[REVIEW] Open Setup Reviews

[Plotinus]

Because of combinatorics, the odds are slightly different than those listed. These are the correct numbers:

5% - 1
10% - 2-3
14% - 4-6
14% - 7-9
18% - 10-13
18% - 14-17
22% - 18-22
22% - 23-27
26% - 28-33
29% - 34-40

Explanation:

for 1 to be scum, the first dice has to be a 1 (1/40) and the second dice has to be not 1 (39/40), or visa versa (*2) = 1/40*39/40*2= 4.875%
for 2-3 to be scum, the first dice has to be 2 or 3 (2/40) and the second dice has to be not 2 or 3 (38/40) or visa versa (*2) = 2/40*39/40*2 = 9.5%

and so on.

so the odds are 4.875%, 9.5%, 13.875%, 18%, 21.875%, 25.5%, and 28.875%

[Plotinus]

I got the numbers wrong but after a long talk with my girlfriend, I've got them right now.

There are 40² combination of 2d40, or 1600.

The chance that the dice have to be rerolled is 190/1600. This is 1² + 2² + 3² + 3² + 4² + 4² + 5² + 5² + 6² + 7².

We really only care about the last roll of the dice. It doesn't matter how many dice rolls it takes to get there. There are 1600-190 = 1410 numbers that don't lead to a reroll.

There are 1*39 numbers that make player a scum (1 dice has to be a 1 and the other dice has to be a number between 2 and 40 inclusive).

There are 2*38 numbers that make player b scum (1 dice has to be a 2 or a 3 and the other dice has to be a 1 or a number between 4 and 40 inclusive)

etc all the way down to player j which is 7*33 = 231.

The chances of any given player being scum are thus:

a: 39/1410 = 2.76%
b: 76/1410 = 5.39%
c: 111/1410 = 7.87%
d: 111/1410 = 7.87%
e: 144/1410 = 10.21%
f: 144/1410 = 10.21%
g: 175/1410 = 12.41%
h: 175/1410 = 12.41%
i: 204/1410 = 14.47%
j: 231/1410 = 16.38%

These numbers add up to 100% exactly but I think 200% might be the percentage we're looking for because there are two scum.

a: 5.53%
b: 10.78%
c: 15.74%
d: 15.74%
e: 20.43%
f: 20.43%
g: 24.82%
h: 24.82%
i: 28.94%
j: 32.77%

[Plotinus]

If that's too many significant digits, this works too:

Odds of being scum	dice number
6%	1
11%	2-3
16%	4-6
16%	7-9
20%	10-13
20%	14-17
25%	18-22
25%	23-27
29%	28-33
33%	34-40