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.

**Events = **__log(1 - 0.9999999926913)__ = __-7.387216144__ = 26
log(1 - 0.486486486486) -0.289448123

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**