On a certain island, everyone is either a truth-teller or a liar.
logic/smullyan/two.fifths.truthful New
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Problem
On a certain island, everyone is either a truth-teller or a liar. If a person is chosen at random, the probability of his/her being a truth-teller is 1/2. You meet a randomly-chosen group of three people coming down the street, ask each of them the same question, and all three answer "No". Given this outcome, each now has probability 2/5 of being a truth-teller. What was the question? -- Robert Israel <israel@math.ubc.ca>
Show solution
"If you are a truth-teller, is one of your companions a truth-teller and the other a liar?" Let F = liar and T = truth-teller. The question rules out (in any order) FTT and keeps in FFF, FFT , TTT. The space is: FFF FFT FTF (FTT) out TFF (TFT) out (TTF) out TTT 9 to 6 or 6/15 = 2/5 -- Nathaniel Silver <mathelp@worldnet.att.net> What about asking "Is it true that you, or exactly one of the other two people, is a liar?". A liar will always answer 'no', whereas a truth-teller will answer 'no' only if the other two are either both liars or both truth-tellers. That will leave five equally probable combinations where everyone answers 'no' (eg. TTT, TLL, LTL, LLT, or LLL) and from that it's easy to see that any particular person has a probability 2/5 of being a truth-teller - although they are not independent: all three are truth-tellers with probability 1/5, not 8/125. What's more, any _one_ of the three persons can be deaf or mute (or otherwise unable to answer) but as long as any two answers are 'no' you still learn enough to know the changed probability. The solution to an opposite problem (where the probability of being a _liar_ is 2/5) is, in my opinion, slightly more elegant. I could not invert that solution to this problem, however. ObPuzzle: What is the elegant solution to the opposite problem? -- Risto Lankinen <rlankine@us.oracle.com> Your solution follows from the previous one, using the fact that A => B and -A v B are logically equivalent. For the "opposite problem" ask "Are you a liar and do (either) both of your companions lie or both tell the truth?" -- Nathaniel Silver <mathelp@worldnet.att.net>