Probability of at least one success in a long string of connected events

Question

I have N events (i from 1 to N), each with an estimated probability of success, p(i).

If all my events were independent I'd be able to calculate the probability of at least one success as (1 - product of (1 - p(i))). But some events are not independent, such as {3,4,5}, {4,5,6} and {5,6,7}.

Within each of these groups, I know all conditional probabilities, such as p(3|4) =70%, p(3|5) = 30%, etc.

Now to the big question: Does it exist a formula to calculate the overall probability of at least one success? (A "closed" formula would be nice. But efficiency in calculation is more important!)

Answer 1

Think I found the solution: We'll use the chain rule of bayesian networks to calculate the probability of every event being unsuccessful. The answer is 1 minus that number.

Is there a better answer out there?