[calebclark]

Although the supplied answer guarantees that no prisoner will die from alligators, it does not guarantee their freedom.

The answer includes a major (and I believe flawed) assumption, which is that the visits will be uniformly distributed. However, the puzzle states that "I may choose the same guy three times in a row".

The Counter could visit the switch room 44 times without flipping the switch if such visits were the first 44 chosen. After everyone visits (which would be 1,012 visits since "given enough time, everyone will eventually visit the switch room as many times as everyone else"), the Counter will be no closer to knowing the truth.

Of course, it is true that the greater the number of visits the greater the probability that the Counter's final visit will fall after everyone has visited, but there is no guarantee.

Regardless of how many visits you assume for each prisoner, there will always remain a probability that the Counter's visits will be clustered early in the visitations. In such a scenario, the Counter's role becomes useless and all prisoners will die in prison.