I would like to understand where the following reasoning is misleading or not applicable or maybe wrong:
Suppose that, as in the model proposed here, that the arrangements of the triples (0,1,2) on, say, 1000 positions are all equally probable.
Let’s consider the columns with the first 100 elements all 0.
of these columns {0,0, …, 0, e_101, e_102, …, e_1000}, there are 3 ^ 900.
So the probability of having 2 of this type in 2 columns is about 1/3 ^ 100.
The probability of having 2 or more in m columns is less than m / 3 ^ 100