bozzball: Your analysis, while accurate in the mathematical sense (generally the best kind of sense, but... maybe not here), is a little flawed in terms of the actual game.
Firstly is understanding the actual role interactions, which is kinda important. The doctor is extremely valuable, but only if there's someone good to protect. This could be a cleared townie, a cop claim, or otherwise someone very likely to be killed in absense of a doctor. In terms of random protection, doctors don't do much, occasionally saving someone, which kinda helps a little but less than you first think (as in this setup, it would probably lead to us wanting to no lynch pretty soon after - but the doctor can be more confident the person he saved was town. Not certain as there's a jailkeeper and the mafia might choose to no kill (very rare but possible)). The Jailkeeper I'm less familiar with, but he's probably a lot better than the Doctor - he can prevent a kill by choosing one of TWO people, which is double odds, and also might stop the mafia rolecop. However, he could also stop the doctor/cop, and unlike the doctor, he gets pretty much no information from his jailing.
The cop, on the other hand, doesn't need to hit a mafioso to be successful. Finding out someone is town can be helpful as well - less so, but still helpful. A cop can easily clear a player (or two) before his death, if he thinks he's going to be lynched, and that can be a big help. The doc/Jailkeeper should protect that player, and then the town has someone to unquestionably trust the next day. If that player was the doc/Jailkeeper, or there isn't one, that kinda sucks (bear in mind the mafia rolecop, which means the mafia pretty much knows everyone's roles by day 3), but otherwise, the cop is always useful if they find out information.
Rules question: Are roles revealed on death? If no, is alignment revealed on death? I presume yes to both, but it'd be useful to know.