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Have you seen Knight-Knave puzzles based on probability rather then facts?

For example a puzzle with condition like this: Knight tells the answer, probability to which according to his knowledge is large that 50%. Knave - the opposite. So to the question "Will a dice show 1 or 2?" knight answers "No" and Knave "Yes".

I understand that the same effect can be achieved with usual characters and questions like "Is probability to get 1 or 2 bigger than 50%?" and I would like to see a puzzle, which can not be solved without such a probability questions.

Can you figure out a non-trivial puzzle like this?

klm123
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    This is still based on facts. Just facts about probability. As such, it would have the standard solutions. – frodoskywalker Oct 20 '14 at 14:51
  • @frodoskywalker, This is still based on facts, correct. But I don't understand, which solutions to which puzzles are you talking about? And my question stays the same. – klm123 Oct 20 '14 at 15:20
  • OK, your edit makes it clear what you're after. I meant that since these knights/knaves are functionally identical to regular ones, the set of puzzles and solutions are identical. I now realise you want a puzzle where such probabilistic questions are required to solve it. Maybe something with 1 knave, 1 knight and 1 that could be a knave or a knight? – frodoskywalker Oct 20 '14 at 15:49
  • So, you want a puzzle where a probability-based question is needed to solve one of these problems... (like maybe it's needed to beat a question limit?) – d'alar'cop Oct 20 '14 at 19:27
  • @d'alar'cop, yes. – klm123 Oct 20 '14 at 19:33
  • Asking questions about probability doesn't sound interesting; what would seem more interesting (and I while don't recall seeing any knight/knave puzzles of this form, I have seen hat puzzles) would be to have a situation where a person confronted with some number of knights, knaves, jokers (who answer "yes" or "no" with equal probability), and devils (who will deliberately give whatever answer will be least helpful), can maximize the probability of successfully determining something. – supercat Nov 19 '14 at 23:47
  • @supercat But we already have loads of those. This is a very tricky (and personally, interesting) variation that probably needs some kind of special ingenuity. Actually, it reminds me of the one about "I'm thinking of a number between 1 and 3"- http://puzzling.stackexchange.com/questions/313/you-have-one-question-to-tell-whether-the-number-im-thinking-of-is-1-2-or-3 – d'alar'cop Nov 20 '14 at 09:26
  • @d'alar'cop: I don't recall any puzzles of the form: "Ten people, consisting of 2 knights, 2 knaves, 2 jokers, and 4 devils, stand before two doorways. If the arrangement of the people is presumed random, and one of the doors holds a prize and the other certain death (also presumed random), how many yes/no questions must one ask to have a 99.9% chance of identifying the prize? Assume jokers determine how a knight would answer, and then randomly (50% probability) decide to whether to answer the same or the opposite (so one can't identify jokers by asking unanswerable questions). – supercat Nov 20 '14 at 16:33
  • @supercat yep, but we have techniques for deducing who the jokers are in such arrangements. something like http://puzzling.stackexchange.com/questions/2588/faulty-computers – d'alar'cop Nov 20 '14 at 16:41
  • @d'alar'cop: In that puzzle, a majority of computers are working, which makes a 100% solution possible. I would think adding probabilistic aspects could make things interesting provided that the puzzle is small enough not to yield an unresolvable combinatorial explosion. Among other things, I would think that an optimal strategy to a well-formulated puzzle would often require making tentative inferences before things are proven, and being prepared to revisit them if they prove false. – supercat Nov 20 '14 at 16:49
  • @supercat indeed, why not construct a solution to this question? I'd be genuinely interested to see it. I don't see atm how I'd prove that there isn't a nice situation which ties what you say to the question and it seems interesting.. OP doesn't even seem to have answered their own question after a month or so either – d'alar'cop Nov 20 '14 at 17:05

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The best-known type of probability-based Knight-Knave puzzle is one where there are not only Knights and Knaves but also Jokers, who lie or tell the truth at random. A fairly standard example is your own Knight, Knave and Joker puzzle. More abnormal variants include COTO's Automatically a Knight, Knave, and Joker, my own Lies, damned lies, and statistics, and last but not least, Mike Earnest's brilliant Past, Present and Future.

For a puzzle where the questions you ask the Knights/Knaves/Jokers have to be probabilistic in nature, two excellent examples are Emrakul's You have one question to tell whether the number I'm thinking of is 1, 2, or 3 and Joe Z.'s extension Differentiate between the numbers from 1 to 5 with one single yes/no question.

Does this answer your question? (Even if it doesn't, it makes a nice collection of some fantastic Knight-Knave puzzles!)

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Rand al'Thor
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Not that I know of... But it'd be easy to come up with that kind of thing.

"You arrive at an island one night where you find 2 types of people, intelligent, and people who often think randomly. You go up to one and ask them if you're thinking of a number between 1 and 3. How can you tell who's who after they tell you their guess?"

warspyking
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  • I do not see how it is related to my question. Explain this explicitly, please. – klm123 Oct 20 '14 at 18:56
  • Maybe I misunderstood your question, what is incorrect about my puzzle? – warspyking Oct 20 '14 at 19:06
  • Incorrect? I told you - I do not see how it is related to my question. You have not explained it enough. – klm123 Oct 20 '14 at 19:08
  • http://puzzling.stackexchange.com/questions/313/you-have-one-question-to-tell-whether-the-number-im-thinking-of-is-1-2-or-3 - yeah this might the right track – d'alar'cop Oct 20 '14 at 19:42