Power analysis can exploit the instantaneous power consumptions of Elliptic curve cryptography ECC) devices and retrieve secret keys. Many countermeasures have been proposed to make ECC implementations secure. One of the approaches is the randomized algorithms proposed by Oswald et al., which combine two scalar point multiplication algorithms and use random variables to decide which algorithm to follow at different stages of the computation. In this paper, we describe a power analysis attack that can break randomized automata proposed by Oswald et al. effectively, even with a small number of power traces.