Colin Fairbrother

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by Colin Fairbrother
As late as May 7, 2011 Ion Saliu wrote preposterously in a newsgroup, "It is totally documented that ‘degree of certainty’ was coined by Ion Saliu and it was first published at" IT IS TOTALLY DOCUMENTED THAT STATEMENT BY ION SALIU IS A LIE.
The question for the Fundamental Formula of Gambling centers around who in history first knew about enumerating the possibilities for 1 event, then for multiple events and to use the words of Jacob Bernoulli calculate the Degree of Certainty from his posthumous publication of Ars Conjectandi in 1705, "Probability is a degree of certainty and differs from certainty as a part from a whole.".
For an interesting article see The Significance of Jacob Bernoulli's Ars Conjectandi for the Philosophy of Probability Today by Glen Shafer
Humans understood Probability going back to our caveman days in working out that more men meant a greater chance of defeating ones enemies.

From the 17th Century a few mathematicians started to use a measuring stick which standardised to a number from 0 to 1 with each respectively designating impossible and certain and the gradations in between represented by a decimal number such as 0.5 meaning 50% or equally likely to go either way. This ratio was arrived at by dividing the favourable chances by all chances.

The general equation for multiple events is then dating from the 17th Century or before : - Probability of Success = Probability of Event ^ Number of Events or
We can solve simply for Number of Events, NE, in basically two ways by using logs. Logarithms date back to the early 17th Century and amongst the mathematicians working on clarifying the idea was John Napier who encountered a constant that subsequently was named by Euler as simply e and ranks in importance alongside pi.

The first way is simply where PE is the base for the log: -

NE = logPEPS

The second way is to log both sides of the equation and then use a log rule : -

log(PS) = log(PENE)
log(PS) = NE x log(PE)
NE = log(PS) / log(PE)

If we are more interested in the complement for the probabilities we deduct each probability from 1 as in : -

NE = log1-PE(1 - PS)
NE = log(1-PS) / log(1 - PE)

Substituting a name such as Jacob Bernoulli's "Degree of Certainty" for PS and then describing the equation as the Formula of Bankruptcy and inviting others to call it The Fundamental Formula of Gambling as Ion Saliu did in 1997 shows neither creativity nor ingenuity. In a stupendous act of plagiarism Ion Saliu went on in later modifications of the original article to embellish it with this statement, "The Fundamental Formula of Gambling (FFG) is an historic discovery in theory of probability, theory of games, (sic) and gambling mathematics.". THE FUNDAMENTAL FORMULA OF GAMBLING IS BLATANT, FLAGRANT AND PREPOSTEROUS PLAGIARISM BY ION SALIU.

As previously mentioned the constant e which is approximately 2.71828 was encountered by John Napier in 1618 when working with logarithms. Dividing 1 by e we get 0.3678796 ...  or approximately 0.37 or 37%. If we subtract a probability of 0.37 from 1 we get 0.63 or 63%. Recently I encountered  1 - 1/e when enumerating 14 million random selections for a Pool of 49 and Pick of 6 using the MersenneTwister algorithm as 63% were distinct as expected and I noted it at the time.
Included in 14 million from 13,983,816 possibilities: 8,845,381 or 63%
Not included
in 14 million from 13,983,816 possibilities: 5,138,435 or 37%
Combinations appearing once only in 14,000,000: 5,143,806 or 37%
Distinct Combinations with appearance more than once
in 14,000,000: 3,701,575 or 26%
Combinations including repeats with appearance > 1
in 14,000,000: 8,856,194 or 63% 
1x 5143806 5143806
2x 2576162 5152324
3x 859154 2577462
4x 214693 858772
5x 43121 215605
6x 7256 43536
7x 1034 7238
8x 139 1112
15 135
10x 1
       Total: 14000000
See : -
Random Lotto Numbers using the Mersenne Twister
It is worth noting that the breakdown of 13,983,816 random selections as for Pool 49 and Pick 6 was hopelessly misunderstood by Ion Saliu and confirmed recently in a newsgroup rec.gambling.lottery April 29, 2011 where he stated, "If you RANDOMLY generate 13983816 combinations, the probability of appearance for one combo is only {1 – 1/e} or 63.2%. That also means that only 63.2% of the combos generated are UNIQUE; 36.8% of the outcome represents duplicates."
Needless to say at the outset and for each of the 14,000,000 events the chances of any one combination appearing is as per probability 1 in 13,983,816 but for a large number of possibilities there comes a point for a large number of events where the likelihood of the number randomly selected being already in previous selections is greater and for the 14 million combinations I generated this occurred around the 9,693,000th selection or 36% of the possibilities due to repeats. 
Contrary to the understanding of Ion Saliu only 37% are "unique" or with a once only occurrence NOT 63% as he states. Of the 14 million combinations 26% occur more than once ie have duplicates and make up 63% of the combinations NOT 36.8% as Ion Saliu states ie the other way round. This from an idiot that claims he invented 1 - 1/e. Ion Saliu hopelessly suffers from the psychiatric illness referred to as Delusions of Grandeur or Grandiose Delusions.  
The absurdity and naivety by Ion Saliu to state that in generating 13,983,816 random 6 integer combinations from a pool of 49 integers 63% or around 9,000,000 will be "unique" ie ONLY APPEAR ONCE and that 37% or around 5,000,000 will appear more than once is obvious for the total combinations would be in excess of 19,000,000 even for just one duplicate for each of the 5,000,000!  
For a petty numerologist like Ion Saliu to claim 1 - 1/e as his own discovery by renaming it PARADOX OR PROBLEM OF N TRIALS IS ALSO BLATANT, FLAGRANT AND PREPOSTEROUS PLAGIARISM BY ION SALIU.
For further details with a table showing that the number of events may still be obtained using the equation above where the probability of the event is correct but where the probability of success is less than optimum eg 1 toss of coin has probability 0.5 but 1 event is obtained with a probability of success of 0.293. Perhaps I should call this the Paradox of Limited Success. see
Colin Fairbrother
ps The following simple table is an example of the constant e. As N is increased by 10 for each line so 1/n is added and the result is raised to the power of N. It can be seen that at around 100 the result has very little increase although N is still being increased by the same amount.
                n           1/n   (1+1/n)^n
1 1.0000 2.0000
10 0.1000 2.5937
20 0.0500 2.6533
30 0.0333 2.6743
40 0.0250 2.6851
50 0.0200 2.6916
60 0.0167 2.6960
70 0.0143 2.6991
80 0.0125 2.7015
90 0.0111 2.7033
100 0.0100 2.7048
110 0.0091 2.7060
120 0.0083 2.7070
130 0.0077 2.7079
140 0.0071 2.7086
150 0.0067 2.7093


This article first published on the web 10/5/2011 and has been viewed 19805 times.
The author may be contacted at To generate sets of Lotto numbers to play for most Lotto games around the world go to LottoToWin ® where for a US$5.00 annual subscription you can store the numbers as well, to check later for wins. The numbers produced have no duplicate subset combinations of two integers (if applicable) or three integers, use all the integers for the particular Lotto game and maximize the coverage of the potential winning main number. The articles on this site have been selected from the many at the forum LottoPoster.
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