Suppose I have a range of integer numbers varied from 0 to L (L is a positive integer). I now randomly select two random integers from 0 to L to form a sub-range of integers. What is the relationship between the number of these sub-ranges and the probability for all these sub-ranges to cover the entire integer range?
Please note: It does not need to choose all L+1 integers to cover the entire range. For example, assuming L is 10. For the first time, the random machine chooses 2 and 5. so the range [2,5] is covered. For the second time, the machine chooses 3 and 10. Then the accumulative range is now updated to [2,10]. The third time, the machine chooses 0 and 4. The accumulative range will be updated to [0,10], which has completely covered all the 11 integers from 0 to 10 and attempts in this run is 3 times. So I am after the relationship between the probability of the entire range being covered and number of attempts.