business state and math past paper2008

Probability 1

NEXT TO PAGE NO 1 2 3 4 6 7 Probability Introduction We all deal with the concept of probability on a daily basis without sometimes even realizing it. What are the chances that we will come to work on time? What is the likelihood that the check we have just written will be sent to our bank before the bank receives the direct deposit from our employer? What are the chances that it will rain tonight? What Are Chances The New Product Will Capture The Market...etc. So what is Probability? It is the chance, the likelihood that something will happen. In statistics, the words chance and likelihood are seldom used to describe the possibilities for an event to take place, instead, the word probability is used along with some other basic concepts which meaning defer from our every day use. Probability is the measure of the possibility for an event to take place. It is a number between 0 and 1. If there is a 100% chance that the event will take place, the probability will be 1 and if it is impossible for it to happen, it will be 0. Trial The performance of any individual item is called Trial Examples When we toss a coin we take trial. The Result may be head or tail. Outcome The result obtain from trial is called outcome. Example The customer takes trial of a new product of a company. The customer may be satisfied or not satisfied from the trial of this product. Experiment The process of getting information for particular purpose it may be conducted in library, market or field. Random Experiment An Experiment is called random it has at least two possible outcome in it Example In tossing coin there are two possible out come head and tail In die thrown there are 6 possible outcomes Sample Space The sample space is an exhaustive list of all the possible outcomes of an experiment. Each possible result of such a study is represented by one and only one point in the sample space, which is usually denoted by S. Examples Experiment rolling a die once: Sample space S = {1,2,3,4,5,6} Experiment Tossing a coin: Sample space S = {Heads,Tails} Experiment Measuring the height (cms) of a girl on her first day at school: Sample space S = the set of all possible real numbers Quality checker check the product from a lot Sample Space S ={good, defective} NEXT TO PAGE NO 1 2 3 4 6 7

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Event An event is any collection of outcomes of an experiment. Formally, any subset of the sample space is an event. Any event which consists of a single outcome in the sample space is called an elementary or simple event. Events which consist of more than one outcome are called compound events. It is represented by A,B,C …etc Set theory is used to represent relationships among events. In general, if A and B are two events in the sample space S, then (A union B) = 'either A or B occurs or both occur' (A intersection B) = 'both A and B occur' (A is a subset of B) = 'if A occurs, so does B' A' or = 'event A does not occur' (the empty set) = an impossible event S (the sample space) = an event that is certain to occur Example Experiment: rolling a dice once - Sample space S = {1,2,3,4,5,6} Events A = 'score < 4' = {1,2,3} B = 'score is even' = {2,4,6} C = 'score is 7' = = 'the score is < 4 or even or both' = {1,2,3,4,6} = 'the score is < 4 and even' = {2} A' or = 'event A does not occur' = {4,5,6}   Mutually Exclusive Events Two events are mutually exclusive (or disjoint) if it is impossible for them to occur together. And their intersection doesnot exit Formally, two events A and B are mutually exclusive if and only if If two events are mutually exclusive, they cannot be independent and vice versa. Examples Experiment: Rolling a die once Sample space S = {1,2,3,4,5,6} Events A = 'observe an odd number' = {1,3,5} B = 'observe an even number' = {2,4,6} = the empty set, so A and B are mutually exclusive.   Not Mutually Exclusive Events Two events are Not mutually exclusive (or joint) if it is possible for them to occur together and their intersection esit   Examples Experiment: Rolling a die once Sample space S = {1,2,3,4,5,6} Events A = 'observe an odd number' = {1,3,5} B = 'observe an number less than 5' = {1,2,3,4} {1,3} = so A and B are Not  mutually exclusive.

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Equally Likely Event   Events are equally likely if one event can occur as other ca Example   In Tossing a coin head can occur as tail can     Not  Equally Likely Event   Events are not equally likely if one event cannot occur as other can     Example A jar containing 10 white balls and 4 green balls the occurrence of white balls is not equal is green can   Independent Event   Event are said to be independent if the occurrence of one event does not effect the occurrence of the other event   Example  Tossing two Coins Rolling two Dies   Dependent event   Event are said to be dependent if the occurrence of one event  effect the occurrence of the other event   Example A jar containing 10 white balls and 4 green balls we  get two balls from jar and we have to know the chance that two balls are White. When we get one ball from jar and then get other another ball from jar with no replacement  getting first ball from jar with no replacement effect chance of second Ball.   Types of  probability   There are three approaches of probability these are 1. Classical approach The classical approach to probability is to count the number of favorable outcomes, the number of total outcomes (outcomes are assumed to be mutually exclusive and equiprobable), and express the probability as a ratio of these two numbers. Here, "favorable" refers not to any subjective value given to the outcomes, but is rather the classical terminology used to indicate that an outcome belongs to a given event of interest. What is meant by this will be made clear by an example, and formalized with the introduction of axiomatic probability theory. We use this Approach When Events Are equally likely The probability of an event (A) is Denoted ByP(A)  and Define By  Example      A spinner has 4 equal sectors colored yellow, blue, green and red. After spinning the spinner, what is the probability of landing on each color?   Probability of yellow, Blue, Green, and Red Will be

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2. Relative Frequency Approach:   In practical life events are not equally likely  so we use Relative Frequency Approach  for this or we define Relative frequency as it is  another term for proportion; it is the value calculated by dividing the number of times an event occurs by the total number of times an experiment is carried out. The probability of an event can be thought of as its long-run relative frequency when the experiment is carried out many times. If an experiment is repeated n times, and event E occurs r times, then the relative frequency of the event E is defined to be rfn(E) = r/n Example Experiment: Tossing a fair coin 50 times (n = 50) Event E = 'heads' Result: 30 heads, 20 tails, so r = 30 Relative frequency: rfn(E) = r/n = 30/50 = 3/5 = 0.6 If an experiment is repeated many, many times without changing the experimental conditions, the relative frequency of any particular event will settle down to some value. The probability of the event can be defined as the limiting value of the relative frequency: P(E) = rfn(E)     A jar containing 12 red balls and 8 white balls The probability  of  getting red ball is  12/20 And probability of getting white balls is 8/20 Here occurrence of red balls is not equally likely than occurrence of White balls   3. Subjective Approach A subjective probability describes an individual's personal judgment about how likely a particular event is to occur. It is not based on any precise computation but is often a reasonable assessment by a knowledgeable person. Like all probabilities, a subjective probability is conventionally expressed on a scale from 0 to 1; a rare event has a subjective probability close to 0, a very common event has a subjective probability close to 1. A person's subjective probability of an event describes his/her degree of belief in the event. Example A Rangers supporter might say, "I believe that Rangers have probability of 0.9 of winning the Scottish Premier Division this year since they have been playing really well." Medical doctors sometimes assign subjective probabilities to the length of life expectancy for people having cancer.  Forecasting is another example of subjective probability.

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Probability Rules   Addition  Law: 1. Addition Law For Not Mutually exclusive Events when two or more events will happen at the same time, and the events are not mutually exclusive, then: P(X or Y) = P(X) + P(Y) - P(X Y) For example, what is the probability that a card chosen at random from a deck of cards will either be a king or a heart? P(King or Heart) = P(X or Y) = 4/52 + 13/52 - 1/52 = 30.77% Addition Law For  Mutually exclusive Events   when two or more events will happen at the same time, and the events are mutually exclusive, then: P(X or Y) = P(X) + P(Y) For example, suppose we have a machine that inserts a mixture of beans, broccoli, and other types of vegetables into a plastic bag. Most of the bags contain the correct weight, but because of slight variation in the size of the beans and other vegetables, a package might be slightly underweight or overweight. A check of many packages in the past indicate that: Weight.................Event............No. of Packages.........Probability Underweight..........X.......................100...........................0.025 Correct weight.......Y.......................3600.........................0.9 Overweight............Z.......................300...........................0.075 Total................................................4000......................1.00 What is the probability of selecting a package at random and having the package be under weight or over weight? Since the events are mutually exclusive, a package cannot be underweight and overweight at the same time. The answer is: P(X or Z) = P(0.025 + 0.075) = 0.1 B The Multiplication Law: 3  Multiplication Law For dependent Events when two or more events will happen at the same time, and the events are dependent, then the general rule of multiplication law is used to find the joint probability: P(X and Y) = P(X) . P(Y|X) For example, supposes there are 10 marbles in a bag, and 3 are defective. Two marbles are to be selected, one after the other without replacement. What is the probability of selecting a defective marble followed by another defective marble? Probability that the first marble selected is defective: P(X)=3/10 Probability that the second marble selected is defective: P(Y)=2/9 P(X and Y) = (3/10) . (2/9) = 7% This means that if this experiment were repeated 100 times, in the long run 7 experiments would result in defective marbles on both the first and second selections. Another example is selecting one card at random from a deck of cards and finding the probability that the card is an 8 and a diamond. P(8 and diamond) = (4/52) . (1/4) = 1/52 which is = P(diamond and 8) = (13/52) . (1/13) = 1/52. 4 Multiplication law For Independent events when two or more events will happen at the same time, and the events are independent, then the special rule of multiplication law is used to find the joint probability: P(X and Y) = P(X) . P(Y) if two coins are tossed, what is the probability of getting a tail on the first coin and a tail on the second coin? P(T and T) = (1/2) . (1/2) = 1/4 = 25%. This can be shown by listing all of the possible outcomes: T T, or T H, or H T, or H H. Games of chance in casinos, such as roulette and craps, consist of independent events. The next occurrence on the die or wheel should have nothing to do with what has already happened.

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C  The Conditional Law Conditional probabilities are based on knowledge of one of the variables. The conditional probability of an event, such as X, occurring given that another event, such as Y, has occurred is expressed as: P(X|Y) = P(X and Y) / P(Y) = {P(X) . P(Y|X)} / P(Y) Note that when using the conditional law of probability, you always divide the joint probability by the probability of the event after the word given. Thus, to get P(X given Y), you divide the joint probability of X and Y by the unconditional probability of Y. In other words, the above equation is used to find the conditional probability for any two dependent events. When two events, such as X and Y, are independent their conditional probability is calculated as follows: P(X|Y) = P(X) and P(Y|X) = P(Y) For example, if a single card is selected at random from a deck of cards, what is the probability that the card is a king given that it is a club? P(king given club) = P (X|Y) = {P(X) .P(Y|X)} / P(Y) P(Y) = 13/52, and P(king given club) = 1/52, thus P(king given club) = P(X|Y) = (1/52) / (13/52) = 1/13 Note that this example can be solved conceptually without the use of equations. Since it is given that the card is a club, there are only 13 clubs in the deck. Of the 13 clubs, only 1 is a king. Thus P(king given club) = 1/13. Application of Probability In business, probability theory is used in the calculation of long-term gains and losses. This is how a company whose business is based on risk calculates "probability of profitability" within acceptable margins. An example of this is the way in which life insurance companies calculate the cost of life insurance policies and is based on how many policy holders are reasonably expected to die within a year versus revenue generated from other policies extended. In this scenario it is important to point out that in order for a company to mitigate the risk associated with loss of revenue it must issue a substantial number of policies.Ultimately the insurance company charges more when it is at risk of having to pay out. I didn't realize that insurance was inverse for health and auto, but I can see why it would be. First, to understand this concept you have to suspend the want to point out the exceptions. This being statements like, "I knew a guy that had a heart attack at 20." Statistically, things such as heart attacks in your 20s don't happen often. That being said, it might be easily shown that as your age increases your risk of sickness. Now also bear in mind that the insurance company is concerned with "how much" moreso than "how often." They are a company so they watch the bottom line. In youth you may be sick, but most likely it will not cost much. A prescription vs. surgery etc. Now coming to car insurance: It is easily shown that the younger the driver the worse the accident might be. This shows why young males are most often the worst to ensure. Statistically, the young male is prone to accidents and usually might be involved in some of the most not less expensive accidents. Older drivers tend to go slower and when there is an accident it is usually not as severe.

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NEXT TO PAGE NO -- 1 2 3 4 5 6 7 8 9 10 11 EARN Online Money 4. Finding This is most important and longest part of the research report. In this part the researcher define all results and finding of the report in paragraphs. This section should be written that reader can understand the result of the research. 5. Conclusion In this part the writer write fist write its finding in a brief summary. Highlight the important results then write its conclusions about its findings 6. Recommendations. Here the researcher write its recommendations about finding of the problems if a research is about a problems then researcher give its solution in this part. 7. Appendices Complex tables, statistical tests, supporting documents, copies of forms and questionnaires, detailed descriptions of the methodology are place in appendices 8. Bibliography It is the list of works cited by an author at the end of an article, paper, book, or other research-based writing. A list of reference materials such as books and articles used for research are place in it.

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NEXT TO PAGE NO 1 2 3 4 5 6 7 8 9 10 11 BACK TO HOME EARN Online Money A. Problem Statement. The problem statement shows the need for research problem. Here the researcher define what was the problem on which he made research and why the search was made. B. Research Objective It shows the purposes of the research the researcher define objective and purpose of the research some time hypothesis statements are also included in this section. C. Background Background material may be of preliminary results or secondary material from literature review 1. Methodology It has five parts A Sampling design B Research Design C Data Collection D Data Analysis E Limitations A Sampling Design In this section the researcher defines the target population that was studied and method of sampling used in research. B Research Design In this part of report the researcher write about material, test, equipment, control conditions and other devices used in research C Data Collection This Part is about data collection. The researcher defines how data was collected. It depends on the selected design. The writer writes about research team, their qualification and experience. D Data Analysis This section show the method used to analysis the data. This section also explain about handling of data, statistical test and computer programmers used in data analysis E Limitation This section of the report thoughtfully defines the limitation of the research. There may be some difficulties in findings results, limitations of methods used in surveys, limitation of data collection and other limitations. All are written in this section.

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NEXT TO PAGE NO 1 2 3 4 5 6 7 8 9 10 11 BACK TO HOME EARN Online Money Research Report Research report is a report in which researcher define his finding, analysis of finding, interpretations conclusion and some time recommendation on his research on particular topic. Essential of a Research Report Following are the essential of research report 1. Prefatory Items Following are the prefatory items of the report A. Letter of Transmittal A letter of transmittal is written when relationship b/w researcher and client are formal and report is written for an outside organization. B. Title Page It include (1) title of the report (2) date of the report (3) name for which report was written (4).name of Researcher of the report C. Authorization letter This letter shows the authority for undertaking research. It also show that who is sponsored the research D. Executive Summary It is the summary of the report or finding of research it cover all aspects of the body of it should be consists of almost two pages. E. Table of Contents. It should contain table of contain of topic data explain that a in it in the report. 2. Introduction. This section include the following A Problems Statement B Research Objective C Background

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References References cited in the text are included in the reference list; however, exceptions can be found to this rule. For example, committees may require evidence that you are familiar with a broader spectrum of literature than that immediately relevant to your research. In such instances, the reference list may be called a bibliography. Some committees require that reference lists and/or bibliographies be "annotated," which is to say that each entry be accompanied by a brief description, or an abstract. Check with your committee Chair before the fact. Appendixes The need for complete documentation generally dictates the inclusion of appropriate appendixes in proposals (although this is generally not the case as regards conference proposals). The following materials are appropriate for an appendix. Consult with your committee Chair. Verbatim instructions to participants. Original scales or questionnaires. If an instrument is copyrighted, permission in writing to reproduce the instrument from the copyright holder or proof of purchase of the instrument. Interview protocols. Sample of informed consent forms. Cover letters sent to appropriate stakeholders. Official letters of permission to conduct research.

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Data Collection Outline the general plan for collecting the data. This may include survey administration procedures, interview or observation procedures. Include an explicit statement covering the field controls to be employed. If appropriate, discuss how you obtained enter. Provide a general outline of the time schedule you expect to follow. Data Analysis Specify the procedures you will use, and label them accurately (e.g., ANOVA, MANCOVA, HLM, ethnography, case study, grounded theory). If coding procedures are to be used, describe in reasonable detail. If you triangulated, carefully explain how you went about it. Communicate your precise intentions and reasons for these intentions to the reader. This helps you and the reader evaluate the choices you made and procedures you followed. Indicate briefly any analytic tools you will have available and expect to use (e.g., Ethnography, NUDIST, AQUAD, SAS, SPSS, SYSTAT). Provide a well thought-out rationale for your decision to use the design, methodology, and analyses you have selected. Significance of the Study Indicate how your research will refine, revise, or extend existing knowledge in the area under investigation. Note that such refinements, revisions, or extensions may have substantive, theoretical, or methodological significance. Think pragmatically (i.e., cash value). Most studies have two potential audiences: practitioners and professional peers. Statements relating the research to both groups are in order. This can be a difficult section to write. Think about implications—how results of the study may affect scholarly research, theory, practice, educational interventions, curricula, counseling, policy. When thinking about the significance of your study, ask yourself the following questions. What will results mean to the theoretical framework that framed the study? What suggestions for subsequent research arise from the findings? What will the results mean to the practicing educator? Will results influence programs, methods, and/or interventions? Will results contribute to the solution of educational problems? Will results influence educational policy decisions? What will be improved or changed as a result of the proposed research? How will results of the study be implemented, and what innovations will come about?

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Sampling The key reason for being concerned with sampling is that of validity—the extent to which the interpretations of the results of the study follow from the study itself and the extent to which results may be generalized to other situations with other people. Sampling is critical to external validity—the extent to which findings of a study can be generalized to people or situations other than those observed in the study. To generalize validly the findings from a sample to some defined population requires that the sample has been drawn from that population according to one of several probability sampling plans. By a probability sample is meant that the probability of inclusion in the sample of any element in the population must be given a priori. All probability samples involve the idea of random sampling at some stage. In experimentation, two distinct steps are involved. Random selection—participants to be included in the sample have been chosen at random from the same population. Define the population and indicate the sampling plan in detail. Random assignment—participants for the sample have been assigned at random to one of the experimental conditions. Instrumentation Outline the instruments you propose to use (surveys, scales, interview protocols, observation grids). If instruments have previously been used, identify previous studies and findings related to reliability and validity. If instruments have not previously been used, outline procedures you will follow to develop and test their reliability and validity. In the latter case, a pilot study is nearly essential. Because selection of instruments in most cases provides the operational definition of constructs, this is a crucial step in the proposal. For example, it is at this step that a literary conception such as "self-efficacy is related to school achievement" becomes "scores on the Mathematics Self-Efficacy Scale are related to Grade Point Average." Strictly speaking, results of your study will be directly relevant only to the instrumental or operational statements. Include an appendix with a copy of the instruments to be used or the interview protocol to be followed. Also include sample items in the description of the instrument. For a mailed survey, identify steps to be taken in administering and following up the survey to obtain a high response rate.

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