  Let $S_h$ be the even pure spinors variety of a complex vector space $V$ of
even dimension $2h$ endowed with a non degenerate quadratic form $Q$ and let
$\sigma_k(S_h) $ be the $k$-secant variety of $S_h$. We decribe a probabilistic
algorithm which computes the complex dimension of $\sigma_k(S_h) $. Then, by
using an inductive argument, we get our main result: $\sigma_3(S_h) $ has the
expected dimension except when $h\in \{7,8\} $. Also we provide theoretical
arguments which prove that $S_7$ has a defective 3-secant variety and $S_8$ has
defective 3-secant and 4-secant varieties.
