David, you are absolutely correct. And carrying your calcs a little further, I figure after 1000 trials (=250 hours = ~ 10.5 days) they could claim freedom with about a 96% chance of success. Double that time and get a chance of execution of only 3x10^-6. I like those odds! After a month, the answer is 100% chance of freedom, at least as far as my calculator's resolution is concerned.
The original puzzle has an answer that is deterministic, and uses the light and switch. I will present it in another post. However, the original statement of the puzzle has the jailer calling up a person every day. As an actuary, I had to change it to every 15 minutes, since the daily test would have too many prisoners dying of natural causes before the solution is completed, and that would "break" the solution. (And as an actuary, I am embarrassed that I did not catch the probabilistic answer until you presented it!)
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- Late Friday Puzzler -- 100 Prisoners