What is the winning probability of each prize in Mark Six? Actually how small is the winning probability?

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2 Mark Six A legal lottery game in Hong Kong What is the winning probability of each prize in Mark Six? Actually how small is the winning probability?

3 Describe non-certain proposition In terms of a numerical measure and this number, between 0 and 1, we call Probability. probability certainty to occur Given an axiomatic mathematical derivation in probability theory Used widely in such area of study as statistics, etc.

4 A way of selection of several things out of a larger group Notation ncr denotes the number of combinations of choosing r objects from n different objects, without repetition and regardless of their order. (where r n) n!=n (n-1) (n-2) 3 2 1

5 A way of arrangement of things Notation npr denotes the number of arrangements of any r objects from n objects, taken r at a time without repetition. (where r n)

6 In each Mark Six draw, the Mark Six Draw Machine will randomly draw six numbers from 1 to 49, called the Drawn Numbers, by letting go six of the 49 balls that are inside the machine, with each ball representing a different number. After that the machine will draw another number, called the Extra Number, to allow some winners to get a bigger prize.

7 In each unit bet, costing ten dollars, the bettors choose six numbers before the Mark Six draw. If any three or more of the chosen numbers match the Drawn Numbers, prizes will be awarded. The more chosen numbers match the Drawn Numbers and the Extra Number, the bigger the prize is.

8 Prize Criteria Probability Stimulation 1 st All 6 drawn number Key Drawn no. Extra no. 2 nd 5 drawn numbers + extra number No. corresponded to drawn numbers No. corresponded to drawn numbers R No. chosen at random (other than drawn no. and extra no.) There are 49-7=42 R number in total.

9 Prize Criteria Probability Stimulation 3 rd 5 drawn number 4 th 4 drawn numbers + extra number R R

10 Prize Criteria Probability Stimulation 5 th 4 drawn number 6 th 3 drawn numbers + extra number R R R R

11 Prize Criteria Probability Stimulation 7 th 3 drawn numbers R R R

12 Are there any numbers which are more lucky than the others? Some bettors study the statistics on the frequency of each number being drawn to predict the results of the coming draw.

13 Drawn No. Frequency Probability Drawn No. Frequency Probability Drawn No. Frequency Probability By experimental trials, the no. 49 has the highest probability of being drawn!!!

14 Experimental probability Expected probability of each ball being drawn =0.023 Theoretical probability Expected probability of each ball being drawn=1/49=0.020 Experimental probability Theoretical probability

15 In fact, there is no use studying the statistic on past result. As every draw result is independent, which means the previous draw result will not affect the coming one. Each number is simply drawn at random.

16 Top 10 Lucky off course betting branches ()=No. of times selling 1st Prize Mark 6 tickets in corresponding off course betting branches (up to 29 Nov, 2011)

17 Located in densely populated areas Large flows nearby Not much OCBs in its area Residents have gambling habits or are rich

18 For example, Kwun Tong District has 55,000 residents/km2, which is the highest in Hong Kong. People Probability of Gambling Gambling Probability of Winning

19 E.g. Stanley Street in Central district Located near the Queen s Road Central and Central-Mid-Levels Escalator and Walkway System (area with large flows) As mentioned: People Probability of Gambling Gambling Probability of Winning

20 E.g. both OCBs in Tai Po has high winning rate 14,800 hectares but only 2 OCBs in the district OCB Density Density Usage Usage Probability of Winning

21 Stanley Street OCB is in Central district Most transnational companies set their headquarters there People working there and people in the expensive neighborhood are most likely to be the richest people in HK People in Villages of New territories plays Mahjong and have gambling habits. They would like to risk some money for big fortune.

22 There are other external factors, such as the geographical advantage, the population density in the district, leading to the location of ticket-buying off course betting branch and the winning probability.

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