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 two 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 two dies which, for six sided dies goes from 2 through 12.
The vertical scale is the number of separate possible outcomes for that particular sum which, for the median dot sum (7) becomes 6.
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, for 2 dies turns out to be 36.
Thus dot sum 7 has a probability of 6/36 while dot sums 2 and 12 have probability of only 1/36.
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 two 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 as a die does, then for a two proton state sum of 3 we would have just {1,2} and not {{1,2},{2,1}} as we have for two dies.
This causes changes in the shape of the "triangle" which is their state sum probability distribution.