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In the 70s, Goldberg, and independently Seymour, conjectured that for anymultigraph $G$, 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 astronger form) is true for random multigraphs. Let $M(n,m)$ be the probabilityspace consisting of all loopless multigraphs with $n$ vertices and $m$ edges,in which $m$ pairs from $[n]$ are chosen independently at random withrepetitions. Our result states that, for a given $m:=m(n)$, $M\sim M(n,m)$typically satisfies $\chi'(G)=\max\{\Delta(G),\lceil\rho(G)\rceil\}$. Inparticular, 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 $m\geq (1+\varepsilon)n^3\log n$.

Goldberg's Conjecture is true for random multigraphs