Hi everyone. I have a problem on my mind as well as a solution for it, but I want to check if there are some other ways of solving it.

So, let's say I have a Bernoulli experiment like a coin toss with probability of success and I want to do at most independent runs of this experiment until I can get a successful run.

I am trying to calculate what is the probability that I get a successful run. The problem is that when I get a successful run I should stop.

Let's say that is the event of getting a success at the i'th run. Therefore :

And I am trying to get . The "problem" is that these events are not independent. If I get a success from the first run I stop.

So . And there is a general formula for this but it has an exponential number of terms in it and it seems a bit of overkill.

I try not to abuse the term "run" here. It is because when I initially seeing this term, I immediately think of something like you are asking for the probabilities of consecutive successes. Do you actually mean you want to calculate the probability of the first success of a series of independent and identical Bernoulli trials occur on or before the -th trial? If yes then it is simply related to the CDF of geometric distribution, which is just a geometric series.

Hmmm..you made me think. I think I am trying to calculate what is the average probability that I would get a success after k runs, given the fact that for example if I get a success from the first try I stop.