By the "fairness rule" from the warden ("given enough time, everyone will eventually visit the switch room as many times as everyone else"), eventually every normal prisoner has to visit the switches.
I thought the fairness rule was basically just dice roll-equivalence. Which makes this whole "precisely 44 times" thing basically just a guess. It's a probabilities game but they pretend it has an exact solution.
It's not a probabilities issue. What's being counted is state changes---after 44, the counter knows that the condition has been satisfied. Fairness just guarantees that those state changes will happen.
