If each prisoner has their own ordering, it's exceedingly likely that one of those orderings will have a 51-cycle. This strategy only works with one common dic

If each prisoner has their own ordering, it's exceedingly likely that one of those orderings will have a 51-cycle. This strategy only works with one common dictionary. You have the same individual odds, but you've linked each person: each member of a cycle succeeds or fails together. If it wasn't decided beforehand, you could just make up the ordering as you go. This gives exactly the same odds as random selection.