The Dvoretzky covering problem is to cover the circle with random intervals. We consider the covering of the circle with random open sets. We find a necessary and sufficient condition for the circle to be covered almost surely when each open set is composed of a finite number of intervals which are separated by a positive distance.
"Covering the circle with random open sets.." Real Anal. Exchange 29 (1) 341 - 354, 2003-2004.