For our purposes, the appearance (or non-appearance) of sequences is of no consequence. All we need to know is that on the next roll we can't predict what that roll will be with any greater accuracy than probability.
This is confused. The predictability or non-predictability of rolls is inextricably bound up with the appearance of sequences. In order to determine whether a dice generator's next roll is predictable or not, one has no choice
but to examine the sequences it produces.
We have to remember that probability represents knowledge
: With no prior knowledge of what a number generator has come up with in the past, all options of prediction become the same. Suppose a dice generator named DG_1 only ever rolls the number 1 and one named DG_123456 rolls a more or less even distribution of all numbers between 1 and 6. Suppose further that we have no knowledge (yet) of what either DG does. Then DG_1 will initially be just as unpredictable as DG_123456. Obviously that doesn't make the two generators equal with regard to predictability in the long run. After DG_1 has produced 11111111111, we can hypothesize about its future rolls and successfully predict them. With DG_123456 this may be harder (unless it comes up with 123456123456123456 or some other easily discernible pattern). Whatever happens, the analysis of sequences is crucial to the question of predictability.
Even if the next roll is not predictable from the previous one, it may yet be predictable from longer sequences of previous rolls. I gave an example of a sequence with this property in http://www.fibsboard.com/general-chit-chat/the-old-dice-controversy/msg28741/#msg28741
11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53 54 55 56
61 62 63 64 65 66
<repeat from beginning>
(For simplicity, only one die is being rolled here, the corresponding sequence for two dice is a 36x36 matrix.)
In this sequence, the distribution of the next roll is perfectly even with respect to the previous one: Every number n is followed by every number m exactly twice. Hence it is impossible to predict the next roll from its predecessor with "greater accuracy than probability" (with greater accuracy than p = 1/6 for any number). But it obviously does not follow that the sequence is random, meaning that it is entirely
unpredictable. For example, knowing that I rolled 444 previously will allow me to predict the next roll to be a 5 with 100% accuracy. Knowing that I rolled 44 will give me a 50/50% predictability of a 4 or a 5 coming up next, and so on.
What the phrase "with greater accuracy than probability" means is exactly what the question of what constitutes randomness (= predictability) is about. It contains the entire complexity
of that question. The phrase "all we need to know ..." seems to promise that the issue of randomness can somehow be reduced to something very simple, namely, the predictability of the next roll. But the predictability of the next roll continues to be a very big deal, namely, to show patternlessness in the generated dice sequence. Only the absence of patterns can guarantee unpredictability (= randomness).