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Random Intervals

There are \(n\) points on a line. These points are paired up at random to form \(n/2\) intervals.

  • Prove that the probability that among these intervals there is one which intersects all the others is \(2/3\).

  • Prove that the probability that among these intervals there are at least \(k\) intervals which intersects all the others is \(\frac{2^k}{2k+1 \choose k}\). Note that this is independent of \(n\).

Source: from book "Mathematical Puzzles: A Connoisseur's Collection"

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