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Suppose we're looking at a random sequence of digits from 0 to 9. We start off reading the digits until every digit from 0 to 9 has been seen at least once and we mark the count of digits read up to that point (run length). We then reset the run length and continue until the whole random sequence has been read. In the end, for every finite random sequence, there is a corresponding sequence of run lengths.

How would we be able to analytically arrive at average length of such these runs? What is the formal mathematical name given to such a run? What branch of mathematics concerns itself with this?

Monte