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.,

*

*flip all coins ($n_1 = n$) once each (i.e., $t=1$) and count each coin that came up heads ($s_1$),

*flip all coins ($n_2 = n$) twice each (i.e., $t=2$) and count each coin that came up heads on either flip ($s_2$),

*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,

*

*flipping all coins ($n_1 = n$) once (i.e., $t=1$) and counting the number of heads ($s_1$),

*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 ($s_1 + s_2$),

*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).

As per the discussion in @EdM's answer, in the absence of censored data this can be viewed as the problem of estimating the underlying CDF. Nonparameteric confidence intervals of a CDF estimate can be calculated pointwise or simultaneously from the empirical distribution function (EDF). Pointwise bounds are calculated for each point of the EDF as an independent Binomial proportion confidence interval, e.g., the Clopper-Pearson intervals I proposed is a correct approach to calculating CIs for this problem.
In the presence of censored data, survival analysis is more appropriate.
