We study the size of the largest biconnected components in sparse Erdős-Rényi graphs with finite connectivity and Barabási-Albert graphs with non-integer mean degree. Using a statistical-mechanics inspired Monte Carlo approach we obtain numerically the distributions for different sets of parameters over almost their whole support, especially down to the rare-event tails with probabilities far less than $10^{-100}$. This enables us to observe a qualitative difference in the behavior of the size of the largest biconnected component and the largest $2$-core in the region of very small components, which is unreachable using simple sampling methods. Also, we observe a convergence to a rate function even for small sizes, which is a hint that the large deviation principle holds for these distributions.