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Topic: Four Liars (Read 561 times) |
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ThudnBlunder
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Four people, A, B, C, and D, each tell the truth, independently of one another, with a probability of 1/3.
If A states that B denies that C says that D is lying, what is the probability that D is telling the truth?
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towr
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Re: Four Liars
« Reply #1 on: Nov 5th, 2003, 5:38am » |
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Can we assume D actually said anything?
(because C could say D was lying even if D didn't say anything)
this is all terribly complicated..
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ThudnBlunder
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Re: Four Liars
« Reply #2 on: Nov 5th, 2003, 5:51am » |
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Quote:Can we assume D actually said anything?
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Good point.
Assumptions:
1) D said something which can be regarded as either true or false.
2) C can accurately assess the truth or otherwise of D's statement and make his/her own.
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| « Last Edit: Nov 5th, 2003, 5:51am by ThudnBlunder » |
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rmsgrey
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Re: Four Liars
« Reply #3 on: Nov 14th, 2003, 7:17am » |
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Another assumption appears to be needed:
3) B made a statement concerning C's statement
Possibly also:
4) Neither A nor B make statements without knowing whether they are true.
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ThudnBlunder
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Re: Four Liars
« Reply #4 on: Nov 14th, 2003, 8:20am » |
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Quote:Another assumption appears to be needed:
3) B made a statement concerning C's statement |
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"B denies that C says" seems to cover that.
Quote:| 4) Neither A nor B make statements without knowing whether they are true. |
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A's statement relates only to what B said (not to what D said), and B's statement relates only to what C said (not to what D said). In that sense, they both know 'the truth' of what they heard.
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| « Last Edit: Nov 14th, 2003, 8:22am by ThudnBlunder » |
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So does anybody actually plan on answering the question?
The actual probability assigned to D is immaterial, because c knows for a fact whether it's true or false.
The statement is equivalent to
A states that b states that c states that d is telling the truth.
Therefore there are 4 ways d could be telling the truth:
If a,b,c were all right (probability 1/27)
If any two of them were wrong (probability 4/27 times 3).
Total 13/27
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ThudnBlunder
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Re: Four Liars
« Reply #6 on: Nov 14th, 2003, 8:45am » |
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Quote:The statement is equivalent to
A states that b states that c states that d is telling the truth. |
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That is correct, but IMO your answer is wrong. 
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You're right. D's probability does make a difference (as I should have realized from the taxi riddle).
When D is telling the truth, there's a 13/27 chance it will be correctly labeled the truth.
But when D lies (2/3 of the time) there's a 14/27 chance it will be incorrectly labeled the truth.
So out of 81 trials, we'll have the given statement 41 times, and it will be true 13 of those 41.
Am I getting close?
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towr
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Re: Four Liars
« Reply #8 on: Nov 14th, 2003, 10:04am » |
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on Nov 14th, 2003, 8:31am, visitor wrote:| So does anybody actually plan on answering the question? |
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Yes, but not anytime soon  (at least not this weekend)
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ThudnBlunder
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Re: Four Liars
« Reply #9 on: Nov 15th, 2003, 9:49pm » |
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Quote:
Yes, that's what I get. Notice that it is just a little less than if D alone had spoken.
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| « Last Edit: Nov 16th, 2003, 8:38pm by ThudnBlunder » |
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rmsgrey
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Re: Four Liars
« Reply #10 on: Nov 17th, 2003, 6:28am » |
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on Nov 14th, 2003, 8:20am, THUDandBLUNDER wrote:
[ quote ]3)B made a statement concerning C's statement
[ / quote ]
"B denies that C says" seems to cover that.
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But we have only A's word for it - so if B says nothing, he doesn't deny anythiing, so A's statement is trivially false...
Quote:
[ quote ]4) Neither A nor B make statements without knowing whether they are true.
[ / quote ]
A's statement relates only to what B said (not to what D said), and B's statement relates only to what C said (not to what D said). In that sense, they both know 'the truth' of what they heard.
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If they heard anything, then they know what they heard, but if either of them didn't hear, or heard indistinctly, or didn't understand for some other reason (foreign language?) what was said, then they would be unable to assess the truth of their own statement.
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ThudnBlunder
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Re: Four Liars
« Reply #11 on: Nov 17th, 2003, 9:13pm » |
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Quote:| If they heard anything, then they know what they heard, but if either of them didn't hear, or heard indistinctly, or didn't understand for some other reason (foreign language?) what was said, then they would be unable to assess the truth of their own statement. |
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I thought it was obvious that this was a simple exercise in probability and logic - not in semantics, nitpicking, or pedantry. 
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rmsgrey
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Re: Four Liars
« Reply #12 on: Nov 20th, 2003, 6:49am » |
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Yes, but the logic and probability seem conceptually simple and just require some calculation, and since I'm too lazy to actually do them, I decided to follow Towr's lead. Since my comments were challenged, I defended them at the earliest opportunity (my internet access is a little sporadic at present)
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