# 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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• What, ex­actly, would the guard would say in differ­ent situ­a­tions? Us­ing the stan­dard, ut­terly un­re­al­is­tic, in­ter­pre­ta­tion of prob­a­bil­ity prob­lems like this, the guard is sup­posed to say this:

``VM: If you are on Vul­can, you are in the Moun­tainVD: I have noth­ing to tell you.EM: If you are on Vul­can, you are in the Moun­tainED: If you are on Vul­can, you are in the Moun­tain``

in which case the prob­a­bil­ity is 13. But I have a hard time be­liev­ing that the guard is will­ing to talk to you here, but wouldn’t be will­ing to talk if you were in the Vul­can Desert.

Since the guard re­fused to talk at first, but then told you some­thing later, it seems pretty clear that they’re try­ing to help you out. The most ob­vi­ous way for them to com­mu­ni­cate to you where you are is like this:

``VM: If you are on Vul­can, you are in the Moun­tainVD: If you are on Vul­can, you are in the De­sertEM: If you are on Earth, you are in the Moun­tainED: If you are on Earth, you are in the De­sert``

But there are other pos­si­bil­ities. Per­haps there’s a policy of ex­e­cut­ing guards that re­veal in­for­ma­tion about where you are, so the guard wants plau­si­ble de­ni­a­bil­ity by ly­ing to you:

``VM: If you are on Vul­can, you are in the De­sertVD: If you are on Vul­can, you are in the Moun­tainEM: If you are on Earth, you are in the De­sertED: If you are on Earth, you are in the Mountain``

It seems that you’ve ruled that out in the prob­lem state­ment, though.

Al­to­gether, as Da­cyn says, “it de­pends on what you know about the psy­chol­ogy of the guard.”

• I posted a fol­low up/​solu­tion here.

• If you as­sume that the guard’s prob­a­bil­ity of mak­ing this state­ment (and only this state­ment) is the same in all cir­cum­stances where the state­ment is true, then the an­swer is 13. Other­wise, it de­pends on what you know about the psy­chol­ogy of the guard.

• EDIT: This was wrong.

The an­swer varies with the gen­er­at­ing al­gorithm of the state­ment the guard makes.

In this ex­am­ple, he told you that you were not in one of the places you’re not in (the Vul­can Desert). If he always does this, then the prob­a­bil­ity is 14; if you had been in the Vul­can Desert, he would have told you that you were not in one of the other three.

If he always tells you whether or not you’re in the Vul­can Desert, then once you hear him say you’re not your prob­a­bil­ity of be­ing in the Vul­can Moun­tain is 13.

• The whole thing is ba­si­cally the Monty Hall prob­lem.

• How is it re­lated to Monty Hall?

• There are N op­tions, the host re­moves one op­tion with the (pre­sumed) con­straint that they only can re­move one of n-1 op­tions which aren’t true. What’s the re­main­ing prob­a­bil­ity? Monty has a fur­ther con­straint that he can’t re­move the one you picked, but it’s very a similar un­der­ly­ing calcu­la­tion.

Also iden­ti­cal in that com­mon for­mu­la­tions are am­bigu­ous about the rules for guard/​monty to elimi­nate an op­tion, lead­ing to irk­some and un­nec­es­sary con­tro­versy.

• In this ex­am­ple, he told you that you were not in one of the places you’re not in (the Vul­can Desert). If he always does this, then the prob­a­bil­ity is 14; if you had been in the Vul­can Desert, he would have told you that you were not in one of the other three.

That can’t be right—if the prob­a­bil­ity of be­ing in the Vul­can Moun­tain is 14 and the prob­a­bil­ity of be­ing in the Vul­can Desert (per the guard) is 0, then the prob­a­bil­ity of be­ing on Earth would have to be 34.

• P(vul­can moun­tain | you’re not in vul­can desert) = 13

P(vul­can moun­tain | guard says “you’re not in vul­can desert”) = P(guard says “you’re not in vul­can desert” | vul­can moun­tain) * P(vul­can moun­tain) /​ P(guard says “you’re not in vul­can desert”) = ((1/​3) * (1/​4)) /​ ((3/​4) * (1/​3)) = 13

Woops, you’re right; nev­er­mind! There are al­gorithms that do give differ­ent re­sults, such as justin­pom­brio men­tions above.

• It seems like the an­swer is ob­vi­ously 50%, but you say the rid­dle is more com­pli­cated than it looks. What am I miss­ing?

• “If you are on Vul­can, you are in the Moun­tain” is log­i­cally equiv­a­lent to, “You are not in the Vul­can Desert” and “If you are in a Desert, you are on Earth”

• Vul­can has con­sid­er­ably higher grav­ity than Earth. Use your shoelace and any handy weight to con­struct a pen­du­lum. If its os­cilla­tion ap­pears no­tice­ably fast to your Earth-ac­cus­tomed eyes, you’re on Vul­can.

Also, don’t talk to that guard any more. He is un­helpful.