I need help with a dice math problem

[Tyler.Tinsley]

How many successes do you require for a throw to be considered successful ?
one or more ? Because this has a strong impact on the odds.

it's actually a variable. the more success the better, shooting a gun and missing will only distract the target, where getting a massive success will not only hit but cause serious injury.

I will explain how I got my general 80% return so you can let me know if I am doing something wrong.

each die has a 1/3 chance of failure, a 2/3 chance of success and a 1/6 chance of adding a die with the very same odds.

assuming perfect results if we roll 6 dice we will have 2 failure 4 success and an extra seventh die generated by the six. this extra die is subject to the same odds as the other dice so it's not quite as good as having a dice with 5/6 chance of success.

It's my thinking that the average return could not possibly be better then 5/6 but lies somewhere between 4/6 and 5/6

if we make 1/6th a percentage of 100 we get 16.6, times that by 4.6 (4 represents the 4/6 chance of success and point six represents the 2/3 potential for the exploded die to result in a success) we get 76.6

This is flawed because it's not taking into account the potential for the exploded die to roll another six and produce another die of the same average return.

It's hard to calculate because you need to know what the average return of a dice is in order to calculate the value added by the explosion. it requires a level of math that I cant figure out.

my attempts result in values that i round up to 80% but that's mostly optimism at play.

[Jasper Flick]

Personally, I would tell players something like "Roughly two out of three dice will be a success, and about one out of ten will generate two successes. But nothing's certain, except that the more dice you roll, the higher your chances of getting those successes!" I think that's fine for an informed decision.

[Guy Srinivasan]

The average is easy to compute in cases like this if you know the trick. First call the average successes per die A. Then, in your example where 1,2 fail, 3,4,5 succeed, and 6 succeeds and explodes,

A = (2/6 * 0) + (3/6 * 1) + (1/6 * (1+A))

solve for A, you get A = 4/5. So in fact each die is worth expected value exactly 80% of a success, though the results aren't just 0 or 1.

To go further, you can compute the variance of a single die, then multiply by number of dice. If you know how to do this, good - use Wolfram|Alpha to make it easier. If you don't, fine... in this case you can tell your players that rolling N dice, on average you'll get 80% successes, and 90% of the time they'll get at least (80 - 115*sqrt(N)/N)% and at most (80 + 115*sqrt(N)/N)%.

[Tyler.Tinsley]

Thank goodness some people go to college or at least pay attention in math class.

thanks for confirming my suspicions. I can now price various things accordingly.

[evilphd7]

Just for fun, here's another variable to consider: the die size....

The advantage is that the numbers to look for and the amount of dice to roll stay the same. Also, having to roll bigger dice reall feels different. The downside is that you need to have all those dice.

I'm actually working on a game at the moment which leverages this concept of "scalable" dice for a difficulty.  It hadn't occurred to me that gaming dice might be difficult for some people to find (I've just always them). Does anyone out there happen to know of any published games which use scalable dice?

[Tyler.Tinsley]

lots and lots of games use the dice scale 4-12. all to varying degrees of success and complexity. my favorite use of the dice scale was in battle ball. a main stream foot ball board game.

[Wordman]

The exact math to figure this out is pretty painful. Mostly, it involves looking at a bunch of isolated cases and adding them together.

However, this problem has three important characteristics: 1) the outcome is totally non-obvious, 2) it is extremely easy to simulate a single event (die roll) and 3) it has a limited number of starting cases (i.e. dice pools will only start out so large). This means that this problem is an extremely good candidate for Monte Carlo simulation. (I work with a bunch of math PhD's and nearly all of them told me to use simulation like this, if that helps.)

As far as Monte Carlo simulations go, this one is really easy. You just need some code that will generate the result of a single roll. Then run that code, say, 10 million times (that should take maybe five minutes, tops). Graph the results.

It won't be an exact answer, but it will tell you what you want to know.

If you lack coding ability, let me know. I can whip something up.

[Wordman]

I decided to just write this code, out of curiosity (code at the end). Here is the results for pools of size one through 10, running a million rolls at each size. Note that the median and the mean are pretty far apart, as you might guess for an "infinite" tail. Bear in mind that this is a simulation, so the distribution will not be perfect. (As an example, in the 1 die table, the 22 success result probably got a few more hits than a pure computation would figure out, just by chance.)

Code Select Expand


-- Number of rolls generating n successes from a 1 dice pool --
0 successes:          111458 = 11.145789%
1 successes:          345093 = 34.509265%
2 successes:          307151 = 30.715069%
3 successes:           93762 = 9.376191%
4 successes:           66931 = 6.693093%
5 successes:           29757 = 2.975697%
6 successes:           19317 = 1.931698%
7 successes:           10182 = 1.018199%
8 successes:            6516 = 0.651599%
9 successes:            3702 = 0.370200%
10 successes:            2273 = 0.227300%
11 successes:            1442 = 0.144200%
12 successes:             894 = 0.089400%
13 successes:             531 = 0.053100%
14 successes:             364 = 0.036400%
15 successes:             218 = 0.021800%
16 successes:             146 = 0.014600%
17 successes:              94 = 0.009400%
18 successes:              64 = 0.006400%
19 successes:              43 = 0.004300%
20 successes:              24 = 0.002400%
21 successes:               9 = 0.000900%
22 successes:              13 = 0.001300%
23 successes:               4 = 0.000400%
24 successes:               5 = 0.000500%
25 successes:               1 = 0.000100%
26 successes:               2 = 0.000200%
27 successes:               1 = 0.000100%
28 successes:               2 = 0.000200%
29 successes:               1 = 0.000100%
30 successes:               0 = 0.000000%
31 successes:               1 = 0.000100%
Average result: 2.001689

-- Number of rolls generating n successes from a 2 dice pool --
0 successes:           36959 = 3.695896%
1 successes:          173274 = 17.327383%
2 successes:          287574 = 28.757371%
3 successes:          216501 = 21.650078%
4 successes:          107925 = 10.792489%
5 successes:           72522 = 7.252193%
6 successes:           39582 = 3.958196%
7 successes:           25402 = 2.540197%
8 successes:           14852 = 1.485199%
9 successes:            9403 = 0.940299%
10 successes:            5840 = 0.583999%
11 successes:            3655 = 0.365500%
12 successes:            2353 = 0.235300%
13 successes:            1474 = 0.147400%
14 successes:             983 = 0.098300%
15 successes:             564 = 0.056400%
16 successes:             385 = 0.038500%
17 successes:             264 = 0.026400%
18 successes:             164 = 0.016400%
19 successes:             114 = 0.011400%
20 successes:              79 = 0.007900%
21 successes:              42 = 0.004200%
22 successes:              27 = 0.002700%
23 successes:              23 = 0.002300%
24 successes:              21 = 0.002100%
25 successes:               3 = 0.000300%
26 successes:               4 = 0.000400%
27 successes:               2 = 0.000200%
28 successes:               5 = 0.000500%
29 successes:               2 = 0.000200%
30 successes:               0 = 0.000000%
31 successes:               2 = 0.000200%
32 successes:               0 = 0.000000%
33 successes:               0 = 0.000000%
34 successes:               1 = 0.000100%
Average result: 2.999588

-- Number of rolls generating n successes from a 3 dice pool --
0 successes:           12501 = 1.250099%
1 successes:           77238 = 7.723792%
2 successes:          188206 = 18.820581%
3 successes:          233199 = 23.319877%
4 successes:          174189 = 17.418883%
5 successes:          110275 = 11.027489%
6 successes:           74441 = 7.444093%
7 successes:           46089 = 4.608895%
8 successes:           30279 = 3.027897%
9 successes:           19078 = 1.907798%
10 successes:           12220 = 1.221999%
11 successes:            7689 = 0.768899%
12 successes:            5149 = 0.514899%
13 successes:            3354 = 0.335400%
14 successes:            2105 = 0.210500%
15 successes:            1389 = 0.138900%
16 successes:             906 = 0.090600%
17 successes:             609 = 0.060900%
18 successes:             386 = 0.038600%
19 successes:             215 = 0.021500%
20 successes:             176 = 0.017600%
21 successes:             106 = 0.010600%
22 successes:              60 = 0.006000%
23 successes:              48 = 0.004800%
24 successes:              35 = 0.003500%
25 successes:              15 = 0.001500%
26 successes:              16 = 0.001600%
27 successes:               9 = 0.000900%
28 successes:               5 = 0.000500%
29 successes:               7 = 0.000700%
30 successes:               3 = 0.000300%
31 successes:               0 = 0.000000%
32 successes:               1 = 0.000100%
33 successes:               0 = 0.000000%
34 successes:               1 = 0.000100%
35 successes:               1 = 0.000100%
36 successes:               1 = 0.000100%
Average result: 3.993547

-- Number of rolls generating n successes from a 4 dice pool --
0 successes:            4124 = 0.412400%
1 successes:           32211 = 3.221097%
2 successes:          102835 = 10.283490%
3 successes:          180912 = 18.091182%
4 successes:          194005 = 19.400481%
5 successes:          151222 = 15.122185%
6 successes:          107332 = 10.733189%
7 successes:           75950 = 7.594992%
8 successes:           50182 = 5.018195%
9 successes:           34099 = 3.409897%
10 successes:           22523 = 2.252298%
11 successes:           15147 = 1.514698%
12 successes:            9857 = 0.985699%
13 successes:            6600 = 0.659999%
14 successes:            4341 = 0.434100%
15 successes:            2996 = 0.299600%
16 successes:            1950 = 0.195000%
17 successes:            1257 = 0.125700%
18 successes:             809 = 0.080900%
19 successes:             554 = 0.055400%
20 successes:             355 = 0.035500%
21 successes:             265 = 0.026500%
22 successes:             169 = 0.016900%
23 successes:              96 = 0.009600%
24 successes:              66 = 0.006600%
25 successes:              47 = 0.004700%
26 successes:              32 = 0.003200%
27 successes:              22 = 0.002200%
28 successes:               9 = 0.000900%
29 successes:              12 = 0.001200%
30 successes:               9 = 0.000900%
31 successes:               3 = 0.000300%
32 successes:               5 = 0.000500%
33 successes:               2 = 0.000200%
34 successes:               0 = 0.000000%
35 successes:               1 = 0.000100%
36 successes:               1 = 0.000100%
37 successes:               0 = 0.000000%
38 successes:               1 = 0.000100%
Average result: 5.000107

-- Number of rolls generating n successes from a 5 dice pool --
0 successes:            1359 = 0.135900%
1 successes:           12925 = 1.292499%
2 successes:           51276 = 5.127595%
3 successes:          115922 = 11.592188%
4 successes:          166252 = 16.625183%
5 successes:          166993 = 16.699283%
6 successes:          136737 = 13.673686%
7 successes:          103789 = 10.378890%
8 successes:           75276 = 7.527592%
9 successes:           53354 = 5.335395%
10 successes:           37555 = 3.755496%
11 successes:           25521 = 2.552097%
12 successes:           17140 = 1.713998%
13 successes:           11696 = 1.169599%
14 successes:            7829 = 0.782899%
15 successes:            5369 = 0.536899%
16 successes:            3577 = 0.357700%
17 successes:            2400 = 0.240000%
18 successes:            1691 = 0.169100%
19 successes:            1084 = 0.108400%
20 successes:             755 = 0.075500%
21 successes:             493 = 0.049300%
22 successes:             311 = 0.031100%
23 successes:             204 = 0.020400%
24 successes:             158 = 0.015800%
25 successes:             130 = 0.013000%
26 successes:              66 = 0.006600%
27 successes:              44 = 0.004400%
28 successes:              32 = 0.003200%
29 successes:              15 = 0.001500%
30 successes:              17 = 0.001700%
31 successes:              13 = 0.001300%
32 successes:              10 = 0.001000%
33 successes:               3 = 0.000300%
34 successes:               2 = 0.000200%
35 successes:               0 = 0.000000%
36 successes:               1 = 0.000100%
37 successes:               1 = 0.000100%
38 successes:               1 = 0.000100%
Average result: 5.995540

-- Number of rolls generating n successes from a 6 dice pool --
0 successes:             471 = 0.047100%
1 successes:            5024 = 0.502399%
2 successes:           23564 = 2.356398%
3 successes:           66052 = 6.605193%
4 successes:          118920 = 11.891988%
5 successes:          151474 = 15.147385%
6 successes:          148640 = 14.863985%
7 successes:          126129 = 12.612887%
8 successes:           99633 = 9.963290%
9 successes:           75052 = 7.505192%
10 successes:           55323 = 5.532294%
11 successes:           39582 = 3.958196%
12 successes:           27801 = 2.780097%
13 successes:           19430 = 1.942998%
14 successes:           13491 = 1.349099%
15 successes:            9182 = 0.918199%
16 successes:            6414 = 0.641399%
17 successes:            4407 = 0.440700%
18 successes:            2974 = 0.297400%
19 successes:            2080 = 0.208000%
20 successes:            1384 = 0.138400%
21 successes:             938 = 0.093800%
22 successes:             627 = 0.062700%
23 successes:             449 = 0.044900%
24 successes:             301 = 0.030100%
25 successes:             208 = 0.020800%
26 successes:             143 = 0.014300%
27 successes:              82 = 0.008200%
28 successes:              75 = 0.007500%
29 successes:              40 = 0.004000%
30 successes:              36 = 0.003600%
31 successes:              27 = 0.002700%
32 successes:              16 = 0.001600%
33 successes:              11 = 0.001100%
34 successes:               7 = 0.000700%
35 successes:               4 = 0.000400%
36 successes:               5 = 0.000500%
37 successes:               2 = 0.000200%
38 successes:               1 = 0.000100%
39 successes:               0 = 0.000000%
40 successes:               0 = 0.000000%
41 successes:               1 = 0.000100%
42 successes:               1 = 0.000100%
Average result: 6.999319

-- Number of rolls generating n successes from a 7 dice pool --
0 successes:             160 = 0.016000%
1 successes:            1888 = 0.188800%
2 successes:           10624 = 1.062399%
3 successes:           34616 = 3.461597%
4 successes:           75937 = 7.593692%
5 successes:          117204 = 11.720388%
6 successes:          138879 = 13.887886%
7 successes:          135116 = 13.511586%
8 successes:          117412 = 11.741188%
9 successes:           95598 = 9.559790%
10 successes:           74616 = 7.461593%
11 successes:           56149 = 5.614894%
12 successes:           40828 = 4.082796%
13 successes:           29801 = 2.980097%
14 successes:           21329 = 2.132898%
15 successes:           15325 = 1.532498%
16 successes:           10507 = 1.050699%
17 successes:            7425 = 0.742499%
18 successes:            5018 = 0.501799%
19 successes:            3663 = 0.366300%
20 successes:            2454 = 0.245400%
21 successes:            1727 = 0.172700%
22 successes:            1135 = 0.113500%
23 successes:             810 = 0.081000%
24 successes:             576 = 0.057600%
25 successes:             364 = 0.036400%
26 successes:             287 = 0.028700%
27 successes:             165 = 0.016500%
28 successes:             131 = 0.013100%
29 successes:              74 = 0.007400%
30 successes:              53 = 0.005300%
31 successes:              40 = 0.004000%
32 successes:              29 = 0.002900%
33 successes:              22 = 0.002200%
34 successes:              14 = 0.001400%
35 successes:              12 = 0.001200%
36 successes:               3 = 0.000300%
37 successes:               4 = 0.000400%
38 successes:               1 = 0.000100%
39 successes:               2 = 0.000200%
40 successes:               0 = 0.000000%
41 successes:               1 = 0.000100%
42 successes:               0 = 0.000000%
43 successes:               2 = 0.000200%
Average result: 7.994876

-- Number of rolls generating n successes from a 8 dice pool --
0 successes:              56 = 0.005600%
1 successes:             687 = 0.068700%
2 successes:            4564 = 0.456400%
3 successes:           17034 = 1.703398%
4 successes:           44121 = 4.412096%
5 successes:           80772 = 8.077192%
6 successes:          113469 = 11.346889%
7 successes:          128215 = 12.821487%
8 successes:          124703 = 12.470288%
9 successes:          111117 = 11.111689%
10 successes:           91923 = 9.192291%
11 successes:           73464 = 7.346393%
12 successes:           56527 = 5.652694%
13 successes:           42304 = 4.230396%
14 successes:           31228 = 3.122797%
15 successes:           23000 = 2.299998%
16 successes:           16677 = 1.667698%
17 successes:           11905 = 1.190499%
18 successes:            8423 = 0.842299%
19 successes:            6056 = 0.605599%
20 successes:            4169 = 0.416900%
21 successes:            2840 = 0.284000%
22 successes:            2100 = 0.210000%
23 successes:            1412 = 0.141200%
24 successes:            1028 = 0.102800%
25 successes:             657 = 0.065700%
26 successes:             494 = 0.049400%
27 successes:             319 = 0.031900%
28 successes:             238 = 0.023800%
29 successes:             150 = 0.015000%
30 successes:             109 = 0.010900%
31 successes:              89 = 0.008900%
32 successes:              47 = 0.004700%
33 successes:              31 = 0.003100%
34 successes:              26 = 0.002600%
35 successes:              22 = 0.002200%
36 successes:               8 = 0.000800%
37 successes:              10 = 0.001000%
38 successes:               3 = 0.000300%
39 successes:               2 = 0.000200%
40 successes:               1 = 0.000100%
41 successes:               0 = 0.000000%
42 successes:               0 = 0.000000%
43 successes:               0 = 0.000000%
44 successes:               0 = 0.000000%
45 successes:               0 = 0.000000%
46 successes:               1 = 0.000100%
Average result: 8.997355

-- Number of rolls generating n successes from a 9 dice pool --
0 successes:              12 = 0.001200%
1 successes:             257 = 0.025700%
2 successes:            1904 = 0.190400%
3 successes:            8114 = 0.811399%
4 successes:           23852 = 2.385198%
5 successes:           51270 = 5.126995%
6 successes:           82885 = 8.288492%
7 successes:          108380 = 10.837989%
8 successes:          119531 = 11.953088%
9 successes:          117302 = 11.730188%
10 successes:          104956 = 10.495590%
11 successes:           88585 = 8.858491%
12 successes:           72347 = 7.234693%
13 successes:           56695 = 5.669494%
14 successes:           43571 = 4.357096%
15 successes:           33096 = 3.309597%
16 successes:           24101 = 2.410098%
17 successes:           18077 = 1.807698%
18 successes:           12861 = 1.286099%
19 successes:            9374 = 0.937399%
20 successes:            6624 = 0.662399%
21 successes:            4815 = 0.481500%
22 successes:            3415 = 0.341500%
23 successes:            2326 = 0.232600%
24 successes:            1703 = 0.170300%
25 successes:            1187 = 0.118700%
26 successes:             850 = 0.085000%
27 successes:             601 = 0.060100%
28 successes:             404 = 0.040400%
29 successes:             280 = 0.028000%
30 successes:             176 = 0.017600%
31 successes:             122 = 0.012200%
32 successes:             121 = 0.012100%
33 successes:              54 = 0.005400%
34 successes:              43 = 0.004300%
35 successes:              25 = 0.002500%
36 successes:              29 = 0.002900%
37 successes:              14 = 0.001400%
38 successes:              15 = 0.001500%
39 successes:               6 = 0.000600%
40 successes:               5 = 0.000500%
41 successes:               3 = 0.000300%
42 successes:               7 = 0.000700%
43 successes:               1 = 0.000100%
44 successes:               2 = 0.000200%
45 successes:               1 = 0.000100%
46 successes:               1 = 0.000100%
47 successes:               1 = 0.000100%
Average result: 9.997082

-- Number of rolls generating n successes from a 10 dice pool --
0 successes:               5 = 0.000500%
1 successes:              92 = 0.009200%
2 successes:             745 = 0.074500%
3 successes:            3627 = 0.362700%
4 successes:           12172 = 1.217199%
5 successes:           29780 = 2.977997%
6 successes:           55981 = 5.598094%
7 successes:           83511 = 8.351092%
8 successes:          104598 = 10.459790%
9 successes:          111856 = 11.185589%
10 successes:          109573 = 10.957289%
11 successes:          100221 = 10.022090%
12 successes:           85998 = 8.599791%
13 successes:           71306 = 7.130593%
14 successes:           56425 = 5.642494%
15 successes:           44239 = 4.423896%
16 successes:           34210 = 3.420997%
17 successes:           25706 = 2.570597%
18 successes:           19196 = 1.919598%
19 successes:           14202 = 1.420199%
20 successes:           10271 = 1.027099%
21 successes:            7531 = 0.753099%
22 successes:            5357 = 0.535699%
23 successes:            3995 = 0.399500%
24 successes:            2769 = 0.276900%
25 successes:            1968 = 0.196800%
26 successes:            1405 = 0.140500%
27 successes:            1000 = 0.100000%
28 successes:             690 = 0.069000%
29 successes:             452 = 0.045200%
30 successes:             347 = 0.034700%
31 successes:             249 = 0.024900%
32 successes:             148 = 0.014800%
33 successes:             114 = 0.011400%
34 successes:              84 = 0.008400%
35 successes:              63 = 0.006300%
36 successes:              40 = 0.004000%
37 successes:              23 = 0.002300%
38 successes:              16 = 0.001600%
39 successes:               9 = 0.000900%
40 successes:               8 = 0.000800%
41 successes:               5 = 0.000500%
42 successes:               4 = 0.000400%
43 successes:               3 = 0.000300%
44 successes:               2 = 0.000200%
45 successes:               0 = 0.000000%
46 successes:               2 = 0.000200%
47 successes:               3 = 0.000300%
Average result: 11.005374


[Wordman]

Here is the code, written in Python. Settings are at the bottom. Change "Repetitions" at the bottom to increase or decrease the number of repetitions made at each size. Change the MinPool and MaxPool to alter the possible range of pool sizes. Note that, being a simulation, you will get different results each time you run this code. If you want to repeat a result, you'll need to change random.seed() into something that provides a constant seed value.

This code is not particularly efficient, and probably contains bugs.

Code Select Expand

import random

def init_poolsize(poolsize,counts):
   results = {}
   results['totalrolls'] = 0
   results['totalsux'] = 0
   results['sux'] = {}
   counts[poolsize] = results
   
def rollpool(poolsize):
   successes = 0
   for die in range(0,poolsize + 1):
      roll = random.randint(1, 6)
      if roll >= 3:
         successes += 1
         if roll == 6:
            successes += rollpool(1)
   return successes

def run_poolsize(poolsize,counts,reps):
   results = counts[poolsize]
   for rep in range(0,reps + 1):
      sux = rollpool(poolsize)
      results['totalrolls'] += 1
      results['totalsux'] += sux
      try:
         results['sux'][sux] += 1
      except:
         results['sux'][sux] = 1

def print_poolsize(poolsize,counts):
   print
   print '-- Number of rolls generating n successes from a %i dice pool --' % (poolsize)
   results = counts[poolsize]
   sux = results['sux']
   
   # Find the highest number of successes on a single roll
   suxcounts = sux.keys()
   suxcounts.sort()
   maxresult = suxcounts[-1]
   
   # Iterator over all possible results
   totalrolls = results['totalrolls']
   for successes in range(0,maxresult + 1):
      count = 0
      try:
         count = results['sux'][successes]
      except:
         count = 0 # No roll generated this many successes
      percentage = (count * 100.0) / totalrolls
      print '%2i successes: %15i = %0.6f%%'  % (successes, count, percentage)
   
   averageresult = (1.0 * results['totalsux']) / totalrolls
   print 'Average result: %f' % (averageresult)
   
Counts = {}
Repititions = 1000000
MinPool = 1
MaxPool = 10
random.seed()
for poolsize in range(MinPool, MaxPool+1):
   init_poolsize(poolsize,Counts)
   run_poolsize(poolsize,Counts,Repititions)
   print_poolsize(poolsize,Counts)


[Guy Srinivasan]

I think this is a bug:

Code Select Expand

for die in range(0,poolsize + 1):
'cause it looks like it's rolling 1 too many dice here, and that jives with the averages you're getting. If a 6 means roll 2 extra dice, then each die would be worth A = 3/6 * 1 + 1/6 * (1+2*A) = 1 success, and rolling "N dice" would instead roll N+1 dice, yielding on average N+1 successes, like your simulation shows.

[Wordman]

I think this is a bug:

Code Select Expand

for die in range(0,poolsize + 1):
'cause it looks like it's rolling 1 too many dice here

Absolutely right. Serious bug, too. Should have been either range(0,poolsize) or range(1,poolsize + 1). Since you don't actually use the 'die' variable there, the first one would be better.

Since we still can't edit posts (for no satisfactorily explained reason), mentally add one pool sizes of the output data.

That will serve as a lesson to me to not post code I only spent 45 minutes on. D'oh!

[Guy Srinivasan]

Unfortunately it's not just a matter of using a different table...

Code Select Expand

if roll == 6:
            successes += rollpool(1)
makes a 6 explode into 2 dice rather than 1.

[Tyler.Tinsley]

I thought the numbers looked a little optimistic , still I'm quite impressed that you set up simulations to test the math. I would love to see some less optimistic 6 explodes as one die tests.

[Wordman]

Unfortunately it's not just a matter of using a different table...

Code Select Expand

if roll == 6:
            successes += rollpool(1)
makes a 6 explode into 2 dice rather than 1.

D'oh! Missed it by that much. What a difference two bloody characters makes.

Anyway, a rerun with the fix:

Code Select Expand


-- Number of rolls generating n successes from a 1 dice pool --
0 successes:          333360 = 33.335967%
1 successes:          555095 = 55.509444%
2 successes:           92972 =  9.297191%
3 successes:           15511 =  1.551098%
4 successes:            2567 =  0.256700%
5 successes:             417 =  0.041700%
6 successes:              72 =  0.007200%
7 successes:               6 =  0.000600%
8 successes:               1 =  0.000100%
Average result: 0.800406

-- Number of rolls generating n successes from a 2 dice pool --
0 successes:          111192 = 11.119189%
1 successes:          370383 = 37.038263%
2 successes:          370405 = 37.040463%
3 successes:          113023 = 11.302289%
4 successes:           27365 =  2.736497%
5 successes:            6099 =  0.609899%
6 successes:            1223 =  0.122300%
7 successes:             260 =  0.026000%
8 successes:              39 =  0.003900%
9 successes:              10 =  0.001000%
10 successes:               2 =  0.000200%
Average result: 1.599795

-- Number of rolls generating n successes from a 3 dice pool --
0 successes:           37337 =  3.733696%
1 successes:          185670 = 18.566981%
2 successes:          339561 = 33.956066%
3 successes:          278055 = 27.805472%
4 successes:          112507 = 11.250689%
5 successes:           34746 =  3.474597%
6 successes:            9275 =  0.927499%
7 successes:            2186 =  0.218600%
8 successes:             521 =  0.052100%
9 successes:             114 =  0.011400%
10 successes:              23 =  0.002300%
11 successes:               6 =  0.000600%
Average result: 2.399155

-- Number of rolls generating n successes from a 4 dice pool --
0 successes:           12358 =  1.235799%
1 successes:           81706 =  8.170592%
2 successes:          219625 = 21.962478%
3 successes:          299322 = 29.932170%
4 successes:          228133 = 22.813277%
5 successes:          104993 = 10.499290%
6 successes:           37954 =  3.795396%
7 successes:           11659 =  1.165899%
8 successes:            3162 =  0.316200%
9 successes:             811 =  0.081100%
10 successes:             214 =  0.021400%
11 successes:              45 =  0.004500%
12 successes:              14 =  0.001400%
13 successes:               5 =  0.000500%
Average result: 3.201216

-- Number of rolls generating n successes from a 5 dice pool --
0 successes:            4090 =  0.409000%
1 successes:           34337 =  3.433697%
2 successes:          119925 = 11.992488%
3 successes:          229183 = 22.918277%
4 successes:          263922 = 26.392174%
5 successes:          192198 = 19.219781%
6 successes:           98005 =  9.800490%
7 successes:           39006 =  3.900596%
8 successes:           13553 =  1.355299%
9 successes:            4117 =  0.411700%
10 successes:            1198 =  0.119800%
11 successes:             346 =  0.034600%
12 successes:              91 =  0.009100%
13 successes:              24 =  0.002400%
14 successes:               3 =  0.000300%
15 successes:               1 =  0.000100%
16 successes:               2 =  0.000200%
Average result: 4.002238

-- Number of rolls generating n successes from a 6 dice pool --
0 successes:            1379 =  0.137900%
1 successes:           13919 =  1.391899%
2 successes:           59608 =  5.960794%
3 successes:          146485 = 14.648485%
4 successes:          227298 = 22.729777%
5 successes:          233380 = 23.337977%
6 successes:          167841 = 16.784083%
7 successes:           89685 =  8.968491%
8 successes:           38973 =  3.897296%
9 successes:           14627 =  1.462699%
10 successes:            4795 =  0.479500%
11 successes:            1438 =  0.143800%
12 successes:             424 =  0.042400%
13 successes:             106 =  0.010600%
14 successes:              29 =  0.002900%
15 successes:              12 =  0.001200%
16 successes:               2 =  0.000200%
Average result: 4.797797

-- Number of rolls generating n successes from a 7 dice pool --
0 successes:             484 =  0.048400%
1 successes:            5236 =  0.523599%
2 successes:           27428 =  2.742797%
3 successes:           83077 =  8.307692%
4 successes:          163374 = 16.337384%
5 successes:          218731 = 21.873078%
6 successes:          209337 = 20.933679%
7 successes:          148116 = 14.811585%
8 successes:           82506 =  8.250592%
9 successes:           38397 =  3.839696%
10 successes:           15441 =  1.544098%
11 successes:            5369 =  0.536899%
12 successes:            1740 =  0.174000%
13 successes:             545 =  0.054500%
14 successes:             171 =  0.017100%
15 successes:              32 =  0.003200%
16 successes:              13 =  0.001300%
17 successes:               3 =  0.000300%
18 successes:               1 =  0.000100%
Average result: 5.599508

-- Number of rolls generating n successes from a 8 dice pool --
0 successes:             143 =  0.014300%
1 successes:            1969 =  0.196900%
2 successes:           12151 =  1.215099%
3 successes:           43503 =  4.350296%
4 successes:          103426 = 10.342590%
5 successes:          171391 = 17.139083%
6 successes:          207995 = 20.799479%
7 successes:          188533 = 18.853281%
8 successes:          133131 = 13.313087%
9 successes:           76267 =  7.626692%
10 successes:           37045 =  3.704496%
11 successes:           15530 =  1.552998%
12 successes:            5914 =  0.591399%
13 successes:            2047 =  0.204700%
14 successes:             669 =  0.066900%
15 successes:             199 =  0.019900%
16 successes:              65 =  0.006500%
17 successes:              16 =  0.001600%
18 successes:               5 =  0.000500%
19 successes:               0 =  0.000000%
20 successes:               1 =  0.000100%
21 successes:               1 =  0.000100%
Average result: 6.399238

-- Number of rolls generating n successes from a 9 dice pool --
0 successes:              53 =  0.005300%
1 successes:             784 =  0.078400%
2 successes:            5299 =  0.529899%
3 successes:           21505 =  2.150498%
4 successes:           59738 =  5.973794%
5 successes:          118520 = 11.851988%
6 successes:          174653 = 17.465283%
7 successes:          195644 = 19.564380%
8 successes:          172489 = 17.248883%
9 successes:          120243 = 12.024288%
10 successes:           70248 =  7.024793%
11 successes:           35407 =  3.540696%
12 successes:           15638 =  1.563798%
13 successes:            6276 =  0.627599%
14 successes:            2343 =  0.234300%
15 successes:             785 =  0.078500%
16 successes:             276 =  0.027600%
17 successes:              70 =  0.007000%
18 successes:              24 =  0.002400%
19 successes:               5 =  0.000500%
20 successes:               1 =  0.000100%
Average result: 7.198898

-- Number of rolls generating n successes from a 10 dice pool --
0 successes:              15 =  0.001500%
1 successes:             263 =  0.026300%
2 successes:            2193 =  0.219300%
3 successes:           10228 =  1.022799%
4 successes:           32470 =  3.246997%
5 successes:           74648 =  7.464793%
6 successes:          130646 = 13.064587%
7 successes:          174249 = 17.424883%
8 successes:          183572 = 18.357182%
9 successes:          156223 = 15.622284%
10 successes:          110777 = 11.077689%
11 successes:           65321 =  6.532093%
12 successes:           33589 =  3.358897%
13 successes:           15443 =  1.544298%
14 successes:            6373 =  0.637299%
15 successes:            2619 =  0.261900%
16 successes:             944 =  0.094400%
17 successes:             294 =  0.029400%
18 successes:              89 =  0.008900%
19 successes:              34 =  0.003400%
20 successes:               7 =  0.000700%
21 successes:               4 =  0.000400%
Average result: 7.997856


[chaviro]

Hi,

I cannot resist a math problem 

Assuming that your single dice works as follows
1-2 nothing
3-6 1 success + reroll if 6

The simplest way to compute the odds of success is, to me, to compute the
probability distribution for a single die. For n dice, you'll just use the
binomial law (don't worry, I'll do the maths below).
So, let's proceed.

With a single die, the distribution is as follows

2/6 : 0 success
1/6*(3 + 2/6) : 1 success
1/6*1/6*(3 + 2/6): 2 success
...
1/(6 to the n)*(3 + 2/6) : n success

The fact is that, contrary to what one usually thinks (feels), the exploding
die has few consequences on the odds ; a little biais up, no more.

Let's call E(1, n) the above probability of getting n success with a die, and
E(0,n) = 0 (and you know that E(k,0) = (2/6 to the k)).  From this, you may
compute the odds of getting E(k,n), the exact probability of getting n success
with k dice recursively :

E(k,n) = k times the sum for i = 1 to n of E(1,n-i)*E(k-1,i)

Well, I suppose this does not help you as much as a bunch of numbers and a nice
curve, so I give you my results, which I hope are correct (I'm rather sure of myself on this one, though) :

1 die
0       0.3333%
1       0.5556%
2       0.0926%
3       0.0154%
4       0.0026%
5       0.0004%

2 dice
0       0.1111%
1       0.3704%
2       0.3704%
3       0.1132%
4       0.0274%
5       0.0060%
6       0.0012%
7       0.0002%

3 dice
0       0.0370%
1       0.1852%
2       0.3395%
3       0.2795%
4       0.1123%
5       0.0344%
6       0.0092%
7       0.0022%
8       0.0005%
9       0.0001%

4 dice
0       0.0123%
1       0.0823%
2       0.2195%
3       0.2995%
4       0.2271%
5       0.1055%
6       0.0378%
7       0.0116%
8       0.0032%
9       0.0008%
10      0.0002%

5 dice
0       0.0041%
1       0.0343%
2       0.1200%
3       0.2296%
4       0.2637%
5       0.1927%
6       0.0975%
7       0.0390%
8       0.0134%
9       0.0041%
10      0.0012%
11      0.0003%

6 dice
0       0.0014%
1       0.0137%
2       0.0594%
3       0.1464%
4       0.2271%
5       0.2339%
6       0.1678%
7       0.0897%
8       0.0389%
9       0.0146%
10      0.0049%
11      0.0015%
12      0.0004%
13      0.0001%

7 dice
0       0.0005%
1       0.0053%
2       0.0276%
3       0.0831%
4       0.1628%
5       0.2187%
6       0.2093%
7       0.1487%
8       0.0826%
9       0.0381%
10      0.0153%
11      0.0055%
12      0.0018%
13      0.0006%
14      0.0002%

8 dice
0       0.0002%
1       0.0020%
2       0.0122%
3       0.0435%
4       0.1031%
5       0.1715%
6       0.2077%
7       0.1889%
8       0.1334%
9       0.0762%
10      0.0368%
11      0.0156%
12      0.0059%
13      0.0021%
14      0.0007%
15      0.0002%

9 dice
1       0.0008%
2       0.0052%
3       0.0215%
4       0.0597%
5       0.1186%
6       0.1747%
7       0.1959%
8       0.1715%
9       0.1207%
10      0.0705%
11      0.0354%
12      0.0157%
13      0.0063%
14      0.0023%
15      0.0008%
16      0.0003%

10 dice
1       0.0003%
2       0.0022%
3       0.0101%
4       0.0323%
5       0.0748%
6       0.1300%
7       0.1743%
8       0.1842%
9       0.1567%
10      0.1100%
11      0.0654%
12      0.0338%
13      0.0156%
14      0.0065%
15      0.0025%
16      0.0009%
17      0.0003%

At least, they seem to match the numbers from the Monte Carlo method (and
should be more accurate, especially on the lower and higher ranges of the
curve). Note that I truncated probabilities which are less than 0.0001%.

Chaviro