Sorry in advance for the confusing title. I'm not quite sure on the proper way to describe this question.
Suppose I have $n$ events: $A, B, C, ...., N$ (I'm using individual letters instead of proper indexing to make it easier to see what I would like to do). For these events I know the following probabilities
$$P(A), P(A | B), P(A | C) , ... P(A | N)$$ $$P(B), P(B | A), P(B | C) , ... P(B | N)$$ $$:$$ $$P(N), P(N | A), P(N | B) , ... P(N | M)$$
What I'm interested in calculating is $$P(A|B \cap C \cap ... \cap N)$$
Is this possible? I feel as though this question is utterly trivial but I have not been able to figure out how to determine $P(A|B \cap C \cap ... \cap N)$ or show that it can't be calculated with the information I have. If it is possible, how would I do it for $n = 3$? I suspect that if I can do it for $n = 3$ a recursive function could be used to calculate for whatever $n$ I want. Ultimately I would like to be able to solve the above probability for around $n = 50$ and I would like to know whether the information I have is sufficient (sans computational complexity)