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 July 7th, 2019, 08:45 AM #1 Newbie   Joined: Jul 2019 From: kazahstan Posts: 2 Thanks: 0 Number of trials until success for successful participants Hello, I have the following problem: Given a Bernoulli trial with probability of success p and a number of trials x, what is the expected number of trials until success among those that succeed in x trials? e.g. if p = 0.02 and x = 60, (1-0.02)^50 = 0.36 attempts fail all 60 trials, and I would like to find the expected number of trials the successful ones take until success. I understand it is lower than 1/0.02 and will decrease with x but don't know how to proceed. I think it's simple and I'm missing something basic, anyway thanks ahead ! Last edited by mikimiki1; July 7th, 2019 at 08:51 AM.
 July 7th, 2019, 09:52 AM #2 Newbie   Joined: Jul 2019 From: kazahstan Posts: 2 Thanks: 0 Better phrased: I have the following problem: Given a Bernoulli trial with probability of success p and a number of trials x, what is the expected number of trials until success in a case success was achieved in x trials? e.g. if p = 0.02 and x = 60, I would like to find the expected number of trials taken in case of succes within 60 trials. I understand it is lower than 1/0.02 and will decrease with x but don't know how to proceed. I think it's simple and I'm missing something basic, anyway thanks ahead !
 July 7th, 2019, 11:11 AM #3 Senior Member     Joined: Sep 2015 From: USA Posts: 2,500 Thanks: 1372 Assuming we stop after success, and that we know success occurred within $n$ trials we have $P[k] = \dfrac{p(1-p)^{k-1}}{1-(1-p)^n}, ~1 \leq k \leq n$ $E[K]=\sum \limits_{k=1}^n k P[k] = \Large \frac{n p (1-p)^n+(1-p)^n-1}{p \left((1-p)^n-1\right)}$

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