There's no 'start' of a cycle, they are circular. I.e. the cycle 1->2->3->1 is the same as 3->1->2->3). I.e. picking a locker at random is equivalent to picking

There's no 'start' of a cycle, they are circular. I.e. the cycle 1->2->3->1 is the same as 3->1->2->3). I.e. picking a locker at random is equivalent to picking a chain of length N with probability N/100. In fact, the prisoners aren't given any information about the cycles and can't communicate, so they can't choose a cycle "at random". Not sure if your question can be formulated as follows (which requires giving more information to the prisoners): Suppose that we color every group of lockers that form a cycle in a different color, and every prisoner then picks a random color and then picks a random locker among the lockers of that color. What is the chance of winning?