Lets say that you have 2 cups and in 1 of them there is an equal number of white and black say 100 white and 100 black. Lets say that in the other cup you add another 100 black to the 100 white and 100 black you put in there.

Now lets say the black is 0 and white is 1.

These are the transformations:

white from 50/50 back to the same cup

black from 50/50 go to 2/3 black 1/3 white

black from 2/3 black back to same cup

white from 2/3 black go to 50/50

Now what is the probability of each of these outcomes?

With 3 colors base 3

with 4 colors base 4 and so on we can express bases to any number as long as we have that many colors.

We would have for 3 these outcomes:

white from 33/33/33 back to same cup

black from 33/33/33 to 25/50/25

red from 33/33/33 to 25/25/50

black from 25/50/25 back to same cup

red from 25/50/25 to 25/25/50

white from 25/50/25 to 33/33/33

red from 25/25/50 back to same cup

white from 25/25/50 to 33/33/33

black from 25/25/50 to 25/50/25

Now for any number of colors we are going to have x^2 outcomes

How would we figure out the probability of each outcome in the binary black/white, ternary black/white/red, quarternary black/white/red/orange, quinary black/white/red/orange/yellow, base 6 black/white/red/orange/yellow/green, base 7 black/white/red/orange/yellow/green/blue, octal black/white/red/orange/yellow/green/blue/indigo, nonary black/white/red/orange/yellow/green/blue/indigo/violet, and base 10 black/white/red/orange/yellow/green/blue/indigo/violet/gray when we have the colors assigned to these digits:

black = 0

white = 1

red = 2

orange = 3

yellow = 4

green = 5

blue = 6

indigo = 7

violet = 8

and

gray = 9

and for any base we get x^2 possible outcomes?