[extropy-chat] alt dot fair dice

[Adrian Tymes]

--- Spike <spike66 at comcast.net> wrote:
> > Can other shapes be made such that there is
> > equal probability of any face downward?  I can
> > think of one: a five sided pyramid shaped
> > solid (four triangular faces and one square
> > face).  If the pyramid is tall and skinny, it
> > is less likely to land on the square face.  If
> > it is short and flat, the square face is more
> > likely to end downward.  So (I think) the
> > intermediate value theorem demands that there
> > is an aspect ratio somewhere between short and
> > tall that would make the square face equally 
> > likely to land downward, even if the surface
> > area of the square face is different from
> > the triangular... spike
> 
> Wait, I am now realizing this whole problem is a
> lot more complicated than I first imagined.  With
> the above example, the fairness of the square-base
> pyramid would depend on having a level surface
> upon which one rolls the die.  Imagine a family
> of pyramid shaped dice with progressively more
> triangular faces.  As the number of faces increases,
> the probability of landing on any given triangular
> face goes down, so the probability of landing on
> the polygonal base must be adjusted downward, so
> the thing gets taller.  Right?  So if one imagines
> rolling a tall die on an inclined surface, it is
> easy to see that it might be exactly *impossible*
> to make it stand on its base, yet it could still
> land with equal probability on any triangular face.

And that is why they prefer regular solids: because
many real surfaces are slightly irregular.  In fact,
for some rolls, they don't even roll the dice per se,
but rather put them in a container with a hole and
roll the container until the dice come out.  I've also
seen a percentile die (00-99) which was one ten sided
die inside another, mostly transparent ten sided die;
the external one yielded one digit while the internal
one yielded the other.