Let's say I have a set of independent Bernoulli trials each with a different probability:
$$ x_i \sim \operatorname{Bernoulli}(p_i) $$
The number of successes (sum of x) will be distributed according to the Poisson-Binomail distribution:
$$ S = \sum_{i = 0}^n x_i \sim \operatorname{PoissonBinomial}(p) $$
However, I am interested in the the probability that there is at least 1 success. In other words:
$$ S > 0 \sim \cdots\text{?} $$