  In this article we propose a probabilistic framework in order to study the
fair division of a divisible good, e.g., a cake, between n players. Our
framework follows the same idea than the ''Full independence model'' used in
the study of fair division of indivisible goods. We show that, in this
framework, there exists an envy-free division algorithm satisfying the
following probability estimate:$$\mathbb{P}\big( C(\mu_1, \ldots,\mu_n) \geq
n^{7+b}\big) = \mathcal{O}\Big(n^{-\frac{b-1}{3}+1+o(1)}\Big),$$where
$\mu_1,\ldots, \mu_n$ correspond to the preferences of the $n$
players,$C(\mu_1, \ldots,\mu_n)$ is the number of queries used by the algorithm
and $b>4$. In particular, this gives$$\lim_{n \rightarrow +
\infty}\mathbb{P}\big( C(\mu_1, \ldots,\mu_n) \geq n^{12}\big) = 0.$$ It must
be noticed that nowadays few things are known about the complexity of envy-free
division algorithms. Indeed, Procaccia has given a lower bound in $\Omega(n^2)$
and Aziz and Mackenzie have given an upper bound in $n^{n^{n^{n^{n^{n}}}}}$. As
our estimate means that we have $C(\mu_1, \ldots, \mu_n)<n^{12}$ with a high
probability, this gives a new insight on the complexity of envy-free cake
cutting algorithms. Our result follows from a study of Webb's algorithm and a
theorem of Tao and Vu about the smallest singular value of a random matrix.
