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 Ė 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.
1. Exactly two of us are knights.
2. Ben got the first rank and Cal got the third rank.
1. Dan got the fourth rank.
2. Exactly one of us is a knave.
1. The absolute difference between Ben's rank and mine is 2.
2. Dan is not a liar.
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 Danís rank is
(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 Artís 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.
Difficulty 3? I don't think so. There is some ambiguity in this problem, enough so that I am not going to spend my time coming up with an answer. Allow me to illustrate the issue.
One possible Assignment of types has Art and Cal as Knights, and Ben and Dan as Knaves. In this case, the rank on the quiz must be BACD.
A different possible assignment is that they are all Liars. In this case, there are 12 different possible orderings of test ranks, specifically the 12 out of 24 where D is not last and BC do not have an absolute difference of two ranks.
I expect there are a lot of other possible assignments, but let's pretend for this illustration that nothing else works.
So, then, what is the the answer to part (iv), the probability that at least one of the friends is a knight? Is it 1/2 (because there are two possible type assignments) or is it 1/13, because there are twelve times as many ways that they can all be liars?
It depends, I think, on our assumptions about the randomness of the test results, a subject about which we are given no information.
On the one hand, if the test results are random, in that each friend has an equal chance of achieving any rank, then I argue that the probability that they are all liars is 12/13.
On the other hand, if the results are 100% predictable (but unknown to us), then there is only a 1/2 chance that they are all liars.
I have no idea which to assume, and at any rate there are too many possibilities for me to do by hand.
Let the discussion begin! (or continue ...)