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A very old man has a million dollars, which he's going to pass down to you! He has 3 associates who have been ordered to help you. The 3 associates know the location where it is, and they tell you it is in either a) a box, b) a safe, or c) a hole.

One of the associates always tells the truth, one always lies, the third tells truth or lies at random.

The associates are very greedy and will try to take the money for themselves, but they will give you a chance at getting it. They let you pick the box, safe, or hole. Then one of then (you don't know which) comes up to tell you whether you have chosen correctly or wrongly, gives you one question, and then shows you the selection.

They then give you a chance to switch to the other selection. What question should you ask. Then, should you switch?

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    $\begingroup$ When knights and knaves meet the Monty hall... $\endgroup$ – warspyking Oct 21 '14 at 18:08
  • $\begingroup$ Can you clarify "after doing so, he shows selection."? $\endgroup$ – oerkelens Oct 21 '14 at 18:11
  • $\begingroup$ He shows one of the other's, for example if you pick the box he may show in the safe or the hole, depending on his personality and greediness. Knights & Knaves & Monty Hall $\endgroup$ – warspyking Oct 21 '14 at 18:14
  • $\begingroup$ You are only allowed to switch to the undisclosed location? Because then any greedy associate will show you the treasure if possible. If you can switch to any location after one has been opened, the basis is standard MH, so the question is how to raise you chances over 2/3 for robbing the associates. $\endgroup$ – oerkelens Oct 21 '14 at 18:19
  • $\begingroup$ @warspyking To clarify, when they tell you if you have the correct or incorrect selection they are abiding by their honesty constraints, correct? So the liar would tell you you have the correct selection when in fact you don't, right? $\endgroup$ – Rob Watts Oct 21 '14 at 18:40
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For simplicity, I'm going to refer to the associates as knight, knave, and joker. Also, I'm assuming that they won't reveal the location of the money. If they did, then obviously you would switch to that once they reveal it.

Assuming that there is an equal chance of any of the three associates coming up and talking to you, we have a two in three chance of talking to knight or knave. So you can ask the associate "If I were to ask you if you always told the truth, what would you say?" Both knight and knave would reply "Yes" if asked if they tell the truth, so knight will honestly report "Yes", while knave will lie about it and say "No". We don't know what joker would say. In either case, we respond as if we were not talking to joker and switch as appropriate.

Let's look at how this plays out:

Suppose we chose the wrong box (2/3 chance):

  • Knight tells us we chose wrong (1/3), then replies "Yes" to the question:
    • We believe knight and switch to the last choice, which is the correct choice
  • Knave tells us we chose right (1/3), then replies "No" to the question:
    • We don't trust knave and switch to the last choice, which is the correct choice
  • Joker (1/3)
    • Tells us we chose right (1/2?)
      • then replies "Yes" to the question (1/2?):
        • We trust joker and don't switch, and lose :(
      • then replies "No" to the question (1/2?):
        • We don't trust joker and switch, and win
    • Tells us we chose wrong (1/2?)
      • then replies "Yes" to the question (1/2?):
        • We trust joker and switch, and win
      • then replies "No" to the question (1/2?):
        • We don't trust joker and don't switch, and lose

Suppose we choose the right box (1/3 chance):

  • Knight tells us we chose right (1/3), then replies "Yes" to the question:
    • We believe knight and don't switch - win
  • Knave tells us we chose wrong (1/3), then replies "No" to the question:
    • We don't trust knave and don't switch - win
  • Joker (1/3)
    • Tells us we chose right (1/2?)
      • then replies "Yes" to the question (1/2?):
        • We trust joker and don't switch, and win
      • then replies "No" to the question (1/2?):
        • We don't trust joker and switch, and lose
    • Tells us we chose wrong (1/2?)
      • then replies "Yes" to the question (1/2?):
        • We trust joker and switch, and lose
      • then replies "No" to the question (1/2?):
        • We don't trust joker and don't switch, and win

So we see that if we talk to knight or knave, we will always win. If we talk to joker, we win about half of the time. If joker's response are truly random, then the chance will be exactly half. If the joker's response is not random, then we can't know what our chance will be (or come up with a strategy that will definitely be better) unless we know more about the joker's strategy.

So if joker chooses randomly your chance of getting the million dollars can be at least $\frac{2}{3}+\frac{1}{2}\cdot \frac{1}{3}=\frac{5}{6}$.

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  • $\begingroup$ Very well explained! Excellent job! $\endgroup$ – warspyking Oct 21 '14 at 19:17
  • $\begingroup$ This is what happens when a (random) Knight/Knave/Joker hosts a random chance game XD $\endgroup$ – warspyking Oct 21 '14 at 19:18
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    $\begingroup$ I think your chance to be correct can be lower than 5/6. The absolute lower limit is 2/3. (This happens if the Joker always selects answers with the intention to mislead us, although describing this as random is a bit of a stretch) - A random model of the joker that gives less than 5/6 could be; 50% tells the truth, otherwise answers in opposite mode of what he had before this question (50/50 for first answer). $\endgroup$ – Taemyr Nov 28 '14 at 9:52
  • $\begingroup$ @Taemyr the 5/6 is the lower bound on the best you can do if the joker chooses randomly. I made that a little bit more clear in my answer. $\endgroup$ – Rob Watts Nov 28 '14 at 18:47

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