Let's assume that you have three empty boxes. You are given an infinite number of balls, but only of x number of types, and asked to present all possible arrangements (ball of type x1 in box + ball off type x2 etc). Eg, if you only have one ball type, the first arrangement is singular, given you can only fill the boxes in one way:

Box A has ball type x1, box 2 also type x1, same for box3.

If, on the other hand, you were given two types of balls, the full set of possible arrangements would have been 8, because each box could have 2 types and each of those types would form a set with each distinct arrangement for the remaining boxes, ie 2X2X2= 8. Likewise, for type=3, you have 3^3, and for type=n you have n^3.

Now, my question:

Someone asks you to do all these arrangements, up to nx (with x being an arbitrarily large integer). Then asks you to choose one case for each arrangement (so for n=2, you choose just one of the 8 possible cases), and add all those single cases as probabilities. What in essence you would be adding up is the probability of each single case, but for a continuum of cases (from n=1 to n=x). In practice what is that probability about, if expressed as a single effect instead of a continuum?

It obviously is not the probability of getting all those specific single cases if the arrangement was run for the entire continuum, given that this has an upper bound at the smallest singular probability of the individual cases.

Just asking for a stupid game I am making. Or I'll go with that anyway