Solved – Probability in mafia/werewolf party game

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.

There's still one missing bit of information, and that's the probability that the witch would save a dying player other than herself (we're assuming that she would always save herself on the first night if she needed to).

For now, let's assume that she's very selfish and would never use her potion on anyone other than herself. We can come back to this assumption later.

First, what are the odds that no one dies after the first night? Let's call this $P(s)$, for "probability of survival". Under our selfish-witch assumption, survival can only happen if the werewolves tried to attack a player under the protection of the guardian angel, or if they attacked the witch (who would save herself). Let's call the probability of the former happening $P(g)$, for "probability that the attacked player was guarded", and the latter $P(w)$, for "probability that the attacked player was the witch". So, we get the following: $$ P(s) = P(g) + P(w) – P(g cap w) $$

The last term represents the case where the werewolves attack the witch, who is also the one being protected by the guardian angel. The reason we need to subtract this case out is because it would otherwise be counted twice! Once in $P(g)$ and once in $P(w)$. Read up about the inclusion-exclusion principle for more detail on this.

You calculated $P(g)$ correctly as the probability that the guardian angel picked a non-werewolf times the probability that the werewolves attacked that person. That is: $$ P(g) = frac{7}{10}timesfrac{1}{8} = frac{7}{80} = 0.0875 $$

The probability that the werewolves attacked the witch is simply: $$ P(w) = frac{1}{8} = 0.125 $$

The odds that the werewolves attack a guardian-angel-protected witch can be calculated by multiplying together the probability that the guardian angel decided to protect the witch (1 in 10) and the probability that the werewolves decided to attack the doubly protected player (1 in 8). In other words: $$ P(g cap w) = frac{1}{10}timesfrac{1}{8} = frac{1}{80} = 0.0125 $$ So, if we plug these into our equation for $P(s)$, we get: $$ P(s) = 0.0875 + 0.125 – 0.0125 = 0.2 $$

As Greg Snow mentioned, we can use Bayes' theorem to find our answer. Since we want to know how likely it is that the guardian angel is the reason for a first night where no one dies, what we're trying to solve for looks like this: $P(g|s)$. This is read as "the probability of $g$ given that $s$ is true", or just "the probability of $g$ given $s$". As Bayes' theorem states, this is equal to the following: $$ P(g|s) = frac{P(g)times P(s|g)}{P(s)} $$ We know that $P(g)$ is $0.0875$, and we know that $P(s)$ is $0.2$, but what is $P(s| g)$? That is, what is the probability that no one dies the first night given that the werewolves attacked a guardian-angel-protected player? Clearly, that is always $1$. So: $$ P(g|s) = frac{0.0875 times 1}{0.2} = frac{0.0875}{0.2} = 0.4375 $$ And there you go! Assuming the witch always saves her potion for herself, the likelihood that a lucky guess by the guardian angel caused the safe night is $43.75%$.

If we know that the witch may be willing to part with her potion a particular percentage of the time, we can simply update our $P(s)$. Let's let this probability of compassion be called $P(c)$. Then: $$ P(s) = 0.2 times 1 + 0.8 times P(c) $$ That is, there will always be a safe night if either the witch or the protected player is attacked (with chance of $0.2$) and a safe night with probability $P(c)$ otherwise (with chance of $1.0 – 0.2 = 0.8$). Updating our $P(g|s)$, we get: $$ P(g|s) = frac{0.0875}{left(0.2 + 0.8 times P(c)right)} $$ Sorry if that was difficult to read because of poor formatting or if it was too long-winded. This is also my first answer on this Stack Exchange board. 😛

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