When dealing with problems, like aging, or playing games, like Tetris, it
is
tempting to use probability distribution over the intake materials or
falling
bricks. However, just having a probability distribution does not
guarantee,
that a specific type of falling brick will ever occur, it
just says that it's
highly
unlikely for it to not occur.

To solve for infinite lifespans, or infinitely long playing of Tetris, what's
needed, is a guarantees of the frequencies, or even patterns. Hence,
the
idea of a "Determinity Distribution" or "Deterministic Probability
Distribution", which sounds like an oxymoron, but could be defined
simply.

Suppose we have an algebra of events (e.g., like when defining a
random variable), and add
a
requirement, that the events have to repeat at no rarer or more often
than threshold
frequencies. That automatically implies that only specific periodic or
aperiodic patterns of event can qualify.

Surely, you may think a periodic pattern already has a "Determinity
Distribution",
because we have frequency guarantees. However, it is not a probability
distribution, because we already know the pattern with certainty.
However, given the
inequalities
(e.g., thresholds, or upper / lower bounds) for frequencies, it may be
possible to construct many patterns, and even aperiodic patterns (just
like aperiodic fractions, like irrational numbers with
proofs), that guarantee those frequencies, and at the same time are not
exactly predictable (!).

Thus, a determinity distribution would define an equivalence class for all
patterns, that the constraints of the probability distribution with
frequency
guarantees.

This also may be applicable to lottery games. For example, lottery
makers
may claim specific probabilities about their games, but not guarantee
them,
so
that the winning prize actually is never really won. However, giving legal
requirements for determinity distribution would change the games...
And yes, determinity distribution would imply, that the more
observations we have, the more certain other events are, but not
completely certain (until the last event before threshold). It's kind of an
interesting concept mathematically, that you can have such interplay of
indeterminism and determinism.

I was thinking a truncated gaussian. If you only allow n (or
if you're feeling more specific, a couple of) standard
deviations either side, you're effectively guaranteeing
every possibility in the range, given a large (or small) enough
ratio between options, and sample size. If you're talking big
permutational combinations, that number can get so big,
you'll never fetch every individual item, even if you start
at the beginning and aim to work your way right to the
end with a probability of 1 for each.

But I'm confused about the idea of guaranteeing a given
outcome - given enough time, space and chances,
Murphy's Law ought to apply - which is as close to a
guarantee as you're likely to get. With enough options on
the table, it may well take until the end of the universe
for a given number to come up. Certainly longer than any
individual's lifespan - so guarantees seem like they'd be
tricky to offer.

You could define some pseudo-random sequence, or just
pick a particular permutation that covered the entire
range given enough goes - guaranteeing hitting everything
seems like another way of saying you'll avoid double-picks
(or for a softer version, limiting them in some way) - and
this eventually means narrowing down the list of possibles
so much that the odds would be shortened considerably
towards the end of a run.