[extropy-chat] what is probability?

[gts]

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.