The cochromatic number $ζ(G)$ of a graph $G$ is the minimum number of colours needed for a vertex colouring where every colour class is either an independent set or a clique. Let $χ(G)$ denote the usual chromatic number. Around 1991 Erdős and Gimbel asked: For the random graph $G \sim G_{n, 1/2}$, does $χ(G)-ζ(G) \rightarrow \infty$ whp? Erdős offered \$100 for a positive and \$1,000 for a negative answer. We give a positive answer to this question for roughly 95% of all values $n$.