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.

So at the outset, there are 3^4 = 81 possible configurations of Knight-Knave-Liar among the four friends. 

However, I believe only 11 of these configurations are consistent with their statements.  These are:

A B C D
0 1 0 1
1 1 0 1
1 1 1 1
2 2 1 1
2 0 1 2
2 1 1 2
1 2 1 2
1 0 2 2
1 1 2 2
2 1 2 2
2 2 2 2

Where 0 = Knight, 1 = Knave, and 2 = Liar.

I'm going in circles on how to assign probabilities to these 11.

Posted by tomarken on 2014-07-16 10:32:53