Abstract
We determine the probability distribution for relative projective objects in
an exceptional sequence of type $A_n$. We show that these events (the $k$-th
object being relatively projective) are independent from each other and from
the length of the sequence. This gives a probabilistic interpretation of the
product formula for the numbers of exceptional sequences and clusters or
partial clusters since the latter numbers are proportional to the number of
signed exceptional sequences. We also prove the analogous statement for rooted
labeled forests.