  We consider a model of randomness for self-similar Cantor sets of finite and
positive $1$-Hausdorff measure. We find the sharp rate of decay of the
probability that a Buffon needle lands $\delta$-close to a Cantor set of this
particular randomness. Two quite different models of randomness for Cantor
sets, by Peres and Solomyak, and by Shiwen Zhang, appear to have the same order
of decay for the Buffon needle probability: $\frac{c}{\log\frac{1}{\delta}}$.
In this note, we prove the same rate of decay for a third model of randomness,
which asserts a vague feeling that any "reasonable" random Cantor set of
positive and finite length will have Favard length of order
$\frac{c}{\log\frac{1}{\delta}}$ for its $\delta$-neighbourhood. The estimate
from below was obtained long ago by Mattila.
