Let P(S) be the probability (according to the judge's thought process) that the criminal is left handed: P(S) = 0.85
Let P(L) be the proportion of the population that is left handed: P(L) = 0.23.
So, let's start with the left-handedness. We know that there's a fairly good (85%) chance that the suspect is left-handed; however, there's a slight (23%) chance that whomever they put up on the stand--regardless of guilt--would be left-handed. This means that if they picked a random person off the street and put them on the stand, there's a 23% chance that they would have an 85% chance of being wrongly accused, or about 21.25% (since it is most likely that the suspicion that the guilty part is left-handed is most-likely independent of any assumptions on the population). In other words,
P(S | L) = P(S)P(L) = (0.23)(0.85) = 0.2125
Let E be the probability that the evidence shows that he is guilty. Given the above conditional probability--the probability that a left-handed person would be wrongly accused--we know that
P(E) = 1 - (0.2125) = 0.7875
Now, the judge knows there is a fairly good chance that it is no coincidence that this person is the suspect, and he's somewhat (65%) sure that this person is guilty anyway. These are certainly independent, and so we have.
P(A | E) = P(A)P(E) = (0.65)(0.7875) = 0.5119
Therefore, there is about a 51.2% chance that this person is guilty given the evidence and the judge's intuition.
Edit: I think I've read this problem wrong. I'm going to look at it again.