A stick is randomly broken into 3 pieces . Determine the probability that the sum of the lengths of any 2 pieces is greater than the length of the third piece.

  • $\begingroup$ As I understand the problem, the probability is 1, since you can choose the largest two pieces to begin with. $\endgroup$ – Marc Apr 12 '16 at 11:49
  • $\begingroup$ the question probably meant any random 2 pieces chosen $\endgroup$ – user330785 Apr 12 '16 at 11:51
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    $\begingroup$ This is a well known problem...you can read about the usual formulation here or for some variants here $\endgroup$ – lulu Apr 12 '16 at 11:55

Geometric solution (non-strict): let $x$, $y$ and $z$ be lengths of pieces. Then in space with $xyz$ axes you have a triangle ($x+y+z=L, x\ge 0, y\ge 0, z \ge 0$). Total area of triangle corresponds to probability 1. Subsets of fitting pieces also correspond to a triangle, with area $1/4$ of the original one.


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