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]:
0xy1
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 twiceor 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 xy
> 1/2 occurs 25% of the time.
y
1 
.......... 
....... 
..... 
... 
1/2. .
 ...
 ......
 .........
 ............
 x
1/2 1
                                    
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