Published in September 2009.

**• BirthdayParadox.EXE ~ Freeware for a registered member — Permanent Membership.
• Collisions.EXE ~ Free probability software for registered members — Permanent (Lifetime) Membership.
This is the most accurate and comprehensive software for calculations in the probability of duplication (or collisions, or coincidences, or repetition), including the tiny **

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*Birthday Paradox*: Combinatorics, Probability, Software, Pick 3 Lottery, Roulette.**This article specifically deals with the application of***Birthday Paradox*to the lottery, lotto, and roulette (other forms of gambling as well by extension). The original essay touched a few issues of creating winning systems from*Birthday Paradox*, or*probability of repetition, duplication, coincidences, collisions*. Such systems would apply to games of chance such as lotto, lottery, roulette… and more. There are lottery players and gamblers with a strong interest in*Birthday Paradox*. They realize how important the probability of repetition is. The amount of messages I receive clearly indicates that there is a lot of life in this phenomenon. Doesn't life start with*Birth*?!

So, here is a second take on the *Birthday Paradox* — the *probability of repetition, duplication, coincidences*. Can such mathematical knowledge be applied to gambling, especially lottery? I was skeptical when I first heard about it and asked about it. After all, I am the master of probability, they say! That's why they ask me! The difficulty of achieving thorough understanding of this phenomenon was caused by a parameter I call * number of elements*. Indeed, the roulette numbers repeat. You can see them all the time, if the casinos turn on the electronic displays known as the marquees. But is there a rule, a mathematical formula that enables us to calculate the repeat probability? The roulette draws (spins) one number at a time. On the other hand, the birthday paradox assumes a string of numbers (a numerical set).

The * Birthday Paradox* is just one particular case of exponential sets. They are also known as Saliusian sets or Ion Saliu sets. The exponents are sets of numbers where the duplicates are allowed. By contrast, the other mathematical sets consist of unique elements only (e.g. permutations, or combinations). The exponential sets had been neglected by mathematics, although they are the most important! All other sets are derived from exponents.

I thought more deeply on this repetition fact. I reconciled the one-element attribute with strings-of-elements. For example, I look at a sequence of 8 roulette numbers as an eight-element string. The degree of certainty is better than 50% that such string should contain one repetition (duplication). One of the eight numbers should be a repeat with a 50-50 chance. The same is true about lottery drawings. In this case, the element is the index of the combination (or set) drawn. Every lotto combination, for example, is defined by an index or lexicographical order, or lexicographic rank.

With this new knowledge in mind, I studied some real data: Lottery drawings and roulette spins. I was somehow surprised to discover that repetition occurs close to that cutoff point of the 50-50 chance! I should also point out that the strength of the analysis and system creation is stronger at the beginning of the game. For lottery and lotto, the beginning is clear: A game starts with the first drawing of a game format. For roulette, a beginning is the very first spin of the day.

My mathematical analysis has been helped a great deal by new games in the Pennsylvania State Lottery. They started a new pick-5 digit lottery game named * Quinto*. At the time of this writing, the Quinto game offered me for analysis 397 drawings. I used my own software for analysis. I included that type of software only in the horseracing bundle:

I looked at the report for the Quinto game. Once again, I am a Johnny-Come-Late kid! I ran Collisions.EXE, option R = Reversed Duplication Problem. The parameters are 0, 99999, and a degree of certainty DC = 50% (do not type the percentage sign). The program calculates that some 373 Quinto drawings are necessary to reach a 50-50 chance of repetition. Question is, should I wait 373 draws before I play the previous draws? I think of that margin of error the pollsters apply in political polling (especially during electoral seasons). It is 3%, plus or minus. In this case, 3% of 373 is something like 11 drawings. Thus, I start playing the system 11 Quinto draws before 373. In the case of Pennsylvania lottery Quinto, the first repeat occurred right on the nose: Draw #373! Boy, o, boy! Play 373 tickets for 11 drawings for a cost of $4103 and win $50,000!

Pennsylvania lottery has also a new 5-number lotto game (5 from 43). The game had 604 draws at this time of analysis. The 50-50 chance of a repeat-combination leads to a result of 1156 draws, with a 3%-margin of error equal to 35. Thus, the system should be played starting at draw # 1156 – 35 = 1121. About two years worth of drawings…

Anyone say Powerball? Okay. That huge game requires 16,453 drawings as the waiting period (minus 494 draws as the margin of error). A long, long way to go (over a hundred years!) Kids, mark this down … for your kids! BRRRRRRRRAHAHAHAHA!!!!!!!

I looked also at roulette. I believe I got an accurate data file from Hamburg Spielbank (Casino), Germany. The file contains all the roulette spins for the month of January, 2006. I will publish here a fragment of the UnderOver report for this roulette case. The fifty-fifty repeat chance for a single-zero roulette game is 8 spins (numbers). The margin of error is 1. To that effect, I wait 7 spins after the roulette table is open and becomes active.
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* Roulette Winning Pattern Under/Over * * File: HAMB0106.WH1 Line Spin Skip_1 Skip_2 lt+lt lt+GT GT+lt* GT+GT … 7934 16 7 10 - - - + 7935 1 5 18 - - - + 7936 27 27 2 - + - - 7937 2 12 19 - - - + 7938 4 27 26 + - - - 7939 22 12 25 - - - + 7940 31 27 24 - + - - ... 7964 26 4 0 - - - + 7965 27 7 0 - - - + 7966 3 27 0 - + - - 7967 10 6 0 - - - + 7968 21 17 0 - - - + 7969 26 27 0 - + - - 7970 32 27 0 - + - - 7971 16 15 0 - - - + 7972 25 27 0 - + - - ... 7984 17 27 0 - + - - 7985 29 27 0 - + - - 7986 21 27 0 - + - - 7987 16 27 0 - + - - 7988 0 27 0 - + - - 7989 14 27 0 - + - - 7990 15 27 0 - + - -

The spin at the bottom (7990) represents the first spin of the month of January 2006. It is also the first recorded day of the month. Just the beginning of a day is sufficient for roulette systems. I wait 7 spins, then start playing the roulette numbers of the previous 8 spins. The next two spins are not winners, but the third one is. I paid for 8+8=16 numbers and I won 36. I made a profit and I might quit as well. I don't quit because I know so well the Fundamental Formula of Gambling (FFG). The numbers tend to repeat after a number of trials (spins) equal to or less than the FFG median. That great piece of software — SuperFormula.exe — calculates the FFG median for 8 spins to be equal to 5. It means that I can decide to play this birthday paradox system only for 5 spins following a hit. I lost the first situation (the next hit occurred after 13 spins). Then, I paid for 1+2+1+1+4+5+2+5 = 17 spins. I add also the first 7 spins of initial waiting for a total of 24 betting units. Total wins: 11 * 36 = 396 betting units, or a profit of 372 betting units. Just a few hours in the day!

It is very important to know the start of the game. Then, only play for a number of trials under the FFG median. Yes, it is likely you will not cash in all winning situations, but you keep the cost down. Collisions.EXE makes it very easy to do the calculatIONs. You only need to know the total elements (sets, combinations, numbers, etc.) in the game. For example, a lotto 6-49 game has a total of 13983816 combosnations. In Collisions.EXE select R = Reversed Duplication Problem, then:

1) The LOWER bound; default = 0 (the 1st digit);

2) The UPPER bound; default = 9 (the last digit);

3) The degree of certainty = 50% or 0.5

* Do NOT type the % sign -- Type only a number; e.g. 50, 66, 75, 99.999, etc.

For a 6-49 lotto game, the parameters are:

1

13983816

50

The result: 4404 lotto draws for at least one lotto combination to repeat with a degree of certainty equal to 50%.

Read Ion Saliu's first book in print: *Probability Theory, Live!*

~ Discover profound philosophical implications of the ** Formula of TheEverything**, including the Birthday Paradox and repetition of phenomena, gambling, lottery included.

- Theory of Probability: Best Introduction, Formulas, Algorithms, Software.
- The Fundamental Formula of Gambling and Winning.
- Mathematics of the Fundamental Formula Of Gambling.
- Combinatorics: Calculate, Generate Exponents, Permutations, Sets, Arrangements, Combinations for any Numbers and Words.
- Standard Deviation: Theory, Algorithm, Software.

Standard deviation: Basics, mathematics, statistics, formula, software, algorithm. - Standard Deviation, Gauss, Normal, Binomial, Distribution.

Calculate: Median, degree of certainty, standard deviation, binomial, hypergeometric, average, sums, probabilities, odds. - Caveats in Theory of Probability.
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.**Software: Science, Mathematics, Statistics, Lexicographic, Combinatorial**