We study the complexity of randomized solution of initial value problems for systems of ordinary differential equations (ODE). The input data are assumed to be γ -smooth ( γ = r + ρ : the rth derivatives satisfy a ρ -Hölder condition). Recently, the following almost sharp estimate of the order of the nth minimal error was given by Kacewicz [Almost optimal solution of initial-value problems by randomized and quantum algorithms, J. Complexity 22 (2006) 676–690, see also 〈 http://arXiv.org/abs/quant-ph/0510045 〉 ]: c 1 n - γ - 1 / 2 ⩽ e n ran ⩽ c 2 ( ε ) n - γ - 1 / 2 + ε , with an arbitrary ε > 0 . We present a Taylor Monte Carlo method and show that it has error rate n - γ - 1 / 2 , this way establishing the exact order of the randomized nth minimal error.