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 Smart prisoners always get a break #2 (Posted on 2007-03-07)
Note: Read this problem carefully, because it's completely different from the original.

As before, 100 prisoners are put into solitary cells, and there's a room with a light bulb. (No prisoner can see the light bulb from his or her own cell.) Every night, the warden picks a prisoner at random, and that prisoner goes to the living room. While there, the prisoner can toggle the bulb if he or she wishes. but this time, the prisoner needs to assert that he knows, which prisoner was in the living room before him. If the assertion is false, all 100 prisoners will be shot. However, if it is indeed true, all prisoners are set free. Thus, the assertion should only be made if the prisoner is 100% certain of its validity.

The prisoners are allowed to get together one night, to discuss a plan.

But, the prisoners know that after that night, when they will go back to their solitary cells. the warden will choose one prisoner secretly (and this time, not randomly) and will kill him.

What plan should they agree on, so that eventually, someone will make a correct assertion?

 See The Solution Submitted by Assaf Rating: 4.1818 (11 votes)

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 re: only 99% sure | Comment 2 of 11 |
(In reply to only 99% sure by Charlie)

To minimize the risk of falsely assuming that designated prisoner A was the one killed the first night, the prisoners could consult the following chart and judge an acceptable risk for all of them being shot. It's the probability that the designated prisoner, A, will not be the one killed the first night (0.99) times the probability he will go a specific number of days without having been in the room to be given the chance to turn on the light (0.99^d, where d is the number of days allowed), times the probability that after that number of day's the alternate will visit the room before him (0.5).

`days   prob of 100allowed   deaths 100 0.18118619 150 0.10961879 200 0.06632007 250 0.04012406 300 0.02427531 350 0.01468672 400 0.00888556 450 0.00537582 500 0.00325240 550 0.00196773 600 0.00119049 650 0.00072025 700 0.00043576 750 0.00026364 800 0.00015950 850 0.00009650 900 0.00005838 950 0.000035321000 0.000021371050 0.000012931100 0.000007821150 0.000004731200 0.000002861250 0.000001731300 0.000001051350 0.000000631400 0.000000381450 0.000000231500 0.000000141550 0.000000081600 0.000000051650 0.000000031700 0.000000021750 0.000000011800 0.000000011850 0.000000001900 0.000000001950 0.000000002000 0.00000000`

DEFDBL A-Z
p = .01: q = .99
FOR days = 100 TO 2000 STEP 50
pMistake = q * q ^ days / 2
PRINT USING "##### #.########"; days; pMistake
NEXT

 Posted by Charlie on 2007-03-07 11:20:07

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