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July 7th, 2019, 08:45 AM   #1
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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.
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July 7th, 2019, 09:52 AM   #2
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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 !
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July 7th, 2019, 11:11 AM   #3
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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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