Let's say that there are seven players, and two of four specific players are known to be mafia.
Let's call those four potentially-mafia players
A,
B,
X, and
Y. Let's say
A and
B are townies, and
X and
Y are mafia.
It's clearly in the best interest of the mafia to kill players not in this set of four players, so I will not consider those possibilities.
Here are the possible lynch permutations, assuming all vanilla:
TT 1/4 MDay 2, 7 players (5/2); lynch
A; alive:
B,
X,
YDay 3, 5 players (3/2); lynch
B; alive:
X,
YDay 4, 3 players (1/2); mafia win
TMT 1/8 MDay 2, 7 players (5/2); lynch
A; alive:
B,
X,
YDay 3, 5 players (3/2); lynch
X; alive:
B,
YDay 4, 3 players (2/1); lynch
B; alive:
YDay 5, 1 players (0/1); mafia win
TMM 1/8 TDay 2, 7 players (5/2); lynch
A; alive:
B,
X,
YDay 3, 5 players (3/2); lynch
X; alive:
B,
YDay 4, 3 players (2/1); lynch
Y; alive:
BDay 5, 2 players (2/0); town win
MTT 1/8 MDay 2, 7 players (5/2); lynch
X; alive:
A,
B,
YDay 3, 5 players (4/1); lynch
A; alive:
B,
YDay 4, 3 players (2/1); lynch
B; alive:
YDay 5, 1 players (0/1); mafia win
MTM 1/8 TDay 2, 7 players (5/2); lynch
X; alive:
A,
B,
YDay 3, 5 players (4/1); lynch
A; alive:
B,
YDay 4, 3 players (2/1); lynch
Y; alive:
BDay 5, 2 players (2/0); town win
MM 1/4 TDay 2, 7 players (5/2); lynch
X; alive:
A,
B,
YDay 3, 5 players (4/1); lynch
Y; town win
(Format is: lynches in order (M=mafia, T=town), probability of the situation occurring given random chance, eventual win.)
Let's tally up the probabilities:
TT 1/4 MTMT 1/8 MTMM 1/8 TMTT 1/8 MMTM 1/8 TMM 1/4 TM = 1/4 + 1/8 + 1/8 = 1/2 = 50%
T = 1/8 + 1/8 + 1/4 = 1/2 = 50%
If we assume foul is telling the truth, randomly lynching will not tilt the odds in either team's favor. This is what MM (Lilac) was trying to say, Two.
Things are different if we are only looking at three players (one townie, two mafia). Basically, just chop off the TT, TMT, and MTT possibilities:
TMM 1/8 TMTM 1/8 TMM 1/4 TSo what do we get? A guaranteed townie win.
Of course, this all hinges on a few assumptions:
- foul is telling the truth.
- All lynches and nightkills are successful.
- No lynches and nightkills are reversed.
- There is one confirmed townie among the four.
If the "confirmed townie" is mafia, it's a guaranteed mafia win, unless someone wakes up in time.