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Say $X_1,X_2,\ldots$ are independent identically distributed Bernoulli randomvariables with mean $p$. This paper builds a new estimate $\hat p$ of $p$ thathas the property that the relative error, $\hat p /p - 1$, of the estimate doesnot depend in any way on the value of $p$. This allows the construction ofexact confidence intervals for $p$ of any desired level without needing anysort of limit or approximation. In addition, $\hat p$ is unbiased. For$\epsilon$ and $\delta$ in $(0,1)$, to obtain an estimate where$\mathbb{P}(|\hat p/p - 1| > \epsilon) \leq \delta$, the new algorithm takes onaverage at most $2\epsilon^{-2} p^{-1}\ln(2\delta^{-1})(1 - (14/3)\epsilon)^{-1}$ samples. It is also shown that any such algorithm that applieswhenever $p \leq 1/2$ requires at least $0.2\epsilon^{-2}p^{-1}\ln((2-\delta)\delta^{-1})(1 + 2 \epsilon)$ samples. The same algorithmcan also be applied to estimate the mean of any random variable that falls in$[0,1]$.

An unbiased estimate for the mean of a {0,1} random variable with relative error distribution independent of the mean