Sample Work for Dissertation Statistical Analysis Help | Tutors India

The National Lottery –it probably won’t be you? This Sample Work has been completed by ‘Tutors India’ Copyright © Tutors India. All rights reserved. www.tutorsindia.com © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 1 of 33

Table of Contents CHAPTER I INTRODUCTION ........................................................................................................................................................... 3 1.1. Background of the Study ........................................................................................................................................................ 3 1.2. Problem Statement ................................................................................................................................................................ 3 1.3. Objective of the Study ............................................................................................................................................................ 4 1.4. Significance of the study ........................................................................................................................................................ 4 1.5. Description of Thesis.............................................................................................................................................................. 4 CHAPTER II REVIEW OF THE LITERATURE ................................................................................................................................... 6 2.1. History of UK lottery ............................................................................................................................................................... 6 2.2. UK Lotteries Current System .................................................................................................................................................. 6 2.3. Structure of UK Lottery Game ................................................................................................................................................ 7 2.4. Conscious selection of Numbers by gamblers ........................................................................................................................ 8 2.5. Winning of UK lottery Chances............................................................................................................................................... 8 2.6. Good Causes of UK lottery..................................................................................................................................................... 9 2.7. Are the Numbers drawn random? ........................................................................................................................................ 10 2.8. Random Numbers chosen by the gamblers .......................................................................................................................... 11 CHAPTER III RESEARCH METHODOLOGY .................................................................................................................................. 14 3.1. Objective of the Study .......................................................................................................................................................... 14 3.2. Statistical methods adopted to identify the random numbers ................................................................................................ 14 3.3. Proportions of prizewinners .................................................................................................................................................. 16 CHAPTER IV RESULTS .................................................................................................................................................................. 19 4.1. Equality frequency of each number. ..................................................................................................................................... 19 CHAPTER V DISCUSSION & CONCLUSION ................................................................................................................................. 26 BIBLIOGRAPHY .............................................................................................................................................................................. 29 APPENDIX - A ................................................................................................................................................................................. 31 © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 2 of 33

CHAPTER I INTRODUCTION 1.1. Background of the Study In many countries, lotteries are a feature of modern life and even those who do not involved in it would be affected by their pretentious by their survival, perhaps via utilizing art or sports service that has been supported all the way through lottery monies. This probably attracted the statisticians and probabilists to study issues related to lottery (Herman, 1981). In the United Kingdom and the Europe national lotto is more popular the biggest lottery game. The biggest difference exists between the classical and lotto games is that in former players bought the tickets with numbers pre-printed already on them, while in later, players need to chose their won numbers (Simon, 1999-9804862). A draw is held usually once or twice a week and tickets who hold those drawn numbers win large monetary prizes (Simon, 1999). This game is controlled by the Camelot Group and Camelot was allotted as head by United Kingdom lotto and was established in 1994. The National Lottery Commission in the United Kingdom administrated the “The National Lotto”. Its primary responsibility is to assurance that gamblers are handled not unfairly and the operator stays motivated to increase the profits and intensity of excitement that the Lottery gives the country (reference). The structure of the UK lottery game followed is same as ‘lotto’ games that are conducted elsewhere (Haigh, 1997) . The United Kingdom's National Lotto has various kinds of games to choose from, but the most common format is the 6/49 game (Herman, 1981). In order to select the six different numbers from the list, gambler needs to pay £1, thus the game starts with one Euro (Haigh, 1997). A player presently has to select six numbers ranging from one to forty-nine {1, 2, 3……..49} and also one bonus number is chosen. Each player needs to either choose their own set of numbers just by filling out the form in the lottery way out or else they can prefer the lottery retailers system that generates the numbers for them automatically. To win cash price in this game, the total of six numbers the gambler has to get at a minimum three numbers that match the drawing. The player matches more than three numbers they get chances to get more money and in addition, the value of prize is depends on the proportion of sales, which is usually fixed. Suppose player match five numbers they would get another chance to win on the bonus number allow the leading cash prize possible aside from matching all six of the numbers plus the bonus. Between 1 and 49, there are 13, 983, 816 possible combinations thus, the probability is to the encounter actually happening is about one in fourteen million (Simon, 1999). In lotto games, since winning numbers are chosen by the players and there is high probability that no ticket will match the winning numbers and thus, over to the next drew jack pot is carried over and this rollover, would further boost the sales of ticket. Thus, prizes will become still higher (Simon, 1999). 1.2. Problem Statement Since, there is less probability of winning chance as reported earlier (Simon, 1999) as the numbers chosen purely depends on the players. There are number of extensive literatures that provide data on gambler choice in one particular draw, prize structure, © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 3 of 33

number of prize winners and the prizes sizes (Riedwyl, 1990). Even today, the debate exists on true randomness was not being achieved in the game, through there are number of physical checks such as BSI Test are made on the process. However, best assurance comes from the point of view of statistical considerations as number of researchers (Bellhouse, 1982a, b; Joe, 1993; Johnson & Klotz, 1993; Morgan, 1984) has examined the true nature of randomness through their reasonable tests of ‘randomness’. Further, on choices of gamblers only little information exists with relevance to UK game (Haigh, 1993). But this information was provided by the British Columbia Lotteries. Several researchers analyzed the winning combinations where in Florida Lottery, Zaman and Marsaglia (1990) analyzed 4676 players and identified 37 previous winning combinations, similarly, Riedwyl (1990), analyzed Swiss 6/45 lottery in week 6, 1990, identified two combinations ({6 11 16 21 26 31} and {1 8 15 22 29 36}) that were selected by the gamblers more than 2400 times. In addition, authors like Zaman and Marsaglia (1990) and Ziemba et al (1986) speculated several events that influenced the numbers such as celebrities’ birthdays, current events, geometric patterns etc. Although number of researchers has examined the randomness and gambler choice, all research was done fifteen years back and the present thesis highlights particularly analyzed the randomness and gambler choice with relevance to present data of 2009 and 2010 using the different models as proposed earlier (Haigh, 1993). 1.3. Objective of the Study The purpose of this research is to study the randomness of the numbers drawn and choice of gamblers or how they pick their numbers. This study particularly focused on UK lotto lottery game with relevance to 2009 and 2010 data. In view of the above, the research objectives of the present study are: Are the numbers drawn are truly random? How gamblers do chose their numbers? IS there any scope for choosing better numbers? 1.4. Significance of the study Thepresent study throws the light on variety of techniques employed by the different lottery players when pick their numbers, further the study also helps to identify the true randomness using the different statistical approaches. The results of the study would also provide assurance to the gambling public about the fairness of draw and number selected have an equal chance of being selected. In addition, the evidence obtained from present study would also help the UK lottery franchisees; commission an independent body to provide checks on the lottery machines without disclosing the specific statistical outcome. 1.5. Description of Thesis The present research work would be categorized into five chapters; the basic background, problem investigated and significance of © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 4 of 33

the study are presented in the chapter one. Further in chapter one, research question, aims and objectives, limitation would also be discussed. Chapter two dealt with the review of literature with particular reference to true randomness and gambler choice in picking up their numbers. The third chapter deals with the research methodology, which included data collection method, study design, and further reason for adopting such design for the present study and limitation of the same would be acknowledged. In chapter four, study findings would be presented in the form of tables and figures, where each and every question would be analyzed and presented and proper interpretation of the same would also be provided in the same chapter. In the Chapter five, discussion and conclusion would be provided. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 5 of 33

CHAPTER II REVIEW OF THE LITERATURE The researcher went over a number of literature and studied relevant literatures that are relevant to the present study in different public and private libraries. The literature studies which have bearing to the present day study are herein cited. The present review of literature highlighted under following headings. The Development of lottery system in UK Game Structure of UK lotto Good causes of UK lotto Are the numbers drawn on Random? How the numbers are chosen by gamblers To progress their chances of determining a winning strategy many people look for the methods. Overall, the choices of numbers would be nearly at random if the gamblers change their strategies greatly from one and another. Need to define a substitute variable for the information on the selection preferences. Betting numbers of observed distribution is not available. The second objectivity of this study is the purpose of the winning chance. How do people choose their numbers and is there any scope for choosing numbers better? This finding suggests that the overall opportunity to win a UK lotto prize follows the rule of games of chance no matter what the strategy used by gamblers. 2.1. History of UK lottery In 1569, the first national lottery in the UK was initiated mainly for the purpose of raising money of the Cinque Ports repair. Prizes were in the form of plate, tapestries and money and almost 400000 tickets were sold. For the promotion of public or semi-public purposes, in the following century other lotteries were introduced. In Virgina (1612), London (1627-1631) and the Spaniards (1640) are the examples of lottery systems for aid of English Plantation. In addition, several lotteries to swell the government coffers, it was authorized by parliament. Although, lotteries were introduced in good motives, towards the end of the 18th century, opposition to state lotteries were gathered that raised illegal practices and the social evils. This was abolished by the select committee of the House of Commons in 1808 (Moore, 1997). 2.2. UK Lotteries Current System From the year 1920s i.e. early 20th century, in UK there was a change in opinions as all lotteries were legalized and codified in 1976, under the act of Lotteries and Amusement Act. There are four types of lottery system that are currently under use in the UK are 1) To certain entertainments, small lotteries 2) lotteries belong to private 3) Lotteries of society 4) Lotteries of local authority. The lotteries of small and private were small in terms of scope and size. Lotteries that belong to society are not used for commercial or private purpose and utilized only for the charitable, sporting and other purposes. For the last lotteries of local authority, in order to publicize the lottery object, it requires permission. The aim is rather for commercial gambling, it should support of good causes. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 6 of 33

There are many countries exists, that shares their profit for voluntary and government organization. For example, The Irish National Lottery helps sports, culture and finance health and also Germany (Moore, 1994). There are different winning combinations in different countries, in the USA, jackpot winning combinations ranging from 3/36 in IOWA and in Maryland its 6/49. Across various borders, lottery activities 2.3. Structure of UK Lottery Game The basic structure of worldwide is followed by every country while playing lottery games. Gamblers using without replacement, they involved in buying tickets, from a larger set of N numbers they try to choose n numbers. Draw takes place at a later date and randomly winning numbers are chosen from the set of N numbers that are large enough utilizing with number balls machine. For example, the better the prize, the more numbers those players could match the balls drawn. Gambler wins a share of the jackpot prize if player matches all n numbers drawn. With five prize levels the UK lotto is a 6/49 game. From a possible 49 six numbers would be chosen from the gambler ticket. The main balls are the first six balls that are drawn in the draw and as a bonus ball; a seventh ball is also drawn. On matching the first six numbers (the jackpot), prizes are distributed to the winners, we refer five of the first six and in addition the bonus ball as 5+, 4+, for 4+ the four of the first six, and also 3+ three of the first six. In the ‘rollover draw’ this improves value for money, as in the earlier draw as the extra prize money has been paid for by gambler The structure of a lottery game can vary, in three ways n and N choice, where n a player must select the number of numbers from a N numbers of large set (Rather than from just one large set in some recent games combinations of numbers from each of two sets of numbers are chosen by the gambler) Operator take-out rate The structure of prize. Example, the relative amount paid into each prize pool and the number of prize levels. Table 1. Prize structure for the UK lotto Probability of matches of Number Percentage prize pool of winning 6 balls 52% (plus any rollover or bonus) 7.15×10−8 5+balls 16% 4.29×10−7 5 balls 10% 1.80×10−5 4 balls 22% 9.69×10−4 3 balls Fixed £10 prize 0.0177 Adopted from Baker & McHale, 2009 In Table 1, the prize structure of UK lotto has been set and almost sales revenue of the take-out rate is 55%, i.e. 45% goes into the prize fund. A fixed price of £10 for the gamblers matching three balls are paid and from the original prize fund (45% of sales) funds © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 7 of 33

that are required to make these pay-outs are taken to leave the ‘prize pool’. If random numbers are chosen by the lottery gamblers, that is if every numbers has equal chance of being selected then this each would be considered as a Poisson random variable if the number of gambler matching each of three, four, five, 5+ and six balls (the Poisson limit of the binomial distribution). Nevertheless, gambler does not choose randomly, as there is a lot of strong evidence for this. As noted by Farr el et al (1999), high numbers are less popular than the low numbers and due their selection forms a shape on the entry form, particular sets of numbers may be more popular. 2.4. Conscious selection of Numbers by gamblers Ticket sales were typically 65 millions in the early years of the Saturday lotto (one ticket costs £1). the expected number of jackpot winners would have been 4.65 with a standard deviation of 2.16, if gambler were to choose numbers randomly. However, there were a remarkable 133 winners of the jackpot prize on January 21st, 1999, where the winning numbers formed an L-shape on that occasion. Thus, based on this gamblers form familiar shapes on the entry form of their obtained numbers that could result in selection on non-random number (Baker & McHale, 2009). Over dispersed distributions of the numbers of winners is the result of the presence of conscious selection however than would be expected under random selection, for some draws there are many more, or fewer, winners. On March 3rd, 2007 the large number of winners in the South African lottery was mainly due to the conscious selection and over dispersed distribution resulted for the winners of numbers of jackpot. Due to this conscious selection and the occurrence of nine jackpot winners, forced political parties to postpone the lottery calling for an investigation into the ‘extremely suspect’ draw on that particular occasion. But, even with selection due to conscious would not be unduly improbable. 2.5. Winning of UK lottery Chances Six Numbers – The Jackpot [the prize almost £2 million, typical]: from the set of integers between 1 and 49, 6 numbers are drawn at random meaning there are combination of 49! / (6!*(49-6)!) . But it doesn’t matter with the order of draw. Meaning, the opportunity for Jackpot would be 1 in 14 million or chance would be 1 in 13,983,816 Five Numbers and in addition Bonus [£100,000 – Typical prize]: from the 1 to 49 set as above you are still matching 6 numbers, but in 6 different ways (in turn by dropping each of the main numbers), you can now do it. Hence, the chance which works out as 1 in 2,330,636 and is 1 in 13,983,816/6 Five Numbers [£1,500 - Typical prize] : than getting 5 numbers + the bonus number This is 42 times more likely. Because, there are 43 balls left after the first six balls are drawn and of 43 balls without matching the bonus number you can match 42 numbers. Thus, the chance which evaluates 1 in 55,491.33333 and the chance is 1 in 2,330,636/42 Four Numbers [£65 – Typical Prize]: First, take last two not matching and first four of your numbers and in this situation, © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 8 of 33

 1st number chance that matches a winning number would be 49/6  2nd number chance that matches a winning number would be 48/5  3rd number chance that matches a winning number would be 47/4  4th number chance that matches a winning number would be 46/3 1 in 45 (45-2), the chance that your 5thnumber doesn’t match the winning number 5th number chance that matches a winning number would be as there are winning numbers that are still 2 unmatched. 1 in 44 , that your 6thnumber doesn’t match a winning number By multiplying them together, we need to accumulate all those chances that is 1 in (49/6)*(48/5)*(47/4)*(46/3)*(45/43)*(44/42) and this is 1 in 15486.953. For that single case occurring Now this is the chance, matching 4 from 6 [6! /(4!*(6-4)!)]. And there are 15 combinations. If the anwers are divided by 15, the answers 1 in 1032.4 Three Numbers (£10 – Constant Prize): Same scheme as the 4 match, follows exactly the same to get these figures for a single case which is 1 in 1133.119 i.e., 1 in (49/6)*(48/5)*(47/4)*(46/43)*(45/42)*(44/41). The chance of a 3 match is 1 in 1133.119/20 or 1 in 56.7, thus, there are 20 combinations of 3 from 6 [6! / (3!*(6-3)!] The approximately 54 to 1, the chance of you winning any of the above prizes, it is recorded that prize will be wined by average of one million people per draw. 2.6. Good Causes of UK lottery Arts: From corner to corner the UK to enjoy, the aim of the arts helps people and broadcast possible range of many activities it takes part in. On a wide range of capital projects, re-equipping of arts venues and also the renovation, arts Lottery money is being spent. A new scheme has launched the arts council:  In arts activity number of people participate and attend  Encourage to a wider level of audience to reach, more new plays like music and other work to be created  The young people along with the mid age people are also emphasized in order to develop their talents, this project has been supported. Charities: To improve the quality of live in the community and to help those at greatest disadvantage, the National Lottery Charities Board gives grants. Of the Charities Board the first three grant programs are 1) Low income and youth issues 2) care, health and disability and 3) poverty. Benefit from the following grants programs, thousands more will benefit as this will provide choices and new opportunities related to sector development voluntarily and involvement of community. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 9 of 33

Heritage To preserve, restore or acquire the nations most treasured the Heritage Lottery Fund is here, which makes our history and culture a fabric. A care for our countryside, buildings, museums, industrial heritage, archives and parks, the country can able to help many organizations and volunteers. The organization could help many sectors such as  For the public to enjoy by keeping keep 38,500 hectares of countryside and woodland  For important works of art save over 30 nationally as well as locally  Many historic ships, trains, piers, lighthouses and bridges were preserved and restored which played an important in heritage of industrial.  To improve and expand their community activities support over 270 historic buildings and places of worship. Millennium A substantial contribution to the community, through millennium projects through the support of public. By marking a significant moment in their history they will be seen by future generations. In three types, the Millennium projects would fall  Projects related to capital  Awards of Millennium  Exhibition and Festival related to Millennium Nationwide program me of events, large and small would be there in addition, throughout the year 2000, this will take place. While reflecting religious, ethnic and cultural diversity, the Millennium Commission believe that this will help communities together. Sports: Funds, facilities and opportunities are more. For distributing Lottery funds, sports are a good cause. Lives of millions of people of all ages in this country it enriches for many different reasons. By the English Sports Council, Sports Councils in Northern Ireland, Scottish Sports Council and the Wales sports council are distributed from the Lottery Sports Fund. Initiatives will be introduced more in 1997. For all of us Coaching and Leadership, Talent Identification and Development program will be introduced that enhance the sports and encourage also. 2.7. Are the Numbers drawn random? Several researchers have examined the randomness of the numbers in order to give assurance to the public that lottery draw is conducted fairly and all numbers have an equal chance of being selected. However, the best assurance of randomness comes from the statically considerations than whatever physical checks are made on the process and whatever steps are taken to ensure the integrity of the draw. As reported by Bellhouse (1982a,b), in the quality control procedures, Canadian Lottery Corporation varied widely suggesting that true randomness was not being achieved while selecting the winning numbers. Testing of Randomness was examined by Joe (1993) for an m/M, ie,, where at random from the integers, m numbers are need to be selected from the integers {1, 2, 3, ………, M}. He examined the uniformity of tests for the single numbers distribution, triples and also pairs and also independence linking draws. He explicitly used m /M lottos and their distribution was shown asymptomatically x2 . Within the same draw, the for a 6/49 lottery, uniform distribution test statistics would be of pairs α = (I, j), where I <j. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 10 of 33

W= {90.59 ∑ (O α – E)2–1.632 ∑ (O α – E)(Oβ– E)}/ D This formula would help to provide meaningful tests from data made on UK national lottery other than frequencies of individual. To several thousand draws, Joe’s (1993) could be applied and his test was more appropriate for the lottery equipment verification. In addition to test the lucky dip facility, they would be useful. A different approach for data on a similar lottery was done by Johnson and Klotz (1993), they supposed that i ball and pi of probability was being selected first and the second and subsequent selection probability would simply rescaled considering the balls drawn already. Over 200 draws, their data would have numbers drowns in order. Using log-likelihood statistics, the null hypothesis of uniformity was tested and this approach helps to account the order of selection, and if the hypothesis has been rejected, it provided clear alternative model. However, mild evidence against uniformity was provided through this data (p value = 0.084). In each draw, they linked with the observations that the balls enter the mixing machines. On sets of allegedly random numbers, certain specific tests was performed by Morgan (1984) which is ‘fitness for purpose’ apart from the lack of pairwise dependence and equality of frequency, though there are number of tests, till now there are no universally acceptable sets of test of randomness. As suggested by Joe’s work m/M lottos and equality of frequency suggested by Morgon (1984) as former has alternative hypothesis to the null hypothesis of later and to have high power to detect, Alternative Hypothesis 1 : Probability m/M + of selection; the other for some ≠0 , M-1 have probability m/M- / (M-1). Alternative Hypothesis 2 – m /M + delta; m /M –delta for the other half have probability of some delta ≠ 0; Alternative Hypothesis 3 & 4’ given that i has been selected, for each i=1, 2….M, then either HA1 or HA2, with (m, M) replaced by (m-1, M-1) 2.8. Random Numbers chosen by the gamblers In number of literatures from the perspectives of economics, statisticians and psychologists have analyzed the people’s behavior in choosing their random numbers particularly, motivations for their behavior was analyzed. To choose their combinations that employed by the lotto gambler are described in this section. The "illusion of control" was referred by Langer (1982) to mention about the gambler choice. This means that people are liable to be tricked into believing that they are able to exert some control over the chance event by allowing lotto gambler to choose their own numbers (the winning combination drawing). In a German 6/49 lottery, Henze (1997) analyzed the most popular combinations, the third most popular combination was the winning numbers from the preceding draw while from earlier draws a total of 269 combinations represented the winning numbers not only in other lotto games in Germany and in neighboring countries (Switzerland, France, and Austria) but also in the Baden- Wurttemberg lottery. In every draw often pick numbers associated with contemporary events, lottery gambler who does not choose the same combinations. Several examples were provided by Clotfelter and Cook (1989, 1991a) where Italian lottery gambler bet heavily on the numbers when Pope John XXIIl died such as time, data and age. Lottery operators around the US witnessed a surge of bets on that number the following day, after a popular American television series shovv'ed a woman betting on a certain number and © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 11 of 33

winning. Into the numbers that they should bet, the gamblers also look for dream books, that translates occurrences and objects, that appeared in their dream Several heuristics that people employ was given by Tversky and Kahneman (1974) when dealing with probability that can lead to misconceptions seriously about randomness like based on the winning numbers of past history that influence current draw. To the "gambler's fallacy", 246 J- SIMON One of the heuristics, representativeness was a lead. In terms of roulette, this is usually described: “it is the belief that if, say, a long run of reds is observed on the roulette wheel, then black is now due”. "Chance is commonly viewed as a self-correcting process in which a deviation in one direction induces a deviation in the opposite direction to restore the equilibrium" was described by Tversky and Kahneman as one of the attribute of misconception. Clotfelter and Cook (1991b), offered further evidence of the “gambler fallacy” and for lotto, the most popular in the Swiss 6/45 Zahlenlotto (see Riedwyl (1990) with the combination of these characteristics. Three-digit daily numbers game was showed that, in the Maryland, the amount bet fall sharply immediately with the individual numbers taking several months to recover to its former level after the number came up. The most popular combinations in the Swiss 6/45 lotto Arithmetic modifications of the previous winning combination (it does not lend itself to geometric modifications, since the play grid in this game is not square). By lottery gamblers to pick combinations), the popularity of these systems may be attributed to a misguided strategy Gamblers to request a random combination to be generated by the lottery terminal for their ticket a "Quick Pick" option (the name varies) was used by the most lotto games around Europe and the US this has to obtain the random combinations. Around the world, in most lotto games about 10%-20% of tickets sold the Quick Pick were accounted. Some people prefer to choose the numbers themselves who decide to play a random combination rather than using a Lucky Dip, as a result of this random picking of numbers, as dubbed by the "the law of small numbers" by Tversky and Kahneman (1974), a consequence of yet another widespread misconception about randomness. The belief that the past history of winning numbers influences the current draw are the serious misconceptions about randomness that deals with probability was described by Tversky and Kahneman (1974) about the several heuristics that people employs. As pointed out by Clotfelter and Cook (1989), that "choosing six different numbers appears to be a daunting prospect for many lottery gamblers". For their intrinsic aesthetic appeal another possibility is that pattered combinations are chosen. Amongst the most popular combinations in lotto, whatever the reasons, the empirical evidence suggests that the most obvious patters in 6/49 Germany lottery (in the Baden-Wurttemberg region). The third most popular combination is the winning numbers from the preceding draw from earlier draws a total of 269 combinations represented the winning numbers both in lotto games in Germany and in neighboring countries (Switzerland, France, and Austria) and in the Baden-Wurttemberg lottery. In the Swiss lottery (the numbers on the ticket are arranged in a 6 by 8 grid, with a gap along the bottom row), principal diagonals the top-right—*lower-left and top-left lower-right were the two most popular combinations. The equivalent diagonals were the first and 9''' most popular combinations in the Baden-Wurttemberg lottery and the most popular combinations are horizontal, vertical, or diagonal lines on the ticket that related to progression of arithmetic. Yet another widespread misconception about randomness as © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 12 of 33

described by Tversky and Kahneman (1974) was the “the law of small numbers" and other options is “quick pick options” that are followed by lotto games in Europe and the US. Thus, from the above literature it can be concluded that there are number of studies that examined randomness of the numbers and gambler choice. Some of the literatures shows that the number generated are not random and more literatures highlighted that they are not random. Further gambler choice of choosing random numbers is also highlighted. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 13 of 33

CHAPTER III RESEARCH METHODOLOGY 3.1. Objective of the Study The purpose of this research is to study the randomness of the numbers drawn and choice of gamblers or how they pick their numbers. This study particularly focused on UK lotto lottery game with relevance to 2009 and 2010 data. In Appendix A, the data used for the present study are presented. Further the data can be found on the website http://www.connect.org.uk/lottery In view of the above, the research objectives of the present study are: Are the numbers drawn are truly random? How gamblers do chose their numbers? IS there any scope for choosing better numbers? With the following objectives in mind, the first aim of the study is to establish whether lottery drawing appears to conduct randomly. Though it is not possible to detect sources of randomness from this analysis, the primary purpose is to support the public that any unusual occurrences actually correspond with random drawings. The methodology adopted usually consists of applying several statistical significance tests for randomness. With particular emphasis on “m from n” lotteries, we describe appropriate statistical methodology for testing the randomness of lottery draws with the data pertaining to the first 280 draws. The UK nation Lotto uses the 6/49 model. Similar games are also conducted in many other countries. Our sample is obtained from (http://www.connect.org.u/lottery) the organizer, with ranges drawn from January 2009 to December 2010 (Appendix A). During this period, there were 280 drawings. Data on each drawing consist of the winning six numbers combination and one bonus number, the number of winners for the match three, four, five, five and bonus, and six (jackpot) categories and the total number of tickets sold in each lottery game. 3.2. Statistical methods adopted to identify the random numbers Random number is “X1,...,Xn be independent random variables...".without replacement Assume that t an infinite sequence {Xi} is given, and that successive stages of the draw first Nterms of the sequence, in addition, of the statement “law of large numbers”. The average of a numbers fundamentally converges to the mean value. This always independent identically –distributed random variables of infinite sequence. The number chosen by the people poses equality of frequency; The out puts all are independent compared to prior outputs; © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 14 of 33

Till now there is no evidence was found for the non- randomness for Lotto game in the retail network. Our investigations began with an exploratory data analysis with this data. The findings here were that there appear to be no incorrect entries in the database. Perhaps the most important test is that the 49 possible. Testing the randomness of lottery draws, with particular emphasis on “m from n” we develop methods for testing simple hypotheses of randomness lotteries. Numbers for the draws are selected with equal likelihood. P-value for the appropriate goodness-of-fit tests that (chi-square test). The role of the goodness of test is that as a result the test is not significant at the 5% level and we do not reject the null hypothesis of randomness - conclude that there is no evidence of any bias in the selection of exacting numbers in the U.K. National Lottery Lotto Lucky draw generator. P =<0.05. The gamblers need to be assured that the draws are conducted fairly. All the numbers combination has an equal chance of being selected. Statistical methods toned to ensure the integrity of the draw. To chick the quality control procedures using the previous study to show whether drawing numbers are random or non-random. To test the randomness for an m/M lotto, it means when m number are to be selected in draws at random from the integers {1, 2, 4 …49}. Let B1 (d) <…< B m(d) denote the m possible number selected in draw d, for d =1,2,3…….,D. The null hypothesis Ho is that all M* combinations are equally likely, and the draws are independent. The numbers are derived pairs to test the uniformity of distribution to show the usual goodness of fit statistics sum the (observed-expected) square / Expected formulas modified with the chi-square distribution. In the univariate case this is achieved by a scale factor. The data consisted of the numbers drawn, in the order over 280 draws. To test the null hypothesis if uniformity, using the usual log- likelihood statistic, it gives a clear alternative model if the null hypothesis is rejected. For m/M lotto’s, there are several alternative hypothesis to the null hypothesis of equality of frequency that to have high power to detect and several possible tests and report the results of applying on six main numbers drawn in the UK lottery 3.3. Test A. Equality of marginal frequency In All draws, assuming that ball k is drawn X k times under null hypothesis Ho ,E(Xk)=mD/M ,and also the usual goodness-of- fit statistic must be modified, as noted earlier without replacement of m balls . © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 15 of 33

Test B. Independence between Draws For the any fixed number i, let W1 denote the number of draws between later successive appearances of i, under the null hypothesis any Wr has the geometric distribution. Test C: Sum of the numbers Under the null hypothesis Ho , the mean and variance of the theoretical which match against with sample mean and sample variance of {S(d): 1≤ d ≤}, denoting the U and V respectively . 3.3. Proportions of prizewinners The test demonstrating the degree to which lotto gamblers choose their numbers in an extremely non-random fashion. Assuming that data for the first 26 weekly draws is consistent with a Poisson model. First, let the probability θ be equal to 2.906% that lotto player win a prize. With many lotto gamblers and possible choices, if lotto tickets are chosen randomly and independently, the number of prize winners P will follow a Poisson distribution if with parameter T θ, where T is the number of tickets sold for each and every draw. Since T may be really large and θ is small enough, P may be approximated by a normal distribution with mean T θ and variance T θ. If W is the winning chance (which is the number of prize winners divided by the total number of tickets sold), then Z = (P −T θ)/√T θ =√T (W −θ)/√ θ should roughly follows a standard normal distribution. How lotto gamblers choose their numbers The UK national Lottery game offers the gamblers to opportunity to choose their own numbers during the ticket purchase and many of the gamblers do not choose their combination randomly. Choosing the combination of the randomly depend on has their own © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 16 of 33

characteristics. The implications of the nonrandom choice of numbers, both for the gamblers and the machinist of lotteries, are also discussed. The major combination or lucky Almost a lot of lottery gamblers have their own personal sets of numbers that they always play. Some people choose lucky numbers like birthday of family members, pets, or favorite celebrities, anniversaries or their house number. Some of them choose superstitious numbers. For example the number 7 is traditionally viewed as being lucky in many different societies. This number consider as victory. The most popular combination in the UK National Lottery is {7, 14, 21, 28, 35, 42} the multiples of 7. This combination is chosen tens of thousands of times every week. Others consider that their numbers are truly lucky, and can increase their chances of winning. Combinations influenced by previous winning numbers Maximum number of ticket sales - gamblers chooses the numbers by themselves. An extra bonus number is drawn, and the prize tier immediately below the jackpot is for tickets matching five of the main numbers plus the bonus number. This is calculated as , , where is defined as • . n ! / r ! ( n – r ) ! table 4 and 5 Layout of the UK National Lottery The lotto, combinations for e.g.{1,2,3,4,5,6 J) or sequences of numbers (e.g. {3,6,9,12,15,18}), Patterned combinations Combinations which "almost" make patterns or arithmetic progressions are also very popular. Marking these numbers on the play grid gives the pattern shown in table.5.The numbers are clustered in the second and third columns, with no two numbers on the same row. Of why so many gamblers chose this combination is that it is convenient: this pattern could easily be generated by people taking a central line down the grid. "Random” combinations There are some reasons why people choose the Quick Pick option. Some gambler simply have no interest in choosing their own numbers. Others prefer that their numbers are chosen by the lottery terminal in order to protect themselves from entrapment. There are also some people who consider that, since the draw is random, they are more likely to win if they choose their own numbers by a random method, which thus mimics the process by which the winning numbers are chosen. This misperception may be attributed to representativeness heuristic. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 17 of 33

the sets of numbers chosen by these gambler are likely to be far from random, it is the belief that small samples should be highly representative of the distributions from which they are drawn. The common example involves sequences of tosses of a coin. When subjects were asked to compare various sequences, they regarded © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 18 of 33

CHAPTER IV RESULTS 4.1. Equality frequency of each number. The UK national lotto data of 2009-2010 sampling period with 280 draws, and 6 main numbers chosen each time, the expected frequency of each number‘s mean of 181.5 with a standard deviation of 11.45. The actual frequencies range from 140 to 211; just one number (38) has appeared 211 times; the second highest frequency is 203. Not much difference between the next frequencies of this data. But the standard chi-square test of goodness-of-fit, even including the data relating to this number, shows no significant departure from equal frequencies. It is important to discover the independence of the successive draw. The results of each draws could be useful in predicting the results of some later draws. Test 1. Individual frequency of 1 - 49 38 23 11 31 43 44 25 40 09 32 27 39 3 30 47 12 35 02 06 10 03 28 42 48 45 18 24 19 49 22 07 04 01 17 46 26 29 34 08 14 36 05 37 15 21 16 13 41 20 Table 1 show the predictable 5 % significance test applied data from a 6/49 lotto. To evaluate independence of numbers in successive draws is to count how frequently a given number in one draw has been followed by a given number in the next. Any pair of numbers, the chance this occurs in a particular pair of draws is (6/49) x (6/49) , 0.01499375 ; there are 49 x 49 =2401 pairs of numbers with 280 pairs of consecutive draws to look at. Table 1. Marginal frequency of gambler choice Marginal frequency 46 26 29 34 8 14 36 5 37 15 21 16 13 41 26 Table 1 Shows Marginal frequency of gambler choice that particular numbers are constant especially © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 19 of 33

Table 2. Observed and Expected Frequency Gape 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 size Observe 20 19 18 18 17 17 17 15 15 14 12 10 80 70 65 60 51 39 27 d 0 0 0 0 8 6 0 4 2 8 0 0 Expecte 71. 67. 64. 63. 62. 60. 55 54. 52. 42. 35. 28. 25. 23. 21. 18. 13. 9. 7. d 2 9 6 6 9 7 0 3 9 9 7 6 0 2 4 2 9 6 1 Table 2 shows the independence between the draws. The number of draws between later successive appearances of the fixed number under the null hypothesis has the geometric distribution. To calculate how many pairs of numbers reach up exactly k times in successive draws. And to compare this to with expected under random chance. The observed data, along with the expected values, are in the Table2. The goodness of – fit statistic (48) for equal mean frequencies has the value =47.94. Chi-square statistics (18 degree of freedom) = 23.17. Table 3. Marginal frequency of gambler choice Frequency (%) Numbers (in order of increasing popularity) 3.0,3.5 46 26 29 34 8 14 36 5 37 15 21 16 13 41 26 3.6,3.7 45 18 24 19 22 7 4 1 17 3 28 42 48 49 3.8,3.9 32 27 39 33 30 47 12 35 2 6 10 4.0,4.9 38 23 11 31 43 44 25 40 9 Table 3 Shows Marginal frequency of gambler choice that particular number are constant especially 46 26 29 34 8 14 36 5 37 15 21 16 13 41 26 and this was similar to the order of Joe (1987) who drew from 161 and agreed with the Zimba et al, as there would be remarkable constant from draw-to draw popularity. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 20 of 33

Figure 1. Data structure random choose in 6/49 6 5 4 x.poi 3 2 1 0 0 10 20 30 40 50 Index The above figure shows the data structure in 6/49 Table 4 Marginal frequency of gambler choice Frequency (%) Numbers (in order of increasing popularity) 3.0, 3.5 46 26 29 34 8 14 36 5 37 15 21 16 13 41 26 3.6,3.7 45 18 24 19 22 7 4 1 17 3 28 42 48 49 3.8,3.9 32 27 39 33 30 47 12 35 2 6 10 4.0,4.9 38 23 11 31 43 44 25 40 9 Table layout of the UK national lotto of 2008 t0 2010 of during the period from Sep 2009 to March 2010, out of 188,111,409, 92 tickets won shares in jackpot, in 26 week draws © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 21 of 33

Table 5. UK national Lottery UK 6/49 tickets (20) 1 6 13 23 29 34 6 13 27 28 34 42 18 27 32 35 44 47 7 14 17 42 44 45 3 11 18 23 38 49 8 24 27 31 34 36 3 22 27 31 38 49 16 22 24 30 39 44 5 7 15 37 42 44 3 12 15 17 22 29 3 11 12 20 21 28 1 6 13 23 29 34 In the UK National Lottery, the numbers are arranged in a 6 by 10 grid, with a gap in the bottom-right comer 1). All the latest 10 numbers look random in this figure But in the survey of the data of 2009 to 2010, 1439, wed 7Oct 2009 shows nonrandom, because the numbers show s the total of 7,9,10 ,7,9,10. Table 6. Combinations influenced by previous winning numbers 1439 Wed 7 Oct 2009 26 27 28 43 45 46 Table 7. 26 Week draw between 200-2010 Winning numbers jackpot wins Lowest 01 02 04 11 13 19 (01) 264,490 0 Highest 26 36 39 44 46 49 (49) 10,257,210 8 Averages 08 15 22 30 36 42 (23) 2,044,689 12 Tables 6 shows total of the 26 week draws of 2009 to 2010 minimum, maximum and average numbers won the jackpot. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 22 of 33

Table 8. 280th draw Frequency of the winner and percentage of the money based on the category. Odds before the draw Allocation of prize money Prize money allocated No of Winners in the draw Category Prize 1 in 13983816 Jackpot 40.7% £7,291,476 4 £1,822,869 5+bonus 1 in 2330636 1 in 55492 1 in 1033 1 in 57 7.9% £1,417,913 19 £74,627 5 match 4.9% £885,600 984 £900 4 match 10.7% 35.8% £1,914,816 39,892 £48 3 match £6,420,700 642,070 £10 The jackpot prize pool for this, the 246th single rollover draw and the 280th rollover in total, included £6,420,700 (36.8%) rolled over from the previous lottery, in addition to the original jackpot prize pool of £7,291,476 (63.2%). Table 7 shows the 280th draw of the total lottery tickets were sold. Column 1 list five possible types of winning combination that generate a prize. Column 2 gives the odds for the each combination calculated assumption draw from the balls in the drum. Column 3 gives the basis of the allocation of the prizes; column 4 prize money allocated of this draw. The higher prize value was far less frequent, with four winners matching the six numbers on their tickets, popularity referred to as hitting the jackpot. On average one ticket in 54 generates some form of prize with 95 % of the prize in the lowest £10 prize category. The allocation of prize money is calculated in three stages. The number of tickets sold is first established and the prize pool set at 45% of the tickets proceeds. In the second stage all three matching ball winners indicated in column 6 of the table 7 are paid the fixed prize of €10. The remainder of the prize money available is then distributed, as 10.7%(four ball),4.9% (five ball),7.9%(five plus bonus ball)40.7%(for the jackpot) for the winning tickets. This percentage spread, which is constant, has been determined by the organizers, Camelot. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 23 of 33

Figure 3: Distribution of number (1 to 49) in 280 draws The distribution of number (1 to 49) was shown through probability distribution model from 280 draws. Table 9. Layout of the UK national lottery 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 24 of 33

The above tables show the UK lottery layout. Table.10Winning ticket in the UK National Lottery, draw 9. (1483 Wed 10 Mar 2010 1, 6, 13, 23, 29, 34,) X X X X X X Combinations which "almost" make patterns or arithmetic progressions are also very popular. For example, the largest number of jackpot winners so far in the26th week last March 2010 UK lottery was in draw 280. The winning combination was {1, 6, 13, 23, 29, 34}. In table 10 marking these numbers on the play grid give the pattern Table 11. Some of the Lottery tickets Layouts UK 6/49 ticket 3 8 13 18 23 28 33 38 43 48 Lotto 6 /49 (Ontario) 15 16 17 18 19 20 21 Lotto 6/49 (British Columbia) 10 20 11 21 12 22 13 23 14 24 15 25 16 26 17 27 18 28 19 29 Swiss 6/45 ticket 2 4 8 10 14 16 20 22 26 28 32 34 38 40 44 1 6 11 16 21 26 31 36 41 46 2 7 12 22 27 32 37 42 47 5 10 15 20 25 30 35 40 45 1 2 3 4 5 6 7 8 9 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1 7 13 19 25 31 37 43 5 11 17 23 29 35 41 6 12 18 24 30 36 42 Above table shows the physical layouts of different country tickets. The swiss ticket comes from Riedwyl’s data and shows some compelling evidence and top left right principal diagnosis and top right lower left are the two most popular combinations. Each chosen over 2500 times, 28 further combinations are chosen and in a column, {40→45}, the near diagonal {5,10, 15, 50, 25, 30}, {1,2,3, 43,44,45}, of 11 sets of six consecutive numbers are observed. For the previous year the Swiss winning combinations at least 295 times were all chosen and from the previous year (215 draws), only one winning combination would be chosen. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 25 of 33

CHAPTER V DISCUSSION & CONCLUSION Certain specific tests on sets of random numbers, in addition to obvious tests like equality of frequency and but not the pair wise dependence, all these testes are not universally acceptable and there are no expectation of full agreement with any of the tests. We suggest that the balls numbered 38 be physically examined to give assurance that there is indeed no reason to suspect they are different; and that the frequency with which 38 appears, in both dummy draws and future live draws, be carefully monitored. To interpret independent between draws, look at the entry in the column 1, observed 200, and expected 71.2", this means that there are 200 pairs of numbers (a, b) that arose exactly one time in consecutive draws (a in the first, b in the second), while random chance would give 71.2, such that the pairs on the average. It shows no evidence against independence. The differences between the observed and expected values are sufficiently small as to be attributable to random chance. According to the result there-is no evidence that knowledge of the results of one draw could help to predict the results of the next one. However the lack of independence makes exact calculation of the expected number of gaps of size k in D draws a complex matter, so to test this data of UK 2009 to 2010 simulated the 100000 sets of 280 random draws to obtain the estimates of these values shown in table 1. (The standard error of each is less than 0.02). A significant proportion of lotto gambler choose the same numbers in each draw (45%- 50% in the British National Lottery"). Many gambler regards "ownership" for a set of numbers as an integral part of their lottery experiences. For those who believe that their numbers are lucky, this can lead to trap: they become more and more influenced that they must win as successive weeks go by without success. “This can lead to a form of compulsion in which gambler cannot give up their set of numbers. Lottery operators are well aware of this phenomenon; one US State lottery company ran an announcement exhorting Gambler: "Don't let your number win without you". The sets of numbers chosen by these gamblers are likely to be far from random; it is the belief that small samples should be highly representative of the distributions from which they are drawn. The common example involves sequences of tosses of a coin. When subjects were asked to compare various sequences, they regarded simply have no interest in choosing their own numbers. Others prefer that their numbers are chosen by the lottery terminal in order to protect themselves from entrapment. There are also some people who believe that, since the draw is random, they are more likely to win if they choose their own numbers by a random process, which thus mimics the process by which the winning numbers are chosen. The prize for three correct numbers on a ticket was originally fixed by Camlet at €10.In March 2010, the regulator allowed a rule change such that, if there were more winners than could be provided with €10prizes in any one week from prize pool, all prizes would be reduced in value pro rata .The same proportion of the prize pool none of the week observed other than this pool in the data. No draw to date has come anywhere near to invoking the rule change. It would seem reasonable for Camlet to agree to maintain some level of reserve before invoking such an essential rule change. A number of gamblers are inclined by how frequently the numbers on the lottery ticket have come up in previous draws. Since each draw is (assumed to be) random and independent, it follows that in order on the relative frequency of occurrence of the numbers in © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 26 of 33

earlier draws can bring in no information about the winning numbers in the current draw it is the belief that numbers which have come up less often in previous draws are more possibility to be drawn now, leading gamblers to select those numbers which have previously been drawn least often. Meanwhile, other gamblers select precisely those numbers that have come up most frequently in the past. One reason for this behavior is that these numbers come most easily to mind. Alternatively, gamblers may believe the numbers that have come up most often in previous draws to be lucky. Gamblers get idea by searching for patterns in the previous winning combinations by net. The experimental facts suggest that information on the previous winning combinations does indeed have both a positive and a negative impact on the subsequent requirement for these numbers and combinations. Also amongst the most popular combinations were the set of 6 numbers with the longest waiting time (defined as the number of draws since that number last came up), which were therefore perceived as being the most "overdue"; and the 6 n . The numbers of jackpot winners selecting { } are unlikely to be chosen by gamblers selection their own random numbers, because these combinations do not look random. The shape of the play grid is also likely to affect the combinations chosen by these gamblers. In addition to avoiding sequences and runs, any combination which is perceived as forming any kind of pattern on the play grid may be considered not to be random -looking, and is therefore less likely to be chosen. Combinations influenced by previous winning numbers In this data the last combination is {1, 6, 13, 23, 29, and 34} and also the {26,27,28,43,45,46} the group of combinations is those that form some sort of pattem on the play grid. Some lotto gamblers are not interested in playing lucky or personally significant numbers, nor in choosing their combinations according to false systems based on the previous winning numbers, but do not wish to choose a random set of numbers. One reason for choosing patterned combinations is that it requires very little effort but certainly this combination does not look random. The probability of any combination being chosen so many times by people playing genuinely randomly is approximately zero. With only one number less than 12, and only three numbers less than 31, the combination contains at most one birthday. Indeed, it would be remarkable if this combination were either significant or lucky for so many people. The most likely explanation, if the six numbers on a lottery ticket fare denoted is defined as the number of different spaces. So for any ticket in a 6/49 lottery, X(t) can vary from 1 to 5. 99.7% of all the possible combinations in 6/49 lotto have 3 or more different spaces, and just 0.3% of the combinations have 1 or 2 different spaces. By contrast, nearly 43% of the popular tickets that were unconnected to previous winning combinations ("without memory") had only 1 or 2 different spaces. So these tickets were either arithmetic progressions {X{t) = 1, e.g. {2,4,8,10,12,14) or "very near" to an arithmetic progression {X{t) = 2, e.g. {7,8,9,28,29,30} or {3,4,16,17,29,30}) are some of them straight away choose random numbers. In most lotto games around the world, Quick Pick accounts for about 10%-20% of tickets sold. It was deemed by Camelot, the operator of the UK National Lottery, to be sufficiently widespread to justify naming their random number generator "Lucky Dip" when they introduced it. Some people who decide to play a random combination prefer to choose the numbers themselves, rather than use Lucky Dip. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 27 of 33

Suggestions of gamblers' behavior The greater is the proportion of gambler who chooses non-random sets of numbers; the result shows more skewed is the resulting distribution of the combinations. Because the expected value of a lottery ticket depends partly on how many other gamblers have selected the same combination. The main suggestion of this for gambler such as lotto is that there is substantial difference in the expected value of tickets with different combinations. As a result, a strategic element enters into the choice of numbers. The probability of winning the jackpot is, of course, the same for each combination. But since the jackpot prize is shared between all the tickets that match the winning combination, lottery gambler who seeks to make the most of their expected return should try to pick the combinations that are least likely to be chosen by other gambler. I the general techniques outlined above for choosing numbers generally do not appear to be based on any strategic considerations. Lottery operators tend to encourage people to play lucky or personal numbers in their advertising, for two reasons: first, gamblers who have their own numbers become captured. These gamblers are thus more likely to play regularly than those who choose different numbers each time they play. Second, a more skewed distribution of combinations implies that the coverage (the proportion of all combinations which are chosen by at least one player) in each draw is lower than it would be if all gambler chose randomly. Since, for a given draw, the event "A rollover occurs" is equivalent to the event "The winning combination is not chosen by anybody", the probability of a rollover is equal to one minus the coverage. So a lower coverage increases the probability of rollovers. Not only do rollovers generate extra sales in the rollover draw itself (about 18% more tickets are sold on average in the UK lottery in rollover draws compared to the previous draw), but they also tend to boost sales in the normal (i.e. non-rollover) draw immediately following the rollover, relative to the previous normal draw. So, rollovers generally lead to increased sales, provided they do not occur too frequently (in which case regular lottery gambler might choose to play only in rollover draws). Given that lottery sales are entirely demand-driven, increased sales are generally beneficial both to gambler and to the lottery operator. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 28 of 33

BIBLIOGRAPHY Bellhouse, D.R (1982a) Fair is fair: new rules for Canadian lotteries. Can. Publ. Poly, 8, 311-320 Bellhouse, D. R (1982b). The need for a federal lotteries review board. Can J. Statist, 10, 213-217 Camelot (1995). Annual Report and Accounts. Watford: Camelot. Moore, P.G (1997). The Development of the UK National Lottery: 1992-96. J.R.Statist. Soc. A, 160, pp169-185 Moore, P.G (1994). A National Lottery in the UK. J. Appl. Statist., 21, 607-622. Baker, R.D & McHale, I.G (2009). Modelling the probability distribution of prize winnings in the UK National Lottery: Consequences of conscious selection J.R. Statist. Soc. A, 172, Part 4, pp813-834 CbemofT, Herman. (1981). "How to Beat the Massachusetts Numbers Game," Mathematical Inlelligencer 3(4), 166-172. CbemofT, Herman. (1981). "How to Beat the Massachusetts Numbers Game," Mathematical Inlelligencer 3(4), 166-172. Clotfelter, Charles, and Philip Cook. (1989). Selling Hope: Stale Lotteries in America. Cambridge, MA: Harvard University Press. Clotfelter, Charles, and Philip Cook. (1991a). "Lotteries in the Real World," Journal of Risk and Uncertainly 4(3), 227-232. Clotfelter. Charles, and Philip Cook, (199tb). "The "Gambler's Fallacy" in Lottery Play," National Bureau of Economic Research Working Paper No. .1769. Cook, Philip, and Charles Ciolfelter, (1993), "The Peculiar Scale Economies of Lotto," American Economic Review 83(3), 634-643- Farrell, Lisa et al. (1996). "It Could be You: Rollovers and the Demand for Lottery Tickets," Keele University Department of Economics Working Paper No. 96/17. Greenwood Major, and G. Udny Yule. (1920). "An Inquiry into the Nature of Frequency Distributions representative of Multiple Happenings with particular Reference lo the Occurrence of Multiple Attacks of Disease or of Repeated Accidents," Journal of the Royal Statistical Society 83, 255-279. Haigh, John. (1997). "The Statistics of the National Lottery," Journal of the Royal Statistical Society A 160, 187-206. Halpem, Andrea, and Scott Devereaux. (1989). "Lucky numbers: Choice strategies in the Pennsylvania Daily Number Game," Bulletin of the Psychonomic Society 27(2), 167-170. Henze, N. (1997). "A statistical and probabilistic analysis of popular lottery tickets." Statistica Neerlandica 51(2), 155-163. 276 J- SIMON Hougaard, Philip, Mei-LingTing Lee, and G. A. Whitmore. (1997). "Analysis of Overdispersed Count Data by Mixtures of Poisson Variables an Poisson Processes," Biometrics 53(4), 1225-1238. Langer, Ellen. (1982). "The Illusion of Control." In Daniel Kahneman, Paul Slovic, and Amos Tversky (eds.), Judgment under Uncertainty: Heuristics and Biases. Cambridge; Cambridge University Press. Riedwyl, Hans. (1990). Zahlenlotto: wie man mehr gewinnt. Bern: Haupt. Simon, Jonathan. (1996). "The Expected Value of Lotto when not all Numbers are Equal," European University Institute Economics Department Working Paper No. 97/1. Stem, Hal. and Thomas Cover. (1989). "Maximum Entropy and the Lottery," Journal of the American StatisticalAssociation 84, 980- 985. Tversky. Amos, and Daniel Kahneman. (1974). "Judgment under Uncertainty: Heuristics and Biases," Science 185 (September), 1124-1131. Walker. Michael. (1992). The Psychology of Gambling. Oxford: Butterworth-Heinemann. Ziemba, William et al. (1986). Dr. Z's 6/49 Lotto Guidebook. Vancouver and Los Angeles: Dr. Z Investments,Inc. © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 29 of 33

Joe, H. (1987). An ordering of dependence for distribution of K-tuples, with applications to lotto games. Can J Statist., 15, 227-238 Johnson, R and Klotz, J (1993). Estimating hot numbers and testing uniformity of the lottery. J Am. Statist. Ass., 88, 662-668 Zaman, A and Marsaglia, G (1990). Random Selection of subsets with specified element probabilities communs statist. Theory Meth., 19, 4419-443 © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 30 of 33

APPENDIX - A UK National Lottery Logo] Unexpired UK National Lotto Winning Numbers Winning Numbers No. Day Date Month Year Jackpot Wins Machine Set 1483 Wed 10 Mar 2010 1 6 13 23 29 34 -37 2,348,612 1 Guinevere 4 1482 Sat 6 Mar 2010 6 13 27 28 34 42 -43 1,822,869 4 Guinevere 2 1481 Wed 3 Mar 2010 18 27 32 35 44 47 -23 2,683,247 0 Guinevere 1 1480 Sat 27 Feb 2010 7 14 17 42 44 45 -47 4,310,939 1 Guinevere 6 1479 Wed 24 Feb 2010 3 11 18 23 38 49 -34 739,121 3 Guinevere 5 1478 Sat 20 Feb 2010 8 24 27 31 34 36 -19 4,856,214 1 Arthur 8 1477 Wed 17 Feb 2010 3 22 27 31 38 49 -45 2,566,661 1 Guinevere 6 1476 Sat 13 Feb 2010 16 22 24 30 39 44 -9 4,779,389 1 Arthur 5 1475 Wed 10 Feb 2010 5 7 15 37 42 44 -3 1,178,108 2 Guinevere 8 1474 Sat 6 Feb 2010 3 12 15 17 22 29 -5 733,644 5 Guinevere 7 1473 Wed 3 Feb 2010 3 11 12 20 21 28 -31 1,942,838 1 Arthur 6 1472 Sat 30 Jan 2010 2 3 4 19 23 40 -22 2,122,675 2 Guinevere 5 1471 Wed 27 Jan 2010 1 4 7 11 13 27 -12 264,490 5 Arthur 2 1470 Sat 23 Jan 2010 3 7 8 16 19 30 -9 1,996,431 3 Guinevere 4 1469 Wed 20 Jan 2010 2 16 17 18 27 28 -43 2,257,533 0 Guinevere 2 1468 Sat 16 Jan 2010 4 7 10 14 27 42 -25 1,182,714 5 Guinevere 3 1467 Wed 13 Jan 2010 6 8 14 16 21 43 -19 2,303,805 0 Guinevere 1 1466 Sat 9 Jan 2010 20 24 33 34 37 48 -25 3,913,487 2 Guinevere 4 © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 31 of 33

1465 Wed 6 Jan 2010 4 16 18 43 45 49 -23 2,563,777 0 Arthur 1 1464 Sat 2 Jan 2010 26 29 30 42 43 47 -18 2,581,567 2 Arthur 3 1463 Wed 30 Dec 2009 5 14 27 29 34 35 -4 10,257,210 1 Arthur 4 1462 Sat 26 Dec 2009 2 8 23 25 36 42 -30 6,985,669 0 Guinevere 1 1461 Wed 23 Dec 2009 15 18 26 32 36 37 -22 2,868,480 0 Arthur 3 1460 Sat 19 Dec 2009 1 20 28 33 40 45 -35 2,507,394 2 Arthur 2 1459 Wed 16 Dec 2009 2 20 34 35 40 44 -49 2,900,560 1 Arthur 3 1458 Sat 12 Dec 2009 3 13 22 23 39 45 -4 2,166,793 2 Guinevere 3 1457 Wed 9 Dec 2009 5 13 31 36 40 41 -6 1,408,235 2 Guinevere 4 1456 Sat 5 Dec 2009 8 20 36 43 45 49 -12 2,625,215 2 Guinevere 2 1455 Wed 2 Dec 2009 2 19 22 25 34 47 -28 3,786,303 2 Guinevere 1 1454 Sat 28 Nov 2009 3 4 27 30 36 47 -8 4,680,451 0 Arthur 4 1453 Wed 25 Nov 2009 3 6 11 17 23 29 -44 850,513 8 Arthur 1 1452 Sat 21 Nov 2009 9 24 30 32 36 40 -35 4,883,733 0 Arthur 2 1451 Wed 18 Nov 2009 1 4 12 13 18 19 -38 2,076,538 1 Topaz 4 1450 Sat 14 Nov 2009 1 5 9 20 23 39 -38 844,570 5 Topaz 2 1449 Wed 11 Nov 2009 4 6 12 31 38 48 -29 5,725,012 2 Topaz 3 1448 Sat 7 Nov 2009 19 21 39 44 45 49 -47 8,047,337 0 Topaz 1 1447 Wed 4 Nov 2009 1 2 23 39 42 49 -5 2,681,258 0 Sapphire 4 1446 Sat 31 Oct 2009 12 18 26 35 40 44 -49 3,204,588 4 Topaz 2 1445 Wed 28 Oct 2009 24 25 34 38 41 43 -11 7,354,212 0 Sapphire 1 1444 Sat 24 Oct 2009 5 9 12 31 32 38 -3 4,110,646 0 Sapphire 8 1443 Wed 21 Oct 2009 6 12 16 19 33 44 -20 2,059,106 1 Topaz 7 © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 32 of 33

1442 Sat 17 Oct 2009 11 25 27 32 37 43 -3 4,556,769 1 Topaz 6 1441 Wed 14 Oct 2009 5 11 24 32 35 42 -20 595,593 4 Topaz 7 1440 Sat 10 Oct 2009 17 36 37 40 43 45 -34 2,610,240 2 Sapphire 5 1439 Wed 7 Oct 2009 26 27 28 43 45 46 -5 1,434,182 2 Amethyst 8 1438 Sat 3 Oct 2009 14 21 23 32 36 46 -17 2,430,341 2 Topaz 8 1437 Wed 30 Sep 2009 2 10 15 28 36 41 -8 836,446 3 Sapphire 8 1436 Sat 26 Sep 2009 3 5 17 32 40 43 -29 4,525,058 1 Amethyst 5 1435 Wed 23 Sep 2009 23 25 28 41 46 47 -27 3,681,197 1 Topaz 6 1434 Sat 19 Sep 2009 18 20 31 33 43 45 -1 5,109,794 1 Topaz 8 1433 Wed 16 Sep 2009 18 23 34 44 46 48 -6 1,367,427 2 Sapphire 7 © 2016-2017 All Rights Reserved, No part of this document should be modified/used without prior consent Tutors India™ - Your trusted mentor since 2001 www.tutorindia.com I UK # +44-1143520021, Info@tutorsindia.com Page 33 of 33

Sample Work for Dissertation Statistical Analysis Help | Tutors India