We present an axially symmetric formula to calculate the probability of finding gravitational arcs in galaxy clusters, being induced by their massive dark matter haloes, as a function of clusters redshifts and virial masses. The formula includes the ellipticity of the clusters dark matter potential by using a pseudo-elliptical approximation. The probabilities are calculated and compared for two dark matter halo profiles, the Navarro, Frenk and White (NFW) and the non-singular- isothermal-sphere (NSIS). We demonstrate the power of our formulation through a Kolmogorov-Smirnov (KS) test on the strong lensing statistics of an X-ray bright sample of low-redshift Abell clusters. This KS test allows us to establish limits on the values of the concentration parameter for the NFW profile (c(Delta)) and the core radius for the NSIS profile (r(c)), which are related to the lowest cluster redshift (z(cut)) where strong arcs can be observed. For NFW dark matter profiles, we infer cluster haloes with concentrations that are consistent to those predicted by Lambda CDM simulations. As for NSIS dark matter profiles, we find only upper limits for the clusters core radii and thus do not rule out a purely SIS model. For alternative mass profiles, our formulation provides constraints through zcut on the parameters that control the concentration of mass in the inner region of the clusters haloes. We find that zcut is expected to lie in the 0.0-0.2 redshift, highlighting the need to include very low-z clusters in samples to study the clusters mass profiles.