In a certain island, one-third of the native people are liars who always lie, one-third are knights who always tell the truth, and one-third are knaves that is, people who strictly alternate between speaking the truth and telling a lie, irrespective of order. The chances of encountering any one of the three natives on a road on the island are the same.

Four friends Art, Ben, Cal and Dan – who are natives of this island got the top four ranks in a certain quiz The following statements are made by each of Art, Ben, Cal and Dan.

Art
1. Exactly two of us are knights.
2. Ben got the first rank and Cal got the third rank.

Ben
1. Dan got the fourth rank.
2. Exactly one of us is a knave.

Cal
1. The absolute difference between Ben's rank and mine is 2.
2. Dan is not a liar.

Dan
1. Exactly one of us is a liar.
2. I am a knave.

Assuming no ties, determine the probability that:

(i) The absolute difference between Cal's rank and Dan’s rank is 1.
(ii) Exactly three of the four friends are liars.
(iii) Exactly two of the four friends are knaves.
(iv) At least one of the four friends is a knight.

*** For Art’s second statement - assume the entire statement is a lie if the whole statement or any of its parts thereof is false. For example- A2 is false if in reality Ben got the first rank and Cal got the fourth rank.

(In reply to re(3): One small step but still unclear. by Steve Herman)

I'm inclined to agree with you but I haven't been able to convince myself that this is definitely the case.  How does this differ from the Monty Hall problem, for example, where we start with equiprobable outcomes, and then after eliminating some of them by introducing new information, the remaining outcomes are no longer equiprobable?

We know, for instance, that (prior to being told anything else) there is a 1/3 chance that Art is a Knight.  After inspecting the statements, it can be deduced that if Art is a Knight, then Cal must also be a Knight and Ben and Dan must be Knaves.  So why is the probability of this configuration 1/11 and not 1/3?

My gut tells me you're right but this is the kind of thing that's tripping me up.

Posted by tomarken on 2014-07-17 10:08:45