Adding details to a story makes it seem more probable to humans (it fits together making a better story), when in fact every additional detail reduces the probability. malthrin linked this earlier, you should read: http://lesswrong.com/lw/jk/burdensome_details/

I think your scenario is a conjunction of unlikely events. I think the chance Harry will use a time turner in the solution is quite low, less than 10%, because the set up is such that he won't have access to it until more than 6 hours have passed. I'll give only a 50% chance that Harry will frame anyone at all, and if he does frame someone, I have no reason to believe that it will be Jugson-- I'd say 50% of the time he'll frame Dumbledore (just because it's easy), with decreasing chances for Quirell, Snape, other professors, and finally Jugson occupies such a small bit of my probability mass here that it wouldn't even occur to me.

Just these three items (time turner, frame, Jugson) reduce your scenario's probability to something very small; I put 2% on predictionbook, but my true probability is probably much lower.

(ETA: I realized on a re-read that my usage of "frame" above is non-standard; I was thinking broadly of turning the blame onto someone, not just of retroactively planting evidence, to which I give a much lower probability.)