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We can suppose that we have $n$ processes, each with the same parameters. Further, each successive process is allowed to start only after the previous process has completed.

Now, a process completes with probability $1-(1/2)^t$ at time $t$. In other words, the process always has a 50-50 chance of completing at time $t+1$, if it did not complete at time $t$.

I'm wondering two things. What function gives the probability that $n$ of these processes will complete at time $t$? Also, if we have the time $t$, what is the probability that $n$ processes have completed?

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This is how I think about problem. You have discrete time intervals. You have people arriving at probability 1/2 each time. You want to know the number of people who has arrived at t.

I do not understand how your two questions are different.

The amount of time taken for $n$ arrivals, is given by Negative Binomial distribution with parameters $n$ and $1/2$. so I gues the probability of $n$ arrivals before $t$ is probability of $NP(n,1/2)$ is smaller than $t$. For the probability, just look at negative binomial distribution on http://en.wikipedia.org/wiki/Negative_binomial_distribution

I am not sure what the thing is called, but this is the discrete analogue of a Poisson process, where the arrival rate is an exponential random variable.

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