  In the 70s, Goldberg, and independently Seymour, conjectured that for any
multigraph $G$, the chromatic index $\chi'(G)$ satisfies $\chi'(G)\leq \max
\{\Delta(G)+1, \lceil\rho(G)\rceil\}$, where $\rho(G)=\max \{\frac
{e(G[S])}{\lfloor
  |S|/2\rfloor} \mid S\subseteq V \}$. We show that their conjecture (in a
stronger form) is true for random multigraphs. Let $M(n,m)$ be the probability
space consisting of all loopless multigraphs with $n$ vertices and $m$ edges,
in which $m$ pairs from $[n]$ are chosen independently at random with
repetitions. Our result states that, for a given $m:=m(n)$, $M\sim M(n,m)$
typically satisfies $\chi'(G)=\max\{\Delta(G),\lceil\rho(G)\rceil\}$. In
particular, we show that if $n$ is even and $m:=m(n)$, then
$\chi'(M)=\Delta(M)$ for a typical $M\sim M(n,m)$. Furthermore, for a fixed
$\varepsilon>0$, if $n$ is odd, then a typical $M\sim M(n,m)$ has
$\chi'(M)=\Delta(M)$ for $m\leq (1-\varepsilon)n^3\log n$, and
$\chi'(M)=\lceil\rho(M)\rceil$ for $m\geq (1+\varepsilon)n^3\log n$.
