Spoiler for Hidden:
EBWOP:
If we lynch OTL, and the mafia is not a fire element (probability 22/23*21/22*20/21*19/20=19/23=about 82.6%), the mafia has a chance of killing their own, based on:
a) All 4 mafia have different elements - probability 22/22*20/21*18/20*16/19, which is about 72.18%
b) 2 mafia have the same element, 2 other mafia have different elements - probability 6*22/22*1/21*20/20*18/19, which is about 27.06%
c) 2 mafia have 1 element, 2 mafia have another element - probability 6*22/22*1/21*20/20*1/19, which is about 1.5%
In case a, their probability of killing a mafia is 4/20=1/5=20%
In case b, their probability of killing a mafia is 2/22=1/11=about 9.1%
In case c, their probability of killing a mafia is 0%
Therefore, the expected probability of killing a mafia is about 16.88% (note that this includes the probability of lynching a mafia immediately by the mafia being the other fire element).
If we don't lynch OTL, the probability of killing a mafia is based on:
a) Mafia is the other fire element (about 17.4% chance)
a1) Two mafia have the same element, one has fire, one has something else. Probability 12*1/23*22/22*1/21*20/20=about 2.48%
a2) All mafia have different elements - probability 1/23*22/22*20/21*18/20=about 3.72%
b) Mafia is not a fire element (about 82.6% chance)
b1) All 4 mafia have different elements - probability 1*20/21*18/20*16/19=about 72.18%
b2) 2 Mafia have the same element, the other 2 are different - probability 6*1*1/21*1*18/19=about 27.06%
b3) 2 Mafia have 1 element, 2 mafia have the other - probability 6*1*1/21*1*1/19=about 1.5%
Killing mafia by lynching probabilities:
a1) 3/21=about 14.3%
a2) 7/23=about 30.4%
b1) 8/23=about 35.8%
b2) 6/23=about 26.1%
b3) 4/23=about 17.4%
Weighting these gives a mafia killing probability *just by lynching* (this does not take into account the possibility of the mafia deciding to not kill OTL) of 34.6%
Although I am not too sure in my math at the moment, and any mistakes should be pointed out, the number of 34.6% without killing OTL vs. 16.88% with killing OTL makes the decision clear to me.
The main point here is that we have a better chance by killing 2 unknowns than the mafia do of killing one unknown.
Now, I love statistical analysis just as much as the next guy, but what you've written
isn't actually a good comparison. You first calculate the chance that the mafia kills one of their own members if we decide to lynch OTL. Then, you calculate the chance that we kill a mafia member if we lynch someone other than OTL. You're comparing apples and oranges, bogtro, and
it's not a valid comparison. If you recalculate the statistics to include the chance that OTL (or his partner) is a mafia member and include the (small) chance that the mafia kills one of their own members, I can
guarantee you'll find that lynching OTL gives us a higher chance at killing a mafia member.
Lynching OTL gives us just as good of a chance of lynching a mafia member or a mafia member's partner as it does to lynch anyone else playing this game. But if we lynch anyone other than OTL, the mafia can kill OTL if they don't have a
mafia member to net two civilian kills with extremely low risk of killing one of their own teammates. However, if we lynch OTL, the mafia is forced to risk killing one of their own. By lynching OTL, not only do we have just as high a chance as lynching a mafia member or a mafia member's partner, but we have a much higher chance that the mafia lynches the partner of one of their own mafia members, thus netting us a mafia kill the following round.
What's more, Root's attack was a bit tool strong considering OTL had admitted his mistake twice by the time Root posted.
It isn't easy to be a civ when you're a mafia veteran. The mafia are much more likely to kill me, being one of the more experienced players. So you know what I do? I make myself a target so that I'm more likely to be saved by the doctor, which makes me riskier for the mafia to lynch. Go hard or go home!
