# Confidence Interval for Cumulative Success

I would like to calculate a confidence interval for the probability of a process succeeding after a given amount of time by running the process $$n$$ independent times and recording the time at which each run succeeded. Essentially, I would like to add confidence intervals to a plot like this: I think this can be modelled as the probability of $$n$$ coins having ever come up heads after $$t=1,2,\ldots$$ flips. If I had $$n$$ independent runs of each different number of flips, I would expect to model the confidence interval at each number of flips as a binomial proportion confidence interval, e.g.,

1. flip all coins ($$n_1 = n$$) once each (i.e., $$t=1$$) and count each coin that came up heads ($$s_1$$),
2. flip all coins ($$n_2 = n$$) twice each (i.e., $$t=2$$) and count each coin that came up heads on either flip ($$s_2$$),
3. etc.

and then calculate the Clopper–Pearson interval for each $$s_t/n$$.

But in this scenario the subsequent flips aren't independent and it is equivalent to,

1. flipping all coins ($$n_1 = n$$) once (i.e., $$t=1$$) and counting the number of heads ($$s_1$$),
2. flipping only the coins that weren't heads ($$n_2 = n-s_1$$) again (i.e., $$t=2$$) and counting the cumulative number of heads ($$s_1 + s_2$$),
3. etc.

In this cumulative scenario, I could still calculate the Clopper-Pearson interval at each $$t$$ from $$\frac{\sum_{i=1}^t s_i}{n}$$, but is that correct? It seems like it ignores the time relationship...

I'm sure this is a common problem and I am just thinking about it wrong or not using the right terms/language, so any direction is appreciated! If it is relevant, I am actually interested in the problem when $$t$$ is a real valued number (e.g., the amount of time a process has been running).