The Defining-Order Paradox

[falkor2k15]

Seems some cyclists are still active here? Recently, I posted about the First Repeat Challenge and also the about the new Reverse Order constant being used on the normal Order constant re: Non-Random Cycles (first introduced by our resident guru Priyanka prior to the end of 2015).

Today I am going to talk about how the probability of an event doesn't necessarily dictate it's behaviour in terms of clustering and gaps.

Red vs. Black is 50/50, but the maximum in a row has been observed by some to reach as high as 17 in both real play and simulation. But what if I told you there are similar events to EC, but with different properties.
Same vs. Different, like Red vs. Black, is 2 sides of the same coin - but in terms of some SD events that I've created, "different" reaches a maximum of 26 in a row and "same" reaches a maximum of 36 in a row! How can that be!? Using such events all we would need to do is wait for a virtual win on "same" and then bet same, right?

In the world of Non-Random, probabilities and averages are no longer useful unless applied to sequences broken down into different stages. RO on Order has two such stages:
Typical sequence: 11111111111211111122111111111121111111111112111111111113133313
Stage 1: 1111111111121111112211111111112111111111111211111111111
Stage 2: 3133313

Clustering: 3s all come together; likewise 1s all come together (with the occasional 2s).
Gaps: there is a considerable "gap" between each cluster of 3s - sometimes 50 cycles - that is not representative of it's probability of occurring at any time in the full sequence.

You cannot simply apply statistics to the entire sequence at once otherwise the result will not accurately reflect what is happening, so we first need to break it down into different stages, separated by key-frames, and look at each part of the non-random sequence in turn. This brings me onto what I've described as the "Defining-Order Paradox" - sticking out like a sore thumb in the First Repeat challenge topic I posted, based on regular cycles...

[falkor2k15]

Cycle Length 2 for Dozens is "king" - it has more chance (44%) of occurring than CL1 (33%) or CL3 - but it's not strong enough to reach 2 in a row so is referred to as coming "alone". Defined by Same x 2 and Order 1 x 2, however, are each strong enough for a 2 cluster, but not for a 3 cluster it seems...

Cycle Length 2
1 in a row   7093   55.7%
2+ in a row   5636   44.3%
   12729   

1+2 in a row   10189
3+ in a row   2540

Order 1
1 in a row   4543   38%
2+ in a row   7453   62%
   11996   
      
1+2 in a row   7226   60%
3+ in a row   4770   40%
   11996   

Defined by Same
1 in a row   38%   
2+ in a row   62%   
      
1+2 in a row   2381   61%
3+ in a row   1525   39%
   3906   

So it would seem that, in all likelihood, Order 1 comes in pairs and so does Defined by Same - and they are in fact the same event - but that's not completely true, as we shall see...

[falkor2k15]

Referring to the First Repeat Challenge data for regular cycles:

2   2         CL1   o1   e1   d2

2   3   3      CL2   o2   e2   d3

3   1   1      CL2   o2   e2   d1
1   2   1      CL2   o1   e2   d1
1   2   1      CL2   o1   e2   d1
1   1         CL1   o1   e1   d1
1   1         CL1   o1   e1   d1
1   1         CL1   o1   e1   d1

1   3   2   3   CL3   o2   e3   d3

3   1   1      CL2   o2   e2   d1

1   2   2      CL2   o2   e2   d2
2   2         CL1   o1   e1   d2
2   2         CL1   o1   e1   d2
2   1   3   2   CL3   o1   e3   d2
2   3   2      CL2   o1   e2   d2
2   2         CL1   o1   e1   d2

2   2         CL1   o1   e1   d2

2   3   1   3   CL3   o2   e2   d3
...

See parts highlighted in bold above... D1 and D2 happen to both be coming in clusters of at least 2+ since defined by same has a 64% chance and less chance of coming "alone". But look at Order 1 (2 columns adjacent to defining): it's always one behind defining!

If Defining is 3 in a row then Order 1 will be 2 in a row; if defining is 7 in a row then order 1 will be 6 in a row...


So what on earth is happening here!? Does "Order 1" only come alone or is "Defined by same" more likely to come in triplets instead of pairs?

Let's just look at occurrences of Order 1 vs. Defined by Same - do we see a pattern? Can you break the non-random sequence down into separate stages?

[falkor2k15]

...not "alone", not 3 in a row, but 2 in a row standard for both Order 1 and Defined by Same, right? -- even using the data posted in the First Repeat topic

Order 1 (in a row)	Count
1	10		10	40%	1 in a row
2	3		15	60%	2+ in a row
3	7		25
4	2
5	2		13	52%	1+2 in a row
6	1		12	48%	3+ in a row
			25
Defining (in a row)	Count
1	13		13	34%	1 in a row
2	10		25	66%	2+ in a row
3	3		38
4	7
5	2		23	61%	1+2 in a row
6	2		15	39%	3+ in a row
7	1		38

So why is "order 1" always one behind "defined by same" - and why is "defined by same" always one ahead of "order 1"?