SPUG: Stick Riddle

[Sanford Morton]

Choose two points (x,y) on the unit interval [0,1]:

   0-------x----------y----1

Typically, it would be assumed that these points are independent and
identically distributed with uniform probablity on [0,1]. (Imagine the
stick is placed in a darkened room which you enter blindfolded with a
machete, and whack around until you hear the stick break twice--or you
lose a body part.)

Now fold the interval at the chosen points.

   0.......x__________y....1
           |          |
           |          |
           |          |
           |

In this picture, it looks like the segments will form a nice triangle.

There are two cases in which a triangle could not be formed:

-- if both points are on the same half of the stick, then the longer
   end segment will exceed the sum of the other two segments. Since the
   points are i.i.d. unif[0,1] both points will be either above or
   below the midpoint 50% of the time. (Most easily visualized on a unit 
   square.)

-- If the points are on opposite halves of the stick but are too far
   apart (> 1/2) then the middle segment will be longer than the sum
   of the two end segments. On the unit square, you can see that |x-y|
   > 1/2 occurs 25% of the time.

       y
       
     1 |----------------------|
       |..........            |
       |.......               |
       |.....                 |
       |...                   |
    1/2|.                    .|
       |                   ...|
       |                ......|
       |             .........|
       |          ............|
       |----------------------|-  x
    		 1/2          1
    




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