MATH HELP: ability generation - what are the average results for these methods?

[The Cardinal]

to generate the six ability scores:

A) roll 4d6, drop lowest - repeat six times

B) roll 5d6, drop the two lowest - repeat six times

C) roll 6d6, drop the lowest three - repeat six times

D) roll 3d6 - repeat nine times, drop the lowest three

E) roll 4d6, drop lowest - repeat eight times, drop the lowest two

F) roll 4d6, drop lowest - repeat seven times, drop the lowest

G) roll 5d6, drop the lowest two - repeat seven times, drop the lowest

 

[CRGreathouse]

A: 12.24 (stdev 2.85)
B: 13.43 (stdev 2.6)
C: 14.27 (stdev 2.36)

 

[ichabod]

The overall averages (with the averages for the individual ranked rolls from highest to lowest):

A: 4d6 drop x6: 12.24 (15.66, 14.17, 12.96, 11.76, 10.41, 8.50)
B: 5d6 drop x6: 13.43 (16.44, 15.21, 14.14, 13.06, 11.80, 9.93)
C: 6d6 drop x6: 14.27 (16.92, 15.89, 14.96, 13.99, 12.83, 11.05)
D: 3d6 x9: 11.98 (14.84, 13.32, 12.25, 11.34, 10.5, 9.65)
E: 4d6 drop x8: 13.36 (16.02, 14.74, 13.74, 12.82, 11.92, 10.94)
F: 4d6 drop x7: 12.91 (15.86, 14.49, 13.39, 12.37, 11.30, 10.05)
G: 5d6 drop x7: 14.06 (16.60, 15.48, 14.53, 13.62, 12.64, 11.46)

 

[The Cardinal]

thanks!

(I love these boards )

 

[The Cardinal]

P.s.:

...anyone out there who could tell me the average results for this one?

6 abilities, 6 rolls - always add up the best three:

1st roll: 8d6
2nd: 7d6
3rd: 6d6
4th: 5d6
5th: 4d6
6th: 3d6

 

[Heretic Apostate]

I'd like to ask, how do y'all compute these averages?

 

[ichabod]

Well, I already did 4d6, 5d6, and 6d6. Here are the others:

3d6: 10.5 (14.23, 12.45, 11.12, 9.88, 8.55, 6.77)
7d6: 14.90 (17.25, 16.38, 15.55, 14.68, 13.61, 11.95)
8d6: 15.39 (17.48, 16.74, 16.00, 15.21, 14.23, 12.67)

 

[ichabod]

I'd like to ask, how do y'all compute these averages?

Well, first you have to come up with the marginal distribution. For rolls up to about 4d6 this is easy to do brute force on a spreadsheet. You have a column for each die, and you list all possible combinations of the four dice (1296 for 4d6 drop). Then you have a column that calculates the result. From there you figure out the percentage that are 18's, 17's, 16's and so on.

For things more complicated than that you need a more complicated spreadsheet or some code. For example, for 8d6 I have python code that can step through every possible combination and spit out the probabilities. That's 1,679,616 combinations. I also have a spreadsheet with all 56 ordered combinations of three dice. Then it figures out how many possible combinations of 4-9d6 can come up with that ordered combination as the three highest dice. This is a multinomial coefficient whose parameters depend on how many numbers in the three highest equal each other, and what the lowest number of the three highest is.

Once you've got the marginal distribution, you can get the distributions for the ranked individual rolls. You use the binomial distribution here. For the highest ability roll, you figure the chance that you get no 18's, and subtract that from one for the chance of at least one 18. Then you figure the chance that you get no 18's or 17's, subtract that from one for the chance you get at least one 18 or 17, and then subtract from that the chance that you get at least 18, which gets you the chance that you get at least one 17. For the second highest roll you figure the chance of getting 1 or less 18, subtract that from one for the chance that you get at least 2 18's. Then continue as for the highest roll. Then continue for the other four abilities. Obviously, this is a lot of calculations, but that's what computers were made for.

Once you have the probability distributions, the average is just the sum of each value times the probability that value will come up. You can then also calculate all sorts of other statistics, like variance, median, and percentiles.