In this talk we describe a probabilistic methodology to derive the precise asymptotic for the probability of observing a maximal component containing more than n^{2/3} vertices in the (near-) critical Erdös-Rényi random graph. Our approach is mostly based on ballot-type estimates for one-dimensional, integer-valued random walks, and improves upon the martingale-based method introduced by Nachmias and Peres in 2009. We also briefly discuss how our method has been adapted to study the same type of problem for (near-) critical percolation on a random d-regular graph, as well as possible future developments.