## 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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