Twins in a finite word are formed by a pair of identical subwords placed at disjoint sets of positions. We investigate the maximum length of twins in a random word over a $k$-letter alphabet. The obtained lower bounds for small values of $k$ significantly improve the best estimates known in the deterministic case. Bukh and Zhou in 2016 showed that every ternary word of length $n$ contains twins of length at least $0.34n$. Our main result states that in a random ternary word of length $n$, with high probability, one can find twins of length at least $0.41n$. In the general case of alphabets of size $k\geq 3$ we obtain analogous lower bounds of the form $\frac{1.64}{k+1}n$ which are better than the known deterministic bounds for $k\leq 354$. In addition, we present similar results for multiple twins in random words.