# SPUG: Stick Riddle

Sanford Morton smorton at pobox.com
Wed Jan 1 20:55:22 CST 2003

```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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