[extropy-chat] what is probability?
gts
gts_2000 at yahoo.com
Sat Jan 13 22:36:04 UTC 2007
On Sat, 13 Jan 2007 13:25:03 -0500, spike <spike66 at comcast.net> wrote:
> Please I missed a few posts while changing diapers; what is the cube
> factory paradox?
Here is the cube paradox but in a form different from the way I described
it the first time [1]. In this version there is no cube factory and no
unknown distribution of cubes.
====
1. Suppose there is a cube hidden in a box. A label on the box says the
cube has a side length between 3 and 5 cm.
2. We don't know the actual side length, but we might assume that all
values are equally likely and simply pick the mid-value of 4 cm.
3. The information on the label allows us to calculate that the surface
area of the cube is between 54 and 150 cm^2. We don't know the actual
surface area, but we might assume that all values are equally likely and
simply pick the mid-value of 102 cm^2.
4. The information on the label allows us to calculate that the volume of
the cube is between 27 and 125 cm^3. We don't know the actual volume, but
we might assume that all values are equally likely and simply pick the
mid-value of 76 cm^3.
5. However, we have now reached the impossible conclusion that the cube
has a side length of 4 cm, a surface area of 102 cm^2, and a volume of 76
cm^3!
====
This is in essence just another form of the Bertrand paradox of selecting
a random chord.
The contradiction in this case arises from the fact that we invoked the
principle of indifference (the logical, or maybe not so logical, principle
that given no evidence to prefer one outcome over another, we should
assign them equal probabilities) in steps 2, 3 and 4.
-gts
1. The cube factory version is from the Stanford Encyclopedia of
Philosophy. This version is from Wikipedia.
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