Assuming 3 scum: We know there are 7 balls.
I. ai) Assuming scum started with 1 ball, we mislynch a ball-less person today, all townies pass any balls in their possession to a target at random, and scum hold their ball:
Probability that all balls end up with a townie: 0
Probability that all balls end up with the scumteam: (3/11)^6 = 4.1%
Probability that scumteam gains at least 1 ball: 1-(8/11)^6 = 85.2%
--Scum still get a NK. If each player is holding, on average, (6/12) = .5 balls after balls have been passed, then scum control 1+(3*.5) = 2.5 balls and will gain an average of .5 balls on kill. This means that
after a night of passing randomly
, scum will control, on average, 3 balls.
aii) Assuming scum started with 1 ball, we mislynch a ball-less person today,
no
townies pass any balls, and scum hold their ball:
Probability that all balls end up with a townie: 0
Probability that all balls end up with the scumteam: 0
Probability that scumteam gains at least 1 ball: 0
--Scum still get a NK. Let's assume that the remainder 6 balls were distributed randomly among all townies, so each townie is holding, on average (6/9) =.666 balls. Scum gain an average of .666 balls on kill. This means that
after a night of not passing
, scum will control, on average, 1.666 balls.
bi) Assuming scum started with 1 ball, we mislynch a person with 1 ball today, all townies pass any balls in their possession to a target at random, and scum hold their ball:
First, the probability that scum gained the lynched person's ball, if the hammerer is random, is (3/12) = 25%.
Probability that all balls end up with a townie: 0
Probability that all balls end up with the scumteam: (.75)(3/11)^6 + (.25)(3/11)^5 = 6.9%
Probability that scumteam gains at least 1 ball (from passing): (.75)(1-(8/11)^6) + (.25)(1-(8/11)^5) = 83.8%
--Scum still get a NK. If each player is holding, on average, (.75)(6/12) + (.25)(5/12) = .48 balls after balls have been passed, then scum control (.75)(1+3*.48)+(.25)(2+3*.48) = 2.69 balls and will gain an average of .48 balls on kill. This means that
after a night of passing randomly
, scum will control, on average, 3.17 balls.
bii) Assuming scum started with 1 ball, we mislynch a person with 1 ball today,
no
townies pass any balls, and scum hold their ball:
Probability that all balls end up with a townie: 0
Probability that all balls end up with the scumteam: 0
Probability that scumteam gains at least 1 ball (from hammering): (3/12) = 25%
--Scum still get a NK. Let's assume that the remainder 5 balls were distributed randomly among all townies, so each townie is holding, on average (5/9) =.555 balls. Scum gain an average of .555 balls on kill. This means that
after a night of not passing
, scum will control, on average, (.25)(2+.555) + (.75)(1+.555) = 1.805 balls.
II. ai) Assuming scum started with 1 ball, we lynch a ball-less mafioso today, all townies pass any balls in their possession to a target at random, and scum hold their ball:
Probability that all balls end up with a townie: 0
Probability that all balls end up with the scumteam: (2/11)^6 = 0.04%
Probability that scumteam gains at least 1 ball: 1-(9/11)^6 = 70%
--Scum still get a NK. If each player is holding, on average, (6/12) = .5 balls after balls have been passed, then scum control 1+(2*.5) = 2 balls and will gain an average of .5 balls on kill. This means that
after a night of passing randomly
, scum will control, on average, 2.5 balls.
aii) Assuming scum started with 1 ball, we lynch a ball-less mafioso today,
no
townies pass any balls, and scum hold their ball:
Probability that all balls end up with a townie: 0
Probability that all balls end up with the scumteam: 0
Probability that scumteam gains at least 1 ball: 0
--Scum still get a NK. Let's assume that the remainder 6 balls were distributed randomly among all townies, so each townie is holding, on average (6/10) =.6 balls. Scum gain an average of .6 balls on kill. This means that
after a night of not passing
, scum will control, on average, 1.6 balls.
bi) Assuming scum started with 1 ball, we lynch a mafioso with 1 ball today, and all townies pass any balls in their possession to a target at random:
First, the probability that scum gained the lynched person's ball, if the hammerer is random, is (2/12) = 16.7%. If this happens, scum obviously won't pass their ball.
Probability that all balls end up with a townie: (.833)(1/11)^7 = .000004%
Probability that all balls end up with the scumteam: (.833)(2/11)^7 + (.167)(2/11)^6 = .001%
Probability that scumteam gains at least 1 ball (from passing): (.833)(1-(9/11)^7) + (.167)(1-(9/11)^6) = 74.5%
--Scum still get a NK. If each player is holding, on average, (.75)(7/12) + (.25)(6/12) = .5625 balls after balls have been passed, then scum control (.833)(2*.5625)+(.167)(1+2*.5625) = 1.292 balls and will gain an average of .5625 balls on kill. This means that
after a night of passing randomly
, scum will control, on average, 1.85 balls.
bii) Assuming scum started with 1 ball, we lynch a mafioso with 1 ball today, and
no
townies pass any balls:
Probability that all balls end up with a townie: 0
Probability that all balls end up with the scumteam: 0
Probability that scumteam gains at least 1 ball (from hammering): (2/12) = 16.7%
--Scum still get a NK. Let's assume that the remainder 6 balls were distributed randomly among all townies, so each townie is holding, on average (6/10) =.6 balls. Scum gain an average of .6 balls on kill. This means that
after a night of not passing
, scum will control, on average, (.167)(1+.6) + (.833)(.6) = .767 balls.
**Assertion: If scum started with more than 1 ball, they would immediately distribute their balls in order to avoid losing their cache to a lynch.
--In all scenarios under I], the probability that all balls end up with a townie is always 0, while the probabilities of scum gaining at least one additional ball after hammering and/or passing are decreasingly proportional with the probabilities calculated.
--In all scenarios under II], the probability that all balls end up with a townie is now always 0, while the probabilities of scum gaining at least one additional ball after hammering and/or passing are proportional with probabilities calculated under I]
