The Curious Prisoner Puzzle

Here’s an in­ter­est­ing rid­dle that is more com­pli­cated than it looks:

You wake up locked in­side a room with no win­dow. You know your cap­tors have four fa­cil­ities in the fol­low­ing lo­ca­tions: a Vul­can Moun­tain, a Vul­can Desert, an Earth Moun­tain, an Earth Desert. They flip a coin to de­cide which planet to send you to, then they flip a coin to de­cide which fa­cil­ity on that planet to use.

Want­ing to know where you are, you try to get some in­for­ma­tion out of the guard. He re­fuses at first, but even­tu­ally he offers the fol­low­ing: “If you are on Vul­can, you are in the Moun­tain”. What are the chances that you are in the Vul­can Moun­tain fa­cil­ity?

(You can as­sume that the guard is tel­ling the truth and not try­ing to in­ten­tion­ally ma­nipu­late the situ­a­tion)

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