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?

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.

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

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)

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.

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.

Prize Criteria Probability Stimulation

1st All 6 drawn number

2nd 5 drawn numbers + extra number

Key

Drawn no.

Extra no.

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.

Prize Criteria Probability Stimulation

3rd 5 drawn number R

4th 4 drawn numbers + extra number

R

Prize Criteria Probability Stimulation

5th 4 drawn number R R

6th 3 drawn numbers + extra number

R R

Prize Criteria Probability Stimulation

7th 3 drawn numbers

R R R

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.

Drawn No. Frequency Probability

1 5 0.0238

2 3 0.0143

3 6 0.0286

4 4 0.0190

5 4 0.0190

6 3 0.0143

7 4 0.0190

8 4 0.0190

9 3 0.0143

10 5 0.0238

11 4 0.0190

12 7 0.0333

13 4 0.0190

14 7 0.0333

15 5 0.0238

16 4 0.0190

Drawn No. Frequency Probability

17 5 0.0238

18 5 0.0238

19 0 0

20 5 0.0238

21 3 0.0143

22 3 0.0143

23 4 0.0190

24 5 0.0238

25 5 0.0238

26 3 0.0143

27 4 0.0190

28 5 0.0238

29 4 0.0190

30 3 0.0143

31 4 0.0190

32 2 0.0095

Drawn No. Frequency Probability

33 4 0.0190

34 8 0.0381

35 5 0.0238

36 4 0.0190

37 4 0.0190

38 4 0.0190

39 4 0.0190

40 1 0.0048

41 6 0.0286

42 4 0.0190

43 6 0.0286

44 3 0.0143

45 3 0.0143

46 4 0.0190

47 7 0.0333

48 5 0.0238

49 9 0.0429

By experimental trials, the no. 49 has the highest probability of

being drawn!!!

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

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.

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)

Located in densely populated areas

Large flows nearby

Not much OCBs in its area

Residents have gambling habits or are rich

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↑

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↑

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↑

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.

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.