We extend the lower bounds on the depth of algebraic decision trees to the case of randomized algebraic decision trees (with two-sided error) for languages being finite unions of hyperplanes and the intersections of halfspaces. As an application, among other things, we derive, for the first time, $\Omega(n^2)$ randomized lower bound for the {\em knapsack problem} (which was previously only known for deterministic algebraic decision trees).