Part of my solution to Power 2 The Cards involved the likelihood of finding at least one match among the 88,889 groups of 5 digits in each of the three 444,445-digit integral powers of 2.

Of course one match is a gross understatement.

1. How many of the 100,000 possible groups of 5 digits would one expect to find represented 0 times, 1 time, 2 times, etc. within such a large number, assuming that it's digits can be considered stochastically random, (the integer equivalent of a normal number)?

2. As a consequence, what fraction of the 88,889 5-digit groupings into which a 444,445-digit integer can be divided would you expect would have no matching 5-digit group elsewhere in the number (keeping strictly to the 5-digit bounds, non-overlapping)?

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