The object of this program is to show clearly the definition of probability in terms of outcomes of the sums of the dots on the upturned faces of three dies.
To do this all possible outcomes are depicted in the form of a triangular array.
The horizontal scale of the array shows progressively the sum of the dots on the
upturned faces of the three dies which, for six sided dies goes from 3 through 18.
The vertical scale is the number of separate possible outcomes for that particular sum which, for the median dot sum (10 or 11) becomes 27.
The probability of any dot sum is the number of ways that that dot sum can occur divided by the number of all possible ways that dot sums can occur which, to 3 dies turns out to be 216.
Thus both sums 10 and 11 have a probability of 0.125 while dot sums 3 and 18 have probability of only 1/216.
The probability of dot sums less than or equal to a given sum is also provided at the bottom of the chart.
In this program the three dies are distinguishable by their difference in color. All macroscopic objects are distinguishable from each other. However, elementary
particles like electrons, protons, and neutrons are not distinguishable so the probability of their states is not the same as depicted here. If a proton had six
states like a die does, then for a three proton state sum of 4 we would have just {1,1,2} and not {{2,1,1},{1,2,1},{1,1,2}} as we have for two dies.
This causes changes in the shape of the "triangle" or their state sum probability distribution.