Three random dots on an octagon's sides

Each of three dots is dropped at a uniformly random point on the boundary of a regular octagon, independently. Find the probability that no two dots land on the same side.

$21/32$. Each dot picks one of the 8 equal sides uniformly. The second must avoid the first dot's side ($7/8$) and the third must avoid both used sides ($6/8$), giving $\tfrac78\cdot\tfrac68=\tfrac{21}{32}$.