Solving Cut-the-knot's "Unreliable Witness"

("Probability of Lying")

To solve this problem, you need to imagine all the cases of lying and telling the truth that might possibly happen for any two witnesses on this little island. These are all hypothetical cases, of course! You'll uses these cases to calculate the percentage of truth-telling and lying that is possible for each of these two island residents when they get together (that is, you'll determine, how often will both tell the truth? how often will both lie? how often will the first tell the truth but the second lie? and how often will the first lie and the second tell the truth?). Once you've got this figured out, you will see that some cases might really occur and some are logical impossibilities here (see http://en.wikipedia.org/wiki/Logical_possibility, because if both witnesses say the same thing--as you've been told is the case in this instance, they have to either both be telling the truth or both telling a lie; one cannot be lying here and one telling the truth! So you need to eliminate these logically impossible cases to get a better estimate of the actual probabilities.