Law of the third stats

[Teorulte]

I ran about 500 000 cycles of 37 spins and came up with the following:

(Number on left is number of unique numbers and next number is how many times a spin cycle had exactly that many unique numbers in the test)

37 spin cycle:

15   1
16   12
17   202
18   1417
19   6020
20   18309
21   41304
22   73951
23   98780
24   102814
25   80164
26   47276
27   20934
28   6706
29   1685
30   349
31   35
32   5

74 spin cycle:

25   46
26   304
27   1898
28   7536
29   22123
30   50675
31   88059
32   114050
33   107055
34   69801
35   29634
36   7735
37   1011

111 spin cycle:

29   31
30   181
31   1407
32   7570
33   29508
34   83849
35   151889
36   155781
37   69674

148 spin cycle:

32   15
33   828
34   8540
35   52625
36   181344
37   256499

Mean and median were approximately:

24 for 37 spin cycle
32 for 74 spin cycle
35 for 111 spin cycle
37 for 148 spin cycle

[esoito]

Interesting analysis, as is the almost-symmetry in the figures.

Was that live or RNG?

[sqzbox]

Good to see that your simulation was accurate.  Here's the maths of it.

Generally speaking, what people usually mean by "the law of the third" is this – at a certain point in a series of spins, roughly one third of the numbers won't have shown, while two thirds will have been dominant.  The rather nebulous point of this is that the implication exists that if we can identify which ones are active, at the expense of certain others, then we are playing a game with only 2/3's of the numbers active and so we are getting "better than odds" payouts.

You have to be real careful about what you mean when using the phrase "law of third".  It is really not a law at all – it is merely an effect of standard probability maths – a "rule of thumb" if you like.  It is better called "rule of the third" really and probably was once, but over time things get warped and the significant difference between a "rule of thumb" and a "law" has been overlooked.  Furthermore, it is only a "third" at one specific point in a series of events.  Before this point, lots more than a third of the numbers haven't shown, and after that point there are less and less, asymptotically approaching, but never reaching, zero. 

To work out the maths, and hence derive a result, we have to carefully decide – what is the question?  I suggest that the proper question to ask is this – derive a general formula to determine how many numbers have not shown after a specified number of  events - and at the same time bear in mind that the answer to this question will be probabilistic, not "absolute".  This would then allow us to ask the more specific question – at what point in a series of trials do we have roughly 1/3 of the possible outcomes still to show?

The following refers to a single zero wheel.  It can be extended to a double zero wheel quite easily as shown in the examples at the bottom.

As previously discussed, the chance that a number won't come up is called the inverse (or negative) probability, and is 36 / 37.  To extend this for a variable number of spins is therefore going to be (36 / 37) ^N (also previously mentioned) where N represents the number of spins.  The next stage is to extend this for all numbers.  This is simply a matter of multiplying the above result by the number of numbers, i.e. 37.

So now we have a general formula for determining the statistical expectation of how many numbers will NOT appear in N spins –

[(nbr of possible outcomes - 1) / nbr of possible outcomes] ^ (nbr of spins) * nbr of possible outcomes.

Or : ((X-1)/X) ^N) * X    where   X = nbr of possible outcomes, and
               N = nbr of spins

So let's now ask the question – after how many spins will we have 12 numbers not shown? (12 is roughly one third of 37 for a single zero wheel.  I'd use 13 for a double zero wheel).

Or     12 = (36 / 37) ^N * 37

We could use logs to solve it, but it is simpler to just plug in a few numbers to N and work it out by trial and error – after all, we don't really want anything other than a whole number for N since you can't have 36.245 spins!

By experimenting you will find that N = 41 gives the closest result to a nice round number of 12.  To be precise, (36 / 37) ^41 * 37 = 12.03.  So, for a single zero wheel, after 41 spins we should have, on average, 12 numbers not shown yet and therefore 25 which have been busy.

So for a double zero wheel (X = 38) and nbr of spins (N) = 40 we get 13.07.  Work it out for yourself – it's quite easy really.

In terms of playing on a single zero wheel it is probably better to find a number closer to 37 spins – just because we kind of consider 37 spins to be a "cycle".  By experimentation with the numbers in the formula it is easy to show that the best result, i.e. one that gives closest to a whole number, is 38 spins.  This produces 13 numbers that have not shown (13.06 actually) and so it is expected that by spin 38, on average, we should have 24 numbers shown.

Other useful results (for a single zero wheel) are:
•   13.13 shows at spin 16
•   14.40 shows at spin 18
•   17.29 shows at spin 23
•   17.83 shows at spin 24
•   23.94 shows at spin 38

See - that wasn't so hard was it?

[Teorulte]

Interesting analysis, as is the almost-symmetry in the figures.

Was that live or RNG?

Thanks for the comment.  I used RNG numbers!

[XXVV]

Thanks Sqzbox for such a compact and practical summary. Thanks Teorulte for your initiative and quest.

This can be a useful first step in designing an effective and efficient bet. Will be working on this over coming weeks.

[XXVV]

@ Albalaha

Why do you adopt such a negative attitude toward this excellent research, and close down any suggestion that further development of ideas can follow?  For example I am aware of a colleague's work involving 37 spin cycles that  has outstanding practical value and proof of performance. This is long cycle work.

In short cycle work there can be opportunities with regard to remaining targets, or alternatively with awakening sleepers.

No need to close the door Al.

[sqzbox]

As I clearly stated Al, bear in mind that the answer to this question will be probabilistic, not "absolute". And I was merely presenting the mathematical truth.

People are endlessly creative and given this truth who knows what they may come up with? You have made an assumption that the only possible strategy that derives from this relates to, in your own words, "which one will hit in which order" and so you have failed to acknowledge the endless other possibilities. This is short-sighted in the extreme.

Never underestimate the power of human creativity and I would suggest to you that you should not insult the intelligence of the good folk who congregate here. Some are beginners looking for advice it is true, and some are extremely experienced and knowledgeable and, in my opinion, it would be unwise, not to mention insulting, to treat them in so condescending a manner.

A good strategy is not about guessing the next number to come out - this is fruitless as the maths of probability shows.  But it is not unreasonable to work on constructing a more sophisticated game strategy which is based around playing the statistical necessities that must be obeyed over a series. Roulette is a closed game in the sense that there are only 37 possibilities and all 37 exist each and every spin, and so over a series, since only these possibilities exist, if a distortion happens in a certain way then there is a consequential distortion elsewhere.

I'm not going to go into detail here as I do not want to limit or steer in any particular direction, people's thought processes, but consider for example if after 24 spins you have seen only 12 "shows", what do you think is likely to happen in the next 14 spins (making up a cycle of 38)?

Here's another example - what do think is most likely going to happen in the next 14 spins if, after the aforementioned 24 spins, you have 20 unique "shows" and these are imbalanced by 8 blacks and 12 reds?

There are endless possibilities.  Let people do what they must, which is to "create". Such is our nature and I can assure you that your attempt to shut them down by expounding your own particular version of the "truth" is fruitless - thank goodness!

[Bayes]

Is it a research? Even the dumbest members on forums know "law of third". You can not use it for any gain, whatsoever. It is as useless info as Martingale. This reminds me of an analysis written by someone in rf.cc that in 37 spins, there will be atleast two numbers repeating from one dozen. None could tell, what to do with this revelation.
and I am not trying to close the door but open eyes. I would love to be proved wrong.

You could say the same for any probability or statistic, including those which you supposedly use in your own holy grail.  And a Martingale is not "info" but a progression. If you're going to design a progression, it should ideally be based on probabilities and statistics.

I have to echo XXVV - why so negative?

A raw statistic is just the beginning, that's all. As sqzbox points out, you have to use your creativilty in order to apply it, and yes, there are endless possibilities.

By the way, the "law of the third" also applies to other bets, so on average:

In 18 spins, 12 splits will have hit.
in 12 spins 8 streets will have hit.
In 6 spins, 4 double streets will have hit.

[warrior]

As I clearly stated Al, bear in mind that the answer to this question will be probabilistic, not "absolute". And I was merely presenting the mathematical truth.

People are endlessly creative and given this truth who knows what they may come up with? You have made an assumption that the only possible strategy that derives from this relates to, in your own words, "which one will hit in which order" and so you have failed to acknowledge the endless other possibilities. This is short-sighted in the extreme.

Never underestimate the power of human creativity and I would suggest to you that you should not insult the intelligence of the good folk who congregate here. Some are beginners looking for advice it is true, and some are extremely experienced and knowledgeable and, in my opinion, it would be unwise, not to mention insulting, to treat them in so condescending a manner.

A good strategy is not about guessing the next number to come out - this is fruitless as the maths of probability shows.  But it is not unreasonable to work on constructing a more sophisticated game strategy which is based around playing the statistical necessities that must be obeyed over a series. Roulette is a closed game in the sense that there are only 37 possibilities and all 37 exist each and every spin, and so over a series, since only these possibilities exist, if a distortion happens in a certain way then there is a consequential distortion elsewhere.

I'm not going to go into detail here as I do not want to limit or steer in any particular direction, people's thought processes, but consider for example if after 24 spins you have seen only 12 "shows", what do you think is likely to happen in the next 14 spins (making up a cycle of 38)?

Here's another example - what do think is most likely going to happen in the next 14 spins if, after the aforementioned 24 spins, you have 20 unique "shows" and these are imbalanced by 8 blacks and 12 reds?

There are endless possibilities.  Let people do what they must, which is to "create". Such is our nature and I can assure you that your attempt to shut them down by expounding your own particular version of the "truth" is fruitless - thank goodness!

This is what I tried to do on The other forum I gave them something and let them create. I got called a scammer by Iggiv and Steve the owner of the forum ,and all because I liked what one person from years ago had to say about roulette ,and used some saying which lead me to finding something that works.in the game . They ban me  for getting heated up but when you look at who started the heat that would be the moderator  and I get crucified . So Al let people do what the they want with the  info that's givien to them on FORUMS. There is no harm done .as far as I m concerned I will not share anything on forums again. Being Creative is all we have .

[Bayes]

Teo,

You do know that thread starters can remove other member's posts, right? 

[Teorulte]

Teo,

You do know that thread starters can remove other member's posts, right? 

LOL.  I do know that now!  Thanks!   

[Turner]

Teo

You have saved me doing what I have been meaning to do for ages.

Great stats. Just what I wanted.

don't worry about Albalaha.

Its all about posters like Bayes and Sqeez....and many others.

[Nickmsi]

Hello . . .

Here's a little tracker that can help you visualize what happens in 37 spins.  It tracks:

1.  # of Unique numbers

2.  # of Sleepers

3. # of Repeaters

4. Composition of RB, OE and HL

5. Breaks down the RE, RO, BE, BO in all 3 Dozens

Just press Function Key F9 for another set of 37 spins and see if the first repeat usually occurs around the 8th spin.

See how often you get 12 unique numbers, 24 uniques, etc.

As Sqzbox suggest, see if you can use the breakdown of information to help  strategize or help make a more efficient bet.

Cheers

Nick

[Teorulte]

Teo

You have saved me doing what I have been meaning to do for ages.

Great stats. Just what I wanted.

don't worry about Albalaha.

Its all about posters like Bayes and Sqeez....and many others.

Thanks for the kind and helpful words.

[Teorulte]

Good work Nickmsi.