Predicting the Mark Six Result by Lau Tsz Shan 19203578 A thesis submitted in partial fulfillment of the requirements for the degree of BaChelor of SCienCe (Honors) in Mathematics and Statistics at Hong Kong Baptist University Date 3-1-2021 1 ACKNOWLEDGEMENT I would like to thank you Dr. Yau Chin Ko Andy for giving guidance, encouragement and advice. For this, I would like to express my sinCere thanks and appreciation. I am very grateful for his comments and discussions. AdviCe given by Dr. Yau Chin Ko Andy has been a great help in my final year projeCt. _Carrie Lau_ Signature of Student __Lau Tsz Shan___ Student Name Department of MathematiCs Hong Kong Baptist University Date: ___29-12-2020___ 2 Table of contents Abstract-------------------------------------------------------------------------------------------6 1. Introduction-----------------------------------------------------------------------------------7 1.1 Motivation-------------------------------------------------------------------------------------7 1.2 Background-----------------------------------------------------------------------------------7 1.2.1 The Mark Six--------------------------------------------------------------------------7 1.2.2 Hong Kong Jockey Club------------------------------------------------------------7 1.2.3 Type of bets---------------------------------------------------------------------------8 1.2.4 Snowball------------------------------------------------------------------------------10 1.3 Objective-------------------------------------------------------------------------------------10 2. Data Collection------------------------------------------------------------------------------11 2.1 Network Information-----------------------------------------------------------------------11 3. Software--------------------------------------------------------------------------------------12 3.1 Excel-----------------------------------------------------------------------------------------12 3.2 C++ Program--------------------------------------------------------------------------------12 4. More Data------------------------------------------------------------------------------------13 4.1 Bonus Distribution--------------------------------------------------------------------------13 3 Table of contents 5. Data Analysis-------------------------------------------------------------------------------14 5.1 Single Number------------------------------------------------------------------------------14 5.2 Region----------------------------------------------------------------------------------------15 5.3 Odd and Even of the Number-------------------------------------------------------------16 5.4 Consecutive Number-----------------------------------------------------------------------17 5.5 Total Number of different periods--------------------------------------------------------18 6. Mark Six Probabilities---------------------------------------------------------------------19 6.1 The probability of 7 prizes----------------------------------------------------------------19 6.2 Expect Value--------------------------------------------------------------------------------20 6.3 Goodness of fit test-------------------------------------------------------------------------21 7. Possible ways to predict the result more accurately------------------------------------23 7.1 Consider odd number first-----------------------------------------------------------------23 7.2 Do not select more than 3 consecutive numbers----------------------------------------23 7.3 Choose total number is range 91-230----------------------------------------------------23 7.4 Consider the region of the number-------------------------------------------------------23 4 Table of contents 8. Method to statistic -------------------------------------------------------------------------24 8.1 Basic Statistics Model-------------------------------------------------------------------- 24 8.2 Decision Tree Model----------------------------------------------------------------------24 9. Program Model-----------------------------------------------------------------------------25 9.1 Check the result of number in C++-----------------------------------------------------25 9.1.1 Set “Include” in the file----------------------------------------------------------------25 9.1.2 Set Some Guidelines-------------------------------------------------------------------25 9.1.3 Prediction Mark Six Result-----------------------------------------------------------25 9.1.4 Result of Program-----------------------------------------------------------------------26 9.1.5 Example Simulation-------------------------------------------------------------------26 9.1.6 Accuracy---------------------------------------------------------------------------------27 9.2 Predict the result of Mark Six in C++--------------------------------------------------28 9.2.1 Set Include in the file-------------------------------------------------------------------28 9.2.2 Example Simulation--------------------------------------------------------------------28 9.2.3 Accuracy---------------------------------------------------------------------------------28 10. Difficulties and Limitation--------------------------------------------------------------29 11. Conclusion---------------------------------------------------------------------------------30 12. Reference----------------------------------------------------------------------------------31 13. Appendix-----------------------------------------------------------------------------------33 5 Abstract Estimating or counting the results of matches or lottery is a hot topic in various industries, including the Mark Six. The probability of profit depends on the accuracy of predicting the probability of winning the Mark Six. According to the information provided by the Hong Kong Jockey Club, statistical results can be drawn. This report mainly collects data for statistical analysis and data analysis. Also, we will use the program to create a model for result prediction. The report shows a lot of tables and figures which are related to the statistics of the Mark Six. The report would discuss the data analysis of the Mark Six, the probability of Mark Six, possible ways to predict the result more accurately, the program model and the result derived from the model. In this model, we will know the result of Mark Six is losing money or not. It is possible to win the game. Although we make a statistic based on the past results, we can also increase the probability of winning the lottery. 6 1.Introduction 1.1 Motivation The mark six is a popular topic in Hong Kong. Many people overspend or even go bankrupt because of buying the mark six. Also, there are many teaching software in the society nowadays. So, I would like to use this teaching software to predict the Mark Six results. 1.2 Background 1.2.1 The Mark six Mark six is a lottery style game1. In Hong Kong, we can choose 6 numbers in the 49 numbers. Also, it has a seven prize levels. The winning numbers in the Mark Six lottery will be selected 6 numbers and one special number from the lottery machine which the number is 1 to 49. However, there are different types of betting in the Mark Six. There is single entry, multiple entry, banker entry and random entry. 1.2.2 Hong Kong Jockey Club The Hong Kong Jockey Club is one of the oldest mechanisms in Hong Kong.2 It is a non- profit organization providing horse racing, sporting and betting entertainment in Hong Kong. It holds a government-granted monopoly in providing pari-mutuel betting on horse racing, the Mark Six lottery, and fixed odds betting on overseas football events. The organization is the largest taxpayer in Hong Kong, as well as the largest community benefactor. 1 https://en.wikipedia.org/wiki/Mark_Six 2 https://en.wikipedia.org/wiki/Hong_Kong_Jockey_Club 7 1.2.3 Types of bets3 In Hong Kong, there are 4 entry in the Mark Six. The unit investment for each Mark Six Entry is $10 with several types of entry available for selection. For Multiple & Banker entries, a partial unit investment of HK$5 will be accepted; all Prizes will be paid based on the fraction that the Partial Unit Investment bears to the Unit Investment. l Single Entry We can choose 6 numbers from 1 to 49 For example, 1+2+3+4+5+6 l Multiple Entry We can choose at least 7 different numbers from 1 to 49. Also, we can choose a partial unit which is 5 HKD per unit investment. For example, we select the 7 different number which are 1 to 7. It has 7 single entries. 1) 1+2+3+4+5+6 2) 7+2+3+4+5+6 3) 1+7+3+4+5+6 4) 1+2+7+4+5+6 5) 1+2+3+7+5+6 6) 1+2+3+4+7+6 7) 1+2+3+4+5+7 We can see that if we buy $10 per unit, it totally is 70HKD. If we buy $5 per unit, it totally is 35HKD. 3 https://is.hkjc.com/aosbs/help/en/mk6_guide.html 8 Multiple Entry Chance Table:4 Table 1 Multiple Entry Chance Table l Banker Entry We can choose less than or equal to 5 numbers from 1 to 49 as banker(s) and (49- the number(s) of banker) number(s) will be selected as leg(s). Also, we can choose a partial unit which is 5 HKD per unit investment. For example, we select 3 bankers which are 1, 2 and 3 with 4 leg numbers 4, 5, 6 and 7. The selections are 3 banker and 4 leg number entry as below: Banker(s) Other number selections 1+2+3 4+5+6 1+2+3 4+5+7 1+2+3 4+7+6 1+2+3 7+5+6 Table 2 Example of Banker Entry We can see that if we buy $10 per unit, it totally is 40HKD. If we buy $5 per unit, it totally is 20HKD. 4 https://is.hkjc.com/aosbs/help/en/mk6_guide.html 9 1.2.4 Snowball If