How to calculate a sense of error for estimated number of events with known probabilities? I have a long series of entities:
x1, x2, x3, ... xn
for each of which, there is a probability of an event occurring. The probability for each x may be different, but they are all independent, and they are all known:
p1, p2, p3, ... pn
I might provide a point-estimate for how many of said event occurred in that series by simply summing the probabilities p, but is it possible to also calculate a confidence interval or some information about the error in a close-form fashion?
The resistance to using some sampling method is that the series of x many be prohibitively long, and there may be many such series upon which the same estimates need be delivered.

EDIT - Refactoring of the question
For a toy example of the problem, imagine I have a large batch of n widgets, each of which has a known probability of failing within the year. Each widget may have a different known probability of failing, some high, some low (say they belong to different widget classes, those classes well-studied in the past). I'm tasked with estimating the number of widgets that will fail at the end of the year for that batch.
I believe I can provide a point estimate by simply summing the probabilities of each widget in the batch. Stakeholders also request a confidence interval, which may not be possible, but I should be able to provide a sense of error/uncertainty/variance around that point estimate.
My understanding is that I can treat my widgets failing/not-failing as a series of Bernoulli trials, and that likely the "number that fail" can be modelled as a Poisson Binomial Distribution, but from there I get stuck.

The total number of failures will follow a binomial distribution, assuming the probability of failure is constant, and you can just calculate that binomial probability. You can look up “Binomial distribution” for how to do this; Blitzstein and Hwang’s textbook Probability is a great intro for this.
If n is large, the binomial can be approximated as normal.
If the probability is not constant, the calculation becomes more involved. Instead the total number of failures will need to be calculated, either painfully/tediously by hand, or more likely by Monte Carlo methods.
Note that this calculation gives a tolerance interval, not a confidence interval. A confidence interval is for a parameter, e.g. the probabilities, not the observations.
