If I have a domain of 0 to 1000000 (but I doubt the domain matters, just that the ranges can overlap). I then chose 1% ranges uniformly, allowing for ranges to overlap completely or partially.
How do I find the expected percent of the domain that will be covered after K selections?
This seems like a simple selection with replacement but since the ranges can overlap I'm not sure how to find it. Is it as simple as repeatedly adding the useful part of the next selection? The first selection covers 1% if the domain. The second selection also covers 1% of the domain but since only 99% is uncovered, the expected coverage of the second selection is 0.99%, thus after 2 selections it is 1.99%...and so on with decreasing returns on each selection?
An example might be:
Domain: 0-999999 (in a dense sequence)
Percent selected per instance: 1%
Selection count: 50
Each range is chosen uniformly.
And I want to know what the expected % of the whole domain was covered (or remains uncovered as it provides the same information).
So on the first choice maybe the range is 128774-138773. Now after one selection 1% of the values have been covered. The second selection is 250791-260790 and now 2% is selected (but it also could have been 131456-141455 which would have overlapped with the first range resulting in less than 2% currently being covered). And so on for the remaining 48 selections.