  This paper provides a quantitative version of de Finetti law of large
numbers. Given an infinite sequence $\{X_n\}_{n \geq 1}$ of exchangeable
Bernoulli variables, it is well-known that $\frac{1}{n} \sum_{i = 1}^n X_i
\stackrel{a.s.}{\longrightarrow} Y$, for a suitable random variable $Y$ taking
values in $[0,1]$. Here, we consider the rate of convergence in law of
$\frac{1}{n} \sum_{i = 1}^n X_i$ towards $Y$, with respect to the Kolmogorov
distance. After showing that any rate of the type of $1/n^{\alpha}$ can be
obtained for any $\alpha \in (0,1]$, we find a sufficient condition on the
probability distribution of $Y$ for the achievement of the optimal rate of
convergence, that is $1/n$. Our main result improve on existing literature: in
particular, with respect to \cite{MPS}, we study a stronger metric while, with
respect to \cite{Mna}, we weaken the regularity hypothesis on the probability
distribution of $Y$.
