Hypothetical Situation

Medic's options:

Prot self: 100% chance of success
Prot fellow townie: 50% chance of success (might prot mafia instead)

Let's say that the probability the medic will prot self is P(s). The probability the medic will attempt to prot his teammate is 1-P(s).

If the mafia hits the medic, his hit will succeed with a probability of 1-P(s); if he hits the other townie, the hit will succeed with a probability of 1-0.5*(1-P(s)) which is equivalent to 0.5+0.5P(s).

If this scenario was a game you could repeat many times, the mafia will choose the path of least resistance, i.e. hit the target against which he'd have the highest chance of success. If the medic is very likely to prot himself, his teammate will die; if the medic is very likely to prot his teammate, he'll die himself. There is an equilibrium at the point in which the mafia's chances of success are equal regardless of who he hits:

1-P(s) = 0.5 + 0.5P(s)
P(s) = 0.33

And the mafia's chance of success is 67% regardless of who he hits. Likewise, the medic will cover the target that will most likely negate the mafia's hit.

P(m) is the chance the mafia will hit the medic, and 1-P(m) is the chance he'll hit the green townie. The medic's payoff for protecting himself is P(m), and his payoff for protecting the townie is 0.5*(1-P(m)) = 0.5 - 0.5P(m). To keep the medic guessing, the mafia will equate the payoff for the medic's actions:

P(m) = 0.5 - 0.5*P(m)
P(m) = 0.33

The mafia will hit the medic 33% of the time, and the medic will prot himself 33% of the time. Mafia's chance of success is 67%.

I've got an issue with using probability in Mafia games - this is a weighted scenario. Probability of success isn't really useful for the Mafia goon because imo the goon isn't playing to win - he's playing so he doesn't outright lose. So the likelihood of calculating his hits are different.

But ok I'm going to try and follow.

The Medic starts out with a 50% chance of successfully protecting his teammate is throwing me off. Imo shouldn't it just be 33% since it's an equal likelihood that *anyone* can be hit? Remember you have no idea which one is the goon. If you did then yes, you are certainly down to just a 50% chance between 2 players getting hit. However since any of the 2 remaining players can be Mafia that means all 3 of you can be hit. So isn't it a 1/3rd shot?

50% is the chance that he will successfully place his protection upon his fellow townie if he tries to do so, because he will either target the townie or the mafia. This is regardless of who the mafia hits- even if the mafia hits the medic, this doesn't guarantee the medic's success or failure in figuring out who his fellow townie is and protecting him properly.

You hit the medic. If you hit the other player, you risk giving additional information to the medic, in which case you automatically lose. If you hit the medic and the hit gets blocked, the medic doesn't learn any additional information and then you can try to talk your way into the win during the day. If you hit the other player and your hit is blocked, basically the only way to win I can see is if you pull off a fake medic roleclaim after the hit (and get the real medic lynched). I'd personally prefer to have the flexibility to lynch the more suspicious of the 2 players rather than confirming one of the players, thus forcing a lynch attempt of the other.
Obviously, this is subject to change based on the players, RoL as medic in particular comes to mind, but assuming players I don't have history on, I'd hit the medic.

how people lean based on that earlier vote

hitting the medic would be the best, because hitting Townie B runs the risk of additional information being fed into the game, whereas hitting the medic just maintains the status quo in the worst case scenario.

Having not read the scenario carefully enough in my first run, I was at a loss after that. If the Medic protected himself and continued to do so what would happen? The only way to really win the game would be to try to convince Townie B to lynch the Medic. Getting Townie B lynched might set up a draw.

RoL as medic

Shoot the medic obviously. Either I win the game or nothing happens and I have to convince a mislynch. Better than hitting the townie. In that case either I win or I got a medic who saved a townie, I'm insta lynched. Not shooting is the same as shooting the medic except without the chance of insta winning.

Though if I was in a situation where it was LYLO, I was townie and Ace was somehow still alive, it's be an obvious lynch =]

This is a example of a dilemma seen infrequently in chase stories. I can't remember the name of the dilemma (if it even has one) but I know an extremely similar case appears in the movie Collateral, specifically this scene starting at 11:00 and ending at 13:08:

There are two iterations of this dilemma seen in these two minutes, but both are dealt with the same way, and Tom Cruise's character chooses correctly in both situations because he recognizes the strategy involved. Though not 100% the same as the medic / townie case, it is similar enough in that there are two paths for the medic to take, and two paths for the goon to take. The goon must take into account the medic's choice in his own, and it is because the medic has to make a choice "first" that the goon has an advantage.

The first choice happens when the good guys get to the bottom floor. There is a diverging of paths: take the street or the subway? The most important thing here is the potential for escape. The street offers the greatest potential for escape, because the good guys can just keep running and hide in a manhole if necessary, or they can quickly grab a taxi. The subway offers the least potential for escape because there is no way to know whether a train will be there on time for the good guys to keep running, and there is a high chance that they will be caught up to if they take this route. But they know that the "goon," Tom Cruise, is close behind. So Jamie Foxx attempts to trick Tom Cruise by taking the path that offers the least potential for escape, the subway.

The second choice is between the upper (same level) or lower (different level) trains. Again, the medic wishes to fool the goon by picking the least obvious choice. But the medic just chose the least obvious choice. If, then, the goon has picked correctly between the street and subway, then the goon is onto the medic's game and it is to the medic's advantage to pick the most obvious choice, thereby attempting to trick the goon again.

The second choice is more difficult, but the principle is the same. At first, the good guys run down the stairs so as not to be spotted, in case the goon has chosen correctly. Then, the upper train arrives first. The most obvious choice for escape in this situation is the upper train, which the good guys will pick to trick the goon. It is always the medic's advantage to attempt to trick the goon. This is why Tom Cruise is able to choose correctly, though it takes some brainpower as you can tell by the video.

In the first case the street choice represents the medic taking action to defend himself. The subway represents the medic acting to protect the townie. The medic isn't sure whether the goon knows he is the medic. So the medic can assume one of two things:
1. The goon doesn't know that he is the medic, or
2. The goon does know that he is the medic.
The medic notices that in the first case, it doesn't matter who he decides to protect, as the result will be a coin flip. So he decides that it is best to assume [2], that the goon knows he is the medic. The medic will most likely then decide it is in his best interest to trick the goon by protecting the townie. In the medic's mind, it is most likely that the goon will decide to kill the medic if he knows his role.

The goon's situation happens to be far superior. By assuming the medic will try to trick the goon, the choice becomes as simple as the one between the street and the subway in Collateral. The medic will pick the least obvious route of escape, which is to protect the townie. The goon can then choose to kill the medic, and he will probably be right.

Also if the final day begins with only the medic and the goon alive, it's a draw.