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Consider a sequence of independent trials with success probability $p$. The formula for eventual success, i.e., there will be at least one success eventually, is $$ q = 1 - \prod_{n=1}^{\infty}{1-p(1-p)^{n-1}} $$

Is there a closed form expression for the above formula?

Eracnet
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1 Answers1

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Let $p\ne 0$. The probability of no success in $n$ trials is $(1-p)^n$. This has limit $0$ as $n\to\infty$. So the probability of at least one success "eventually" is $1$.

André Nicolas
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  • Of course, thanks!. The mistake I made in coming up with the above formula is assuming that the following events 1) "probability of no success in exactly $k$ trials" and 2) "probability of no success in exactly $k+1$ trials" , are independent. But, they are not. – Eracnet Jun 09 '15 at 18:25
  • You are welcome. It is the monkeys and Shakespeare thing: A monkey typing at random will (with probability $1$) eventually write Hamlet. – André Nicolas Jun 09 '15 at 19:04