I realized when trying to run the numbers on choosing between all 3 products, it's no longer binomial and it's been a long time since I did stats and I don't want to run all the permutations of like 0-0-5, 0-1-4, 0-2-3, 0-3-2, 0-4-1, 0-5-0, 1-0-4, 1-1-3, etc.In post 26, ssbm_Kyouko wrote: I used a binomial calculator to get these numbers on the probability of these shipment configurations occurring (only counting the town players because this is the POV that scum will have if we follow this strat):
0L-5C: 0.03125
1L-4C: 0.15625
2L-3C: 0.3125
3L-2C: 0.3125
4L-1C: 0.15625
5L-0C: 0.03125
Treat these configurations as distinct rather than collapsing them and consider the Linen to be inspected every time (in actuality this will be random, but you could reverse the L and C and the numbers are the same. Scum are guessing the PA from amongst the off-wagon players (the Copper number plus 2) and if they are correct they gain a nightkill to POE their TD guess down to the Copper Number (if they guess the PA then the PA is removed from the denominator and the NK removes one more, or in the case of 5-0 wagon, scum just win due to the PA guess + the TD guess).
Here are the odds of scum guessing the TD in each configurationif we mislim on D2, which includes their ability to POE by guessing the PA:
0L-5C: 100% (If we mislim with 5 clears it means we killed a PR so there is only 1 potential PR left for scum to guess)
1L-4C: (1/6 guess * (1/5 NK TD + (4/5 miss NK * 1/4 guess TD))) + (5/6 fail * 1/6 guess TD) = 20.5%
2L-3C: (1/5 guess * (1/4 NK TD + (3/4 miss NK * 1/3 guess TD))) + (4/5 fail * 1/5 guess TD) = 26.0%
3L-2C: (1/4 guess * (1/3 NK TD + (2/3 miss NK * 1/2 guess TD))) + (3/4 fail * 1/4 guess TD) = 35.4%
4L-1C: (1/3 guess * (1/2 NK TD + (1/2 miss NK * 1/1 guess TD))) + (2/3 fail * 1/3 guess TD) = 55.6%
5L-0C: 100% (If we mislim with 5 clears it means we killed a PR so there is only 1 potential PR left for scum to guess)
Multiply the odds of each configuration occurring by scum's chance to guess the TD and we get scum's EV in the D2 mislim scenario. Note that if we eliminate scum on D2 that they do not get to know if their PA guess was correct because we move straight to their guessing game before N2, which is when they would learn whether they guess correctly.
0L-5C: (0.03125 * 1) +
1L-4C: (0.15625 * .205) +
2L-3C: (0.3125 * .26) +
3L-2C: (0.3125 * .354) +
4L-1C: (0.15625 * .556) +
5L-0C: (0.03125 * 1) = 37.3% chance to have guessed or killed the TD by the end of D3 if we fail to eliminate scum on D1 or D2
I need to run some additional numbers on the distribution of wagons if all the townies randomly choose between all 3 products instead of between 2 - the distributions may end up being more favorable that way. I won't have time to do this until tomorrow though.
However, using the numbers from this post and combining with Roden's idea of triple masons and my idea of splitting the 9 living players at the end of D1 into 3 groups of 3, we can compare the 2 scenarios.
Scenario 1 [Town Random + TD Wheat]: Scum has a 37.3% chance to have guessed or killed the TD by the end of D3 if we fail to eliminate scum on D1 or D2. This is not their EV because they also autowin if we fail to eliminate scum by the end of D3.
Scenario 2 [1-3Linen 4-6Copper 7-9Wheat]: This guarantees we either enter the 3-2 scenario from above, or catch scum:
3L-2C: (1/4 guess * (1/3 NK TD + (2/3 miss NK * 1/2 guess TD))) + (3/4 fail * 1/4 guess TD) = 35.4%
So scum have slightly worse odds if we just divide up the playerlist into groups of 3 and let the PA inspect one of the shipment(s) that contains neither the PA nor the TD. Furthermore, scum cannot deviate from this plan without autolosing if they should be in the shipment that the PA decides to inspect
Scenario 2a (Player is missing): If a player is missing from where they should be, they are obviously scum and can just be killed
Scenario 2b (Tea is found): if tea is found, we have a pool of 3 guaranteed to contain 1-2 scum. Let's assume it's one scum because it's easy to see that if there are 2 scum one will be found by D3. On D2 we choose which of the 3 to eliminate and before hammering we leash the surviving POEs to ship Linen and Copper (each ships one). Everyone else ships Wheat. The PA inspects either shipment (L/C) on N2 and if they find tea on a 1p shipment, that player is scum. If they don't find tea on a 1p shipment, the other player is scum. If they inspect a shipment that has 2 players for some reason, then the person that did not obey the leash is scum.
Scenario 2c (Tea is not found): We are in the 3-2 scenario from above where scum have a:
In this case (2c) the D2 shipping plan maybe should be for everyone to ship wheat and for the PA to inspect either L or C