We show that the existence of a perfect set of random reals over a modelM ofZFC does not imply the existence of a dominating real overM, thus answering a well-known open question (see [BJ 1] and [JS 2]). We also prove that \mathbbB ×\mathbbB\mathbb{B} \times \mathbb{B} (the product of two copies of the random algebra) neither adds a dominating real nor adds a perfect set of random reals (this answers a question that A. Miller asked during the logic year at MSRI).