In this paper we study bond percolation on a one-dimensional chain with power-law bond probability $C/ r^{1+sigma}$, where $r$ is the distance length between distinct sites. We introduce and test an order $N$ Monte Carlo algorithm and we determine as a function of $sigma$ the critical value $C_{c}$ at which percolation occurs. The critical exponents in the range $0<sigma<1$ are reported and compared with mean-field and $varepsilon$-expansion results. Our analysis is in agreement, up to a numerical precision $approx 10^{-3}$, with the mean field result for the anomalous dimension $eta=2-sigma$, showing that there is no correction to $eta$ due to correlation effects.
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