Need help with Power Analysis(?)

#1
I got a weak grasp of stats. I'm trying to do something at work that requires me to have a decent grasp over the error rate in a growing sample of phone calls.

I need to know the "success rate" behind a campaign of phone calls. The success rate is usually constant in any given campaign of calls, but can differ wildly between campaigns... and is almost always below 5%

I get samples as often as I want them, but the analysis equation I'm using for error rate calculation only starts working at my first success.

The problem is, with my percent chance so low (with a simulated run that was 1.4% successful, the first success happens really late (say, 400 calls in).

Running a 1.96*sqrt(pq/n) on p=1/410 gives me 0.004774654.
and 1/410 = 0.0024 ... so there's supposed to be a 95% chance that the real success rate is < 0.0072 ... which is horribly wrong. Even with 3 standard deviations, I get 0.0073 for the error rate, making the reasonable max be 0.0097 which is still off by a huge margin.

In this particular sample it became accurate a few steps later...

Now, there's a possible explanation for this inaccuracy. The success happens -after- the call at a variable and random time. I have hardcoded that time in my test, but in the real system it could be between 5 seconds and 5 minutes between cause and effect.

Is there a way to take that into account in statistics?