THE MATHEMATICS OF LOTTERY Odds, Combinations, Systems

THE MATHEMATICS OF

LOTTERY

Odds, Combinations, Systems

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C?t?lin B?rboianu

INFAROM Publishing

Applied Mathematics

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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

material is concerned, specifically the rights of translation, reprinting, reuse of formulas and tables,

recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks.

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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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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