# Boundaries on Probability of Independent Events

Given an integer n, and an event n that happens with $P(\frac{1}{n})$, is the probability that e will happen in n trials bounded by any constant?

For example, if I had an n-sided fair die and a target value t, can I say with certainty that regardless of the value of n, the odds of rolling a t in n rolls are no worse than $\frac{1}{x}$ for some constant x?

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$$1-\left(\frac{n-1}n\right)^n\gt\lim\limits_{k\to\infty}1-\left(\frac{k-1}k\right)^k=1-\frac1{\mathrm e}=0.632$$
@MichaelChernick No idea what it is you are talking about: this is a bound for every $n$, nothing needs to be added and an hour earlier is in fact 13 minutes later. (For your interest, this is the last time I reply to this kind of raving comment from you. From now on, please pick your fights with someone else.) –  Did Jul 9 '12 at 21:18
If P(e)=1/n on any trial and trials are independent then the probability that e occurs at least once in n trials is 1- probability that it does occur in n trials =1 - [(n-1)/n]$^n$. This converges to 1-exp(-1) = [exp(1)-1]/exp(1). The convergence is from above but for large enough n take the limit plus a small ε for the bound.