I found that the probability of an event occurring is an algebraic function of all the probabilities that I want to find;
For small $n$, it would be easy to solve for all $v$ as a system of equations. However, as $n$ grows large (in the order of hundreds to several thousand), analyzing the data in this manner becomes infeasible.
The majority of the variables may be zero (or very, very close) and are negligible. The function is not a polynomial so it might be hard to solve using basic linear algebra.
Is there a way to estimate what these variables could be?