confidence interval for bernoulli(?)-like trials
Let's say we have a counter, set to 0 initially.
Let's say we conduct experiment where in each trial a counter may be increased by 1 with known probability p.
E.g. if p = 0.8 then after the 1st trial the counter will be 1 with probability 0.8 and 0 with probability 0.2. After the 2nd trial, the counter will contain 0 with probability 0.04 (0.2 * 0.2), 1 with probability 0.32 (2 * 0.2 * 0.8) and 2 with probability 0.64 (0.8 * 0.8).
Now, with given probability of success (== counter increment) p and number of trials n, does it make sense to construct a confidence interval around the expected final counter value? If yes, then how? If no, what would be another way to communicate confidence in this case?
This is actually a simplified description of a research experiment I'm involved in. I need to give prediction on the counter value after a large number of trials. Intuitively I think that the expected value of the counter would be p^n, but how do I state my level of confidence in it?
Suggestions and links are appreciated. Thanks!