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How Many Random Intervals Intersect All Others?
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DOI：10.1007/s12045-026-1997-0.png)
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Abstract
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Pick two points independently and uniformly in (0, 1), and consider them as endpoints of a random interval. In this manner, pick n random intervals independently. How many of these n random intervals intersect all (n - 1) other intervals? The answer is in {0,1,2,& mldr;,n}, but with what associated probabilities?From [1], I learned the stunning result that, for any n >= 2, no interval intersects all others with probability 1/3, free of the number n of intervals! Its original solutions-algebraic and combinatorial-are found in [2]. In contrast, our proof here, by mathematical induction, is new, broader, and simpler.
Keywords:
Continuous random variable
weak law of large numbers
random pairing
pigeonhole principle
recursive relation
mathematical induction
Journal
R
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0.4
Papers:
101
Citations:
848