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

One small step but still unclear. by tomarken)

Tomarken:

Thanks for doing the heavy lifting, or at least starting it.

As I pointed out in my first post, the probabilities depend on your assumptions.

Under one assumption, the 11 configurations are equally likely.

Under a different assumption, they must be weighted by the number of different test score rankings associated with each configuration. That would make the last configuration 12 times as likely as the first configuration.

I think the 2nd, more challenging assumption, is more correct. But the puzzle difficulty of only 3 makes me think that the puzzle author disagrees.