between accuracy and precision, and will include the recommendation for the decision to be made. a. Include appropriate probability concepts and your application of them to find resulting data to limit the uncertainty in this decision. The probability concept that will be used to make the decision to open a Pediatric Afterhours Clinic is the Poisson distribution. Administration wants to be 90% certain that opening the clinic will alleviate congestion in the emergency department...
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...following: (a) Descriptive Statistics - Concepts / Applications(b) Inferential Statistics – Concepts / Applications(c) Scope and Limitation of Use | Evaluation Criteria: Mid Term Test: 30 marks Test 2 / Quiz: 10 marks Test 3 / Quiz: 10 marks End Term Exam: 50 marks | Faculty: Prof. (Dr.) Tohid Kachwala (email contact tkachwala@nmims.edu, Cabin number: 729, Mobile: 9869166393, Extension: 5871) | Pedagogy: 1. Use of problem solving for all the topics. 2. Use of Statistics in Practice / Case lets / Case studies. 3. Use of Software like Excel / SPSS. | Session Outline: | Session | Topic / Description | 1 | Introduction to Probability – Experiments, Assigning Probabilities, Some basic relationships of Probability Read ASW Chapter 4 or LR Chapter 4 | 2 | Theories of Probability - Classical theory, Relative Frequency theory, Axioms, Addition rule, Multiplication rule, Rule of at least one, Concept of Expected number of Success – Numerical Problems & Applications Case Problem: Hamilton County JudgesSIP: Morton International - Chicago, Illinois Read ASW Chapter 4 or LR Chapter 4 | 3 | Bayes Theorem – Theory, Problems & Applications, Probability revision using tabular approach Read ASW Chapter 4 or LR Chapter 4 | 4 | Probability Distribution - Meaning of Probability Distribution, Type of Probability Distribution, Need (Application) for Probability DistributionRead ASW Chapter 5 or LR Chapter...
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...Lesson 2: Review of basic concepts of probability theory Coverage: Basic probability rules Random variables and associate concepts Normal distributions Reading: Chapter 2 (1-7), Chapter 3 (1-5, 10-11) and Chapter 4 (1-8) (1 8) Homework: Replicate and complete all the classroom exercises. Print answers of 2.1 (b-c-d), 2.3 (b-c-d) and 2.4(b-c-d) in one (1) page. 1 Business Statistics Lesson 2 - Page 2 Objectives At the end of the lesson, you should be able to: Define and apply the basic probability rules Describe the basic concepts related to random variables D ib th b i t l t dt d i bl Describe and use the properties of means and variances Recognize and understand the most commonly used probability distributions Use the basic data manipulation and descriptive statistical features of SPSS and transfer between SPSS and Excel SPSS, 1 Business Statistics • • • • Review of probability concepts Lesson 2 - Page 3 Probability: is defined on random events (occurrences), takes values between 0 and 1, and can be interpreted as limit of relative frequency (objective probability) Note: In everyday usage, probability might mean the extent of our belief in the occurrence of the event (subjective probability). However, statistics mostly deals with objective interpretation based on relative frequency. j p q y Basic probability rules: P( Sure event) = 1 and P( Impossible event) = 0 P(A or B) = P(A) + P(B) – P(A and B) Consequences: P( not A) = 1 - P(A)...
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...Institute of Management Sciences Peshawar Bachelors in Business Studies Course Plan Course Title: Statistics for Business Instructor: Shahid Ali Contact Email shahid.ali@imsciences.edu.pk Semester/Duration: 16 Weeks Course objectives : To introduce students to the concepts of statistics and to equip them with analytical tools to be used in business decision making. The course is intended to polish the numeric ability of the students to identify business problems, describe them numerically and to provide intelligible solutions by data collection and inferential principles. Course pre-requisites Intermediate statistics Attendance Policy: Late arrivals are highly discouraged. Any student coming late to a class late by 5 minutes after the scheduled start time will be marked as absent for the day. The teacher reserves discretion, however, to allow or disallow any student, to sit in the class in case of late arrivals. Attendance is not be entertained once the attendance register is closed. Class Project Students will be divided in groups for a class project. Each group will have to nominate a group leader. The details of the project will be made available to the group leader. Class Presentations Each student will have to make at least one individual presentation and one group presentation in the class. The group presentation will be on the project explained earlier. The individual presentations will...
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...Teaching Statistics and Probability is great for promoting problem solving and critical thinking, enhancing communication, developing number sense, and applying computation. As it applies to every day situations and appeals to our sense of fairness, it is very close in nature to inquiry based learning. Children encounter ideas of statistics and probability outside of school every day. The data students see are often represented graphically, statistically, or probabilistically. Weather reports are just one example of probability data we hear on the news. Begin teaching probability by formulating questions. “How many children in this class prefer to eat apples?” Children are familiar with line plots, which they learned earlier, review and build on that knowledge. Next step in teaching probability is to teach to collect data: observations, survey and questionnaires, experiments, interviews, simulations, poles, examining records, and searching info sources. It is important to teach kids to use appropriate methods of collecting data. Next step is to analyze data, represent it graphically. Representing data is done in a concrete way first (laying objects on the graph), and moving towards pictorial representation (drawing a chart with pictures of items being compared), and then symbolic (line plot, pie chart). Help students understand graphic representations by asking questions about the chart. Different ways to represent graphically: line plots, stem and leaf plots, box plots, picture...
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...has taught me many concepts on how to work with students and to help them achieve success in the classroom. The objective of the course Math for Elementary teachers two is to assist the learner with understanding day to day applications of mathematics, and to give different ideas on how to differentiate learning. There were several ideas covered in this course but there are several of the major mathematical concepts that stand out to me. The National Council of Teachers of Mathematics principals and standards was a very big help in guiding me to understand what was required to teach specific grades based on the common core. The major mathematical concepts that are in this course are, data analysis, probability, introduction to geometry, applications of geometry, application of measurement and mathematical connections. In data analysis we used appropriate statistical methods to analyze data such as detecting patterns, developing explanations and testing hypotheses. Teaching the children how to sort the data such as using tally marks for how many people like a certain object over something else. Bar graphs, pie graphs, line graphs and pictographs. Learning data is a very important concept that the students will need for the rest of the educational career. Probability which is the measure of how likely something is was also a concept that was covered in the class. With probability you have the experiment which is the situation that involves the probability. The outcome is the...
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...and statistically. Statistical Techniques are applied in all the functions of business like Operations, Marketing, HR, Finance etc. The aim of this course is to learn when a technique is appropriate and what it can achieve. The emphasis throughout the course is on concepts and reasoning rather than technical details. You should acquire some basic data analysis skills but most importantly, become a more informed and critical producer and user of business Statistical analyses. Learning Objectives: Ser. # | Course Learning Objectives | Link with Program Learning Objectives | 1 | To understand the basic concepts and principles used in Business Statistics. | To inculcate business knowledge and analytical skills in graduates to think decisively in order to develop innovative solutions to problems in a business environment. | 2 | Organizing qualitative and quantitative data into a frequency table, displaying the data through charts and graphs, describing and exploring data through different numerical measures. | To provide a progressive and structured framework to graduates that enables them in developing and applying knowledge set of critical and ethical evaluation. | 3 | To know and understand probability and its different rules. | To craft graduates’ expertise in order to increase their resourcefulness. | 4 | To...
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...determine a decision on a daily basis. A decision to buy rental car insurance will be answered by using the concept of probability. This paper will focus on the application of various probabilities to formulate the decision under uncertainty. Discreet outcome from statistical analysis as well as trade-offs between accuracy and precision obtained by different probabilities concepts shall be evaluated. According to car accident statistic stats, auto, fatal, and drunk driving, the estimation of having an accident is of one in 16 cars. It has provided useful information to make important decision. There are a number of probability concepts that can be used in determining the results from the research data that was given. Probability is used to limit the uncertainty of the decision on whether to buy the rental car insurance. The probability concept that works the best and meets all of the criteria from the information that was gathered is the Bayes’ Theorem. The application of Bayes' theorem helps to interpret the data because it is most relevant to the itinerancy of the whole trip. The Bayes' theorems follow the method of logical inference by establishing the degree of assurance in every decision (Lind, Marchal & Wathen, 2008). Consequently, Bayes' theorem is best utilized for the purpose of predicting confidence levels for purchasing rental car insurance based on the probability of...
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...Probability Distributions Individual Project Tammy Lynn Ayers AIU Online American Inmates Union Data Collection Results Submitted by Tammy Lynn Ayers On November 27, 2011 Dear Mr. Smith, We administered a survey to our inmates in order to measure their satisfaction with their incarceration. From this survey, we collected nine different sections of data. They include: gender, age, types of offense, type of facility, length of sentence, satisfaction with criminal justice system, satisfaction of legal services, sentence satisfaction, and incarceration services satisfaction. This data-set was used to profile the satisfaction of future inmates. This email represents our analysis of the data for the same purpose using the concepts of probability. Before we address the results of the survey, we will discuss the importance of inmate satisfaction, the concept of probability, and the uses of probability in business. Research has shown that inmates who are satisfied with their incarceration are more cooperative. Therefore the number of incidents involving inmates are reduced. The term probability is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long term proportion with which a certain outcome will occur in a specific situation with short term uncertainty. Probability deals with experiments that yield random short term results or outcomes but they reveal long term predictability. Perfect information is available...
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...Chapter 2 Probability Concepts and Applications Teaching Suggestions Teaching Suggestion 2.1: Concept of Probabilities Ranging From 0 to 1. People often misuse probabilities by such statements as, “I’m 110% sure we’re going to win the big game.” The two basic rules of probability should be stressed. Teaching Suggestion 2.2: Where Do Probabilities Come From? Students need to understand where probabilities come from. Sometimes they are subjective and based on personal experiences. Other times they are objectively based on logical observations such as the roll of a die. Often, probabilities are derived from historical data—if we can assume the future will be about the same as the past. Teaching Suggestion 2.3: Confusion Over Mutually Exclusive and Collectively Exhaustive Events. This concept is often foggy to even the best of students—even if they just completed a course in statistics. Use practical examples and drills to force the point home. The table at the end of Example 3 is especially useful. Teaching Suggestion 2.4: Addition of Events That Are Not Mutually Exclusive. The formula for adding events that are not mutually exclusive is P(A or B) = P(A) + P(B) – P(A and B). Students must understand why we subtract P(A and B). Explain that the intersect has been counted twice. Teaching Suggestion 2.5: Statistical Dependence with Visual Examples. Figure 2.3 indicates that an urn contains 10 balls. This example works well to explain conditional...
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...this 6.3 Probabilities Using Counting TechniquesThis is a featured page In a number of different situations, it is not easy to determine the outcomes of an event by counting them individually. Alternatively, counting techniques that involve permutations and combinations are helpful when calculating theoretical probabilities. This section will examine methods for determining theoretical probabilities of successive or multiple events. Permutation? or Combination? The following flow chart will help determine which formula is suitable for any given question. By simply following a series of "yes" or "no" questions, the appropriate formula can be determined. Flow Ex. 1 - Using Permutations: 6.3 Probabilities Using Counting Techniques - MDM4U1@FMG 6.3 Probabilities Using Counting Techniques - MDM4U1@FMG The specific outcome of Mike starting in lane 1 and the other two starting in lane 2 and lane 3 can only happen one way, so n(A) = 1. Therefore, 6.3 Probabilities Using Counting Techniques - MDM4U1@FMG The probability that Mike will start in the first lane next to his other brothers in lane 2 and 3 is approximately 0.00101. Ex. 1(a) - Using Permutations: Exactly Three People form a line at a grocery store. What is the probability that they will line up in descending order of age? (I.e. oldest, middle and youngest) →Solution using the blank like method: n(A): # of ways they will line up in descending order of age, thus: 6.3 Probabilities Using Counting...
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...Big Idea/Unit OverviewStudents will gather data, conduct investigations, and apply mathematical concepts and models to solve problems in mathematics and other disciples. Students will demonstrate reasoning skills in developing, explaining, and justifying sound mathematical arguments, and analyze the soundness of mathematical arguments of others and communicate with mathematics orally and in writing as part of independent and collaborative work, including making accurate and clear presentations of solutions to problems. The students will determine and interpret conditional probabilities and probabilities of compound events by constructing and analyzing representations, including tree diagrams, Venn diagrams, and area models, to make decisions in problem situations. The student use probabilities to make and justify decision about risks in everyday life and calculate expected value to analyze mathematical fairness, payoff, and risk. | Concepts and Key Understandings * Analyze and construct representations of events, including tree diagrams * Construct Venn diagrams and determine probabilities * Analyze and construct area models to determine probabilities * Explore the use of probabilities such as playing computer games or selecting classes * Explore and make decisions and justify their decisions about risk involved in a situation * Calculate expected values to analyze payoffs in variety of situations * Apply their understanding of expected values to determine the...
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...Careful predictions of outcomes and estimates of risk • The latest decision tools and techniques Linear Programming Operations management often presents complex problems that can be modelled by linear functions. The mathematical technique of linear programming is instrumental in solving a wide range of operations management problems. Linear programming is used to solve problems in many aspects of business administration including: • product mix planning • distribution networks • truck routing • staff scheduling • financial portfolios • corporate restructuring Decision Trees In many problems chance (or probability) plays an important role. Decision analysis is the general name that is given to techniques for analysing problems containing risk/uncertainty/probabilities. Decision trees are one specific decision analysis technique. Sensitivity analysis Queuing Theory (Waiting Line...
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