Okay, first post on this Stack Exchange board, and I'm no mathematician so I kind of lack the proper jargon. But here goes.
I'm currently playing a game known as werewolves, which is based on the better-known mafia party game, on a forum. There's a number of players (11 in our case), among which a number of werewolves (let's assume 3; actual number unknown but can be a variable in the calculations). Among the non-werewolves there could be a witch which has a life potion, and a guardian angel who can choose someone (but not herself) to protect during a night. At night, the werewolves choose someone to attack. If that person was protected by the guardian angel, they fail. If not, then the witch can save the dying victim with her potion.
After the first night, we observed no-one had died. This means either the angel had made a lucky choice, or the witch saved the victim. Note that the angel doesn't have any information about anyone and might accidentally protect a werewolf (who wasn't in any danger). Nor do the werewolves know anything about anyone's roles but their comrades'.
It's easy enough to calculate the odds that the angel would make a lucky choice. It's the chance that she chooses a non-wolf multiplied by the chance that the attacked person was chosen. The chance she chooses a non-wolf is (11 - 1 - 3)/10. The chance that the person protected happens to be the one that gets attacked is 1/8. So this ends up with 7/80 or 0.0875. I ran a little program that simulated this to verify this outcome.
Now we're trying to make out what the odds are that the result after the first night, namely no-one died, are due to the guardian angel. I believe that it is simply the chance that the angel happened to protect the right person. But another player in the discussion believes the chance to be much greater, because we observed that no-one died and thus the scenario of a successful murder is excluded.
According to me that is falsely using the outcome as a condition. Such as stating that a coin flip had a 100% chance of being heads after doing the flip and seeing it land as heads. However, this is somewhat more complex and I might be missing something.
What is the correct way here to determine the odds that given the current situation (no-one died last night) this was brought on due to a lucky choice of the angel?
EDIT: added some more info. The witch is told at night who was attacked if that person did not happen to be the angel's protégé. It is assumed that if the witch is attacked and not protected, she will always use the life potion on herself. Otherwise, the choice to use the potion is hers. Werewolves can't target each other. They know each other's identities and can communicate. They collectively agree upon a single target.