# Is there an analytical way of determining the probability for a specific outcome given a joint p.m.f.?

I'm looking for a way to determine the probability for a specific outcome based on (what I think should probably be) a joint probability mass function. I'll try to put into words my specific case:

I have 10 probability mass functions (X1, X2, ..., X10) of 10 discrete and independent random variables. Each describes the probabilities for rolling either one, two or three s-sides dice, where s can vary.

Is there a way to determine

$$\Pr\left( \sum X_k \ge n \right), \text{ with } \sum \min(X_k) \le n \le \sum \max(X_k) \text{ and } k \in \{1,\ldots,10\}$$

that doesn't involve computationally constructing what basically is a giant 10-dimensional lookup table? From what I have found so far, most solutions for the case of two random variables involve constructing a distribution matrix, and hardly anything I found dealt with more than two variables.

I assume 'flattening' marginal distributions down to a single distribution (that spans across the sums of all possible combinations of the variables' values) is not allowed, since that would somehow violate the variables' independence?

To illustrate what I mean by 'flattening' I made the following example (two random variables, one represents rolling two 6-sided dice, the other one represents rolling two 4-sided dice). 'Flattening' would be the step from figure 1 to figure 2.

Should I be mistaken and a procedure like that is indeed allowed, am I then right to assume that the process can be applied iteratively for all of the 10 distributions? That would be an acceptable solution also (ensuring any given lookup table is two-dimensional at most).