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Mark Six
◦ A legal lottery game in Hong Kong
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What is the winning probability of each prize
in Mark Six?
Actually how small is the winning probability?
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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.
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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)
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n!=n× (n-1)×(n-2)×… ×3×2×1
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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)
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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.
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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
1st
All 6 drawn number
Probability
Stimulation
Key
Drawn no.
Extra no.
 No. corresponded
to drawn numbers
 No. corresponded
to drawn numbers

2nd
R No. chosen at
random (other than
drawn no. and extra
no.)
5 drawn numbers
+ extra number
There are 49-7=42 R
number in total.

Prize
Criteria
3rd
5 drawn number
Probability
Stimulation
R
4th
4 drawn
numbers +
extra number
R
Prize
Criteria
5th
4 drawn number
Probability
Stimulation
RR
6th
3 drawn
numbers +
extra number
RR
Prize
Criteria
7th
3 drawn
numbers
Probability
Stimulation
RRR
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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
1
5
2
3
3
6
4
4
5
4
6
3
7
4
8
4
9
3
10
5
11
4
12
7
13
4
14
7
15
5
16
4
Probability
0.0238
0.0143
0.0286
0.0190
0.0190
0.0143
0.0190
0.0190
0.0143
0.0238
0.0190
0.0333
0.0190
0.0333
0.0238
0.0190
Drawn No. Frequency
17
5
18
5
19
0
20
5
21
3
22
3
23
4
24
5
25
5
26
3
27
4
28
5
29
4
30
3
31
4
32
2
Probability
0.0238
0.0238
0
0.0238
0.0143
0.0143
0.0190
0.0238
0.0238
0.0143
0.0190
0.0238
0.0190
0.0143
0.0190
0.0095
Drawn No. Frequency
33
4
34
8
35
5
36
4
37
4
38
4
39
4
40
1
41
6
42
4
43
6
44
3
45
3
46
4
47
7
48
5
49
9
Probability
0.0190
0.0381
0.0238
0.0190
0.0190
0.0190
0.0190
0.0048
0.0286
0.0190
0.0286
0.0143
0.0143
0.0190
0.0333
0.0238
0.0429
By experimental trials, the no. 49 has the highest probability of
being drawn!!!
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Experimental probability
◦ Expected probability of each ball being drawn
=0.023
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Theoretical probability
◦ Expected probability of each ball being
drawn=1/49=0.020
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Experimental probability ≈ Theoretical
probability
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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.
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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)
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Located in densely populated areas
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Large flows nearby
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Not much OCBs in its area
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Residents have gambling habits or are rich
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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↑
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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)
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As mentioned:
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◦ ∵People↑ Probability of Gambling↑
◦ ∵ Gambling↑ Probability of Winning↑
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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↑
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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.
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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.