THE MADNESS OF ION SALIU AND HIS SPURIOUS
CLAIM TO THE SO CALLED FUNDAMENTAL FORMULA OF GAMBLING
by Colin Fairbrother

In 1997 Ion Saliu published an html page (FFG) giving a table calculated from
using a simple formula that relied on using the log rule
that log_{b}(m^{n}) = n * log_{b}(m) and purported to show the number of events required
in succession for various degrees of certainty up to 99.9% which he thought defined adequately the upper limit.
He has variously lauded this simple equation as a great achievement despite the "calculated" data given being
variously under-estimated, wrong, mis-applied or distorted. Because the equation is just a transformation from the standard probability
equation for multiple events in any introductory book or paper on probability in reality it would barely warrant a
pat on the head and a wry smile from a 3rd year high school teacher.

Probability is defined in its mathematical sense for a given scenario as the ratio of the events considered favourable
to that of all possibilities and ranges from 0 (impossible) through 0.1, 0.2 etc to 0.5 (50/50 or equally likely to
occur or not occur) and then on with 0.6, 0.7 etc to 1 which is certainty.

Consider a coin toss with the possibilities listed below for 3 throws: -
1 Throw H or T
2 Throws HT or TH or HH or TT
3 Throws HHH or HHT or HTH or HTT or THH or THT or TTH or TTT

With P as the probability, 1 as the number of occurrences of HHH which we will
regard as favourable and 8 as the possible occurrences
for 3 throws we have: -

P (Probability) = Events Favourable or 1 or 1 or 0.125.
Possible Events 8 2^{3}

We see that number of events, which is 3 in the example above, becomes the exponent n in the introductory and fundamental
probability equation for multiple events that's been around for hundreds of years: - Degree of Certainty or Probability of Success = (Probability of Event)^{n}
or for the complement Degree of Certainty or Probability of Non-Success = (1 - Probability of Event)^{n}

As an expression "Degree of Certainty" is recorded in English as being used by Geoffrey Chaucer in the 14th century. You will
probably find the term Probability of Success used more often than Degree of Certainty which was first used in a mathematical paper
by Jacob Bernoulli in his seminal work "The Art of Conjecturing" or in Latin "Ars Conjectandi" published post humously in 1713.
The English translation gives,"Probability is a degree of certainty and differs from certainty as a part from a whole."

The formula or equation as given by Ion Saliu is just a transformation from the standard format and you only need a rudimentary
knowledge of arithmetic to realise this: -

N = log(1 - Degree of Certainty)
log(1 - Probability of Event)

If we have the probability of an event and the nominated probability for success
we can solve to get an approximation of the number of events as a whole number.

The equation can be written: - P_Success = (P_Event)^{Events}

To get the exponent Events we can log both sides and still preserve equality: -

log(P_Success) = log((P_Event^{)Events})

Using one of the three basic log rules we know that the right side of the equation can be substituted with: -

log(P_Event^{Events}) = Events x log(P_Event)

Therefore: - log(P_Success) = Events x log(P_Event)

Events = log(P_Success)
log(P_Event)

If more interested in the complement for the two probability figures then we subtract from 1.

The simple equation now becomes: -

Events = log(1 - P_Success)
log(1 - P_Event)

To apply the equation consider the occurrence of 26 blacks consecutively on a roulette table at
Monte Carlo casino August 18, 1913. This particular occurrence I have known about since a boy and have cited it often in my writings on
the web dating from 2004 referring to it as the Monte Carlo Factor ie as a rule of
thumb if the odds are 1 in 2 as for a coin face or pretty well a color in
Roulette then multiply the 2 by 13 to give a possible and confirmed in real life run on a color of 26. For Pick 3 straights multiply 1000
by 13 to give a possible absence of 13,000 or more. The figures are useful in giving an indication of how long a run can go for and
should dispel
any reliance on something being "due".

Probability 26 consecutive blacks = 18/37 x 18/37 x 18/37 x ... 18/37 = (18/37)^{26} = (0.486486486)^{26}
= 0.0000000073087.
The complement to this is 0.9999999926913.
The inverse of this figure gives 1 in 136,823,184.
It is a simple application of the multiplication principle found in the first chapter of any book on probability and possibly touched on
in primary school.

Some would ask at this point if we are interested in how many events why not
produce a list for the range we are interested in which can be simply done using
the basic formula without transformation in any spreadsheet like Excel? Looking
at the table below you see that for Number Events = 26 - the probability
of success is 0.0000007 ie extremely unlikely - but it has happened. The
exponent is a whole number so all the calculations have been done without
using any logs.

Apart from this method one can also solve or transform for the exponent as in:
DC = PE^{n }by n = log_{PE}DC
where PE becomes the base for the log.
DC is Degree of Certainty or Probability of Success
PE is the Probability for the Event
and n is the number of Events

Apparently Ion Salui knew nothing about this alternative solving for the exponent as it is not
mentioned in any of his maniacal writings up until the date this article was posted ie May 2011.

Ion Saliu wrote regarding Lotto Pick 3,"Self,how many drawings do I have to play so that there is a
99.9% degree of certainty my combination of 1/1,000 probability will come out?".
For Pick 3 this gives 6904 draws and for both a series of same side coin tosses or same color roulette spins it is 10. The 1913 Monte Carlo
experience of 26 blacks in succession, which anyone writing on gambling should have been aware of, shows there is only a 0.0000007%
probabability of 26 spins with the same color run in roulette and using the same probability percentage 27 same side coin tosses
are possible and using the complement probability of 99.9999993% 18,725 draws in Pick 3 are possible before a specific straight occurrs.

The chances are that you will never be part of a long run in roulette and anyone that placed any
reliance on the upper limit figures given by Ion Salui would have wasted a lot of money just like the multi-millions lost in 1913 at
Monte Carlo when the patrons were in a frenzy after 15 blacks in succession with less than a 1% chance as they thought of it continuing -
but continue it did for another 11 spins.

Longshot figures are also hopelessly irrelevant as participation in a game requires something other than making a bet on something that takes years
before occurring.

Generally, with Ion Saliu we are dealing with at the least muddled and distorted
meanderings. When referring to the table below it helps to realise the following
two self evident facts:

The probability of a particular straight being drawn in Pick 3 increases as the number of draws increases.

The probability of getting the same color in roulette or the same side in tossing a coin decreases as the tosses increase.

The table is easily done in Excel and applies the fundamental probability
equation without using logs. In the case of Pick 3 with high exponent values we
calculate the complement and subtract from 1 ie 1-(1-0.001)^{n}.

The figures are to be taken with a grain of salt to show the approximate upper limits for absence and are not to be relied on as far
as any betting scheme is concerned with limits. As an example consider a series of 12 coin tosses. From our starting toss the likelihood
of 12 heads or 12 tails occurring is very unlikely ie HHHHHHHHHHHH or TTTTTTTTTTTT. For the 12th toss if 11 Heads or 11 Tails have occurred the
probability of 12 Heads or 12 Tails is exactly the same. In other words knowing what has occurred previously gives you no advantage in
anticipating what will occur next as the chances for what we regard as success are perfectly counter balanced by the complement or what
we regard as failure for this binomial example.

Compare the accurate figures given with the red coloured wrong figures given by Ion Salui and what a difference. For three tosses he
gives 90% when in fact it is only 87.5% ie 7 in 8 and for 9 tosses he gives 99.9 when it is 99.8 and the 0.1 in the calculation is
meaningful. Whether he intentionally fudged the figures to deceive or to promote his crackpot Pick 3, con-artist scheme where he claims to
reduce the House Edge from 50% to 2.3% or just through plain incompetence I will leave to others to conclude. In my considered opinion it
is a combination of all three.

Probability of Success

Probability Success Exponent Calculation

Probability Single Event

1 - Probability of Success

Events

% Probability

% Probability Ion Saliu

Either Color Roulette 1 Spin

(18/37)^{1} = 0.48648649

0.486486

0.513514

1

48.648649

Same Color Roulette 2 Spins

(18/37)^{2} = 0.236666910

0.486486

0.763331

2

23.6666910

Same Color Roulette 3 Spins

(18/37)^{3} = 0.115136319

0.486486

0.884864

3

11.5136319

Same Color Roulette 4 Spins

(18/37)^{4} = 0.056012264

0.486486

0.943988

4

5.6012264

Same Color Roulette 5 Spins

(18/37)^{5} = 0.027249209

0.486486

0.972751

5

2.7249209

Same Color Roulette 6 Spins

(18/37)^{6} = 0.013256372

0.486486

0.986744

6

1.3256372

Same Color Roulette 7 Spins

(18/37)^{7} = 0.006449046

0.486486

0.993550

7

0.6449046

Same Color Roulette 8 Spins

(18/37)^{8} = 0.003137374

0.486486

0.996863

8

0.3137374

Same Color Roulette 9 Spins

(18/37)^{9} = 0.001526289

0.486486

0.998474

9

0.1526289

Same Color Roulette 10 Spins

(18/37)^{10} = 0.000742519

0.486486

0.999257

10

0.0742519

Same Color Roulette 15 Spins

(18/37)^{15} = 0.000020233

0.486486

0.993551

15

0.0020233

Same Color Roulette 26 Spins

(18/37)^{26} = 0.000000007

0.486486

0.996863

26

0.0000007

Either Coin Face 1 Toss

(1/2)^{1} = 0.500000000000

0.5

0.5

1

50.0

50

Same Coin Face 2 Tosses

(1/2)^{2} = 0.250000000000

0.5

0.75

2

25.0

75

Same Coin Face 3 Tosses

(1/2)^{3} = 0.125000000000

0.5

0.875

3

12.5

90

Same Coin Face 4 Tosses

(1/2)^{4} = 0.062500000000

0.5

0.9375

4

6.25

95

Same Coin Face 5 Tosses

(1/2)^{5} = 0.031250000000

0.5

0.96875

5

3.125

Same Coin Face 6 Tosses

(1/2)^{6} = 0.015625000000

0.5

0.984375

6

1.5625

99

Same Coin Face 7 Tosses

(1/2)^{7} = 0.007812500000

0.5

0.9921875

7

0.78125

Same Coin Face 8 Tosses

(1/2)^{8} = 0.003906250000

0.5

0.99609375

8

0.390625

Same Coin Face 9 Tosses

(1/2)^{9} = 0.001953125000

0.5

0.998046875

9

0.1953125

99.9

Same Coin Face 10 Tosses

(1/2)^{10} = 0.000976562500

0.5

0.999023438

10

0.09765625

Same Coin Face 11 Tosses

(1/2)^{11} = 0.000488281250

0.5

0.999511719

11

0.048828125

Same Coin Face 12 Tosses

(1/2)^{12} = 0.000244140625

0.5

0.999755859

12

0.0244140625

Same Coin Face 27 Tosses

(1/2)^{27} = 0.000000007450

0.5

0.999999992

27

0.0000007451

Specific Pick 3 Straight 1 Draw

(1/1000)^{1} = 0.001000000

0.001

0.9990000000

1

0.10000

Specific Pick 3 Straight 106 Draws

(1/1000)^{106} = 0.100623052

0.001

0.899376948

106

10.06230

Specific Pick 3 Straight 288 Draws

(1/1000)^{288} = 0.250346438

0.001

0.749653562

288

25.03464

Specific Pick 3 Straight 693 Draws

(1/1000)^{693} = 0.500099765

0.001

0.499900235

693

50.00997

Specific Pick 3 Straight 916 Draws

(1/1000)^{916} = 0.600067024

0.001

0.399932976

916

60.00670

Specific Pick 3 Straight 1050 Draws

(1/1000)^{1050} = 0.650246042

0.001

0.349753958

1050

65.02460

Specific Pick 3 Straight 1204 Draws

(1/1000)^{1204} = 0.700188819

0.001

0.299811180

1204

70.01888

Specific Pick 3 Straight 1386 Draws

(1/1000)^{1386} = 0.750099755

0.001

0.249900245

1386

75.00997

Specific Pick 3 Straight 1609 Draws

(1/1000)^{1609} = 0.800073411

0.001

0.199926589

1609

80.00734

Specific Pick 3 Straight 1897 Draws

(1/1000)^{1897} = 0.850124321

0.001

0.149875679

1897

85.01243

Specific Pick 3 Straight 2302 Draws

(1/1000)^{2302} = 0.900056651

0.001

0.099943349

2302

90.00566

Specific Pick 3 Straight 2995 Draws

(1/1000)^{2995} = 0.950038296

0.001

0.049961703

2995

95.00382

Specific Pick 3 Straight 4603 Draws

(1/1000)^{4603} = 0.990001328

0.001

0.009998672

4603

99.00013

Specific Pick 3 Straight 6905 Draws

(1/1000)^{6905} = 0.999000699

0.001

0.000999301

6905

99.90007

Specific Pick 3 Straight 13000 Draws

(1/1000)^{13000}= 0.999997754

0.001

0.000002245

13000

99.99997

Specific Pick 3 Straight 18725 Draws

(1/1000)^{18725}=0.999999993

0.001

0.000000007

18725

99.999999

Any Win 6/49 Lotto 245 Draws

0.018637^{370} = 0.99905176

0.018637

0.000948235

245

99.905176%

Any Win 6/45 + 2 Bonus Lotto 388 Draws(Aus Saturday)

0.011802^{590} = 0.999092461

0.011802

0.000907538

388

99.909246%

Colin Fairbrother

This article first published on the web 5/5/2011 and has been viewed 15361 times. The author may be contacted at fairbros@bigpond.com. 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.