THE MATHEMATICS OF LOTTERY Odds, Combinations, Systems
THE MATHEMATICS OF
LOTTERY
Odds, Combinations, Systems
?
¡Ç
C?t?lin B?rboianu
INFAROM Publishing
Applied Mathematics
office@
ISBN 978-973-1991-11-5
Publisher: INFAROM
Author: C?t?lin B?rboianu
Correction Editor: CarolAnn Johnson
Copyright ? INFAROM 2009
This work is subject to copyright. All rights are reserved, whether the whole work or part of the
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Duplication of this publication or parts thereof is permitted only under the provisions of Copyright
Laws, and permission for use must always be obtained from INFAROM.
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Contents (of the complete edition)
Introduction ......................................................................................
5
The Rules of Lottery ...................................................................¡.
11
Supporting Mathematics ......................................................¡.......
15
Probability space ..............................................................¡.........
16
Probability properties and formulas used .........................¡.........
19
Combinatorics ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡...
22
Parameters of the lottery matrices ¡¡¡¡¡¡¡¡¡¡.........
25
Number Combinations .......¡¡¡¡.¡¡¡¡¡¡¡¡¡.........
27
Probabilities of Winning with Simple Lines ..................................
37
General formula of the winning probability ¡¡¡¡¡¡..........
37
Cumulated winning probabilities ¡¡¡¡¡¡¡¡¡¡¡......
53
Intermediary probabilities ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡.
67
Enhancing the Winning Probability .................¡¡¡¡.¡¡¡.
75
Probabilities of winning with systems ¡¡¡¡¡¡¡¡..........
76
Probability thresholds ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡
91
Compound Lines ................¡¡¡¡¡¡¡¡¡¡¡¡¡..........
107
Unfolding compound lines ¡¡¡¡¡¡¡¡¡¡¡................
108
Winning probabilities ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡........
111
The number of prizes ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡
127
Systems of compound lines ...........................................................
136
Bridgehead Systems ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡.
141
Winning probabilities ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡........
142
The number of prizes ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡
155
Reduced Systems .........¡.¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡
159
Obtaining the reduced systems ¡¡¡¡¡¡¡¡¡¡¡¡¡..
161
Winning probabilities ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡........
173
Probabilities of Repeated Events ¡¡¡¡¡¡¡¡¡¡¡........
177
The Strategy of Choosing ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡
191
References ......................................................................................¡
203
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4
Introduction
Lottery is by far the oldest and the most widely known game of
chance, having been practiced since antiquity. In its various forms,
the lottery preserves a basic structure and technical procedure that
makes it the easiest and most popular game of chance: the random
draw from an urn of some objects (balls, tickets, lots, plates, slips,
etc.) containing predefined symbols (numbers, images, words, etc.),
followed by the distribution of prizes for players who made correct
predictions regarding this draw, according to some pre-established
rules. Particularizing this definition, we find forms of lottery even in
the simple procedures of drawing lots or organizing tombolas.
Nowadays, the most prevalent form of lottery is that with
randomly selected numbers; winning categories are based on the
number of numbers correctly predicted on the playing ticket. The
most popular forms of these games are the national and state
lotteries.
The early history of the lottery can be traced back to the second
millennium B.C. In the Chinese The Book of Songs there is a
reference to a game of chance known as ¡°the drawing of wood¡±,
which in context appears to describe the drawing of lots. The first
signs of a lottery trace back the Han Dynasty between 205 and 187
B.C., where ancient Keno slips (a form of lottery also practiced
nowadays) were discovered. The first known European lottery
occurred during the Roman Empire. The earliest record of a public
lottery offering tickets for sale is the lottery organized by the Roman
Emperor Augustus Caesar. Proceeds went for repairs to the city of
Rome, and the winners were given prizes in the form of valuable
articles.
The first recorded lotteries to offer tickets for sale with prizes in
the form of money were held in the Low Countries during the period
1443¨C1449, and their funds were used for town fortifications. In the
17th century, it was quite common in the Netherlands to organize
lotteries to collect money for the poor; the prizes were paintings.
The Dutch were the first to shift the lottery to solely monetary prizes
and to base prizes on odds (roughly about 1 in 4 tickets winning a
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