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Thread: Question about a Bernoulli experiment.

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    Question about a Bernoulli experiment.




    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 p and I want to do at most k 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 A_i is the event of getting a success at the i'th run. Therefore : P(A_i) = (1 - p)^{i-1} * p

    And I am trying to get P(A_1 \cup A_2 \cup ... \cup A_k). The "problem" is that these events are not independent. If I get a success from the first run I stop.

    So P(A_1 \cup A_2) = P(A_1) + P(A_2) - P(A_1 \cap A_2). 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.

    Can anyone think of a more "elegant" solution?

    Cheers!

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    Re: Question about a Bernoulli experiment.

    See if I understand your problem correctly.

    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 k-th trial? If yes then it is simply related to the CDF of geometric distribution, which is just a geometric series.

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    Re: Question about a Bernoulli experiment.


    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.

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