Math Help

[Vondy]

I'm hoping someone with stronger math skills than me can lend me a hand. I'm fiddling with a dice pool system using D6's whereby the highest die is the result, but additional sixes beyond the first (presuming that you get one), add one to the total. So a 2d6 roll that resulting in 6 and 6 would deliver a result of 7. I have the probabilities for that (and MoS for opposed rolls).

But I want to work with reality distortion levels. I need probabilities for the same system, but where every 5 or 6 adds 1 to the result. So if I rolled four dice and ended up with 3, 5, 5, and six my total would be 8. Or even where 4-6 skewed the number upward.

Do we have any math hounds that can do this faster than me? I don't need chances for each result as much as I do chances of success for each target number up to 12.

[Cancer]

I did this brute-force, only going up to 9 dice and target number of 12. Let me know if, no, how I blundered.

Results for 2 dice with a promotion threshold of 6
Reach target number 6 in 11 of 36 cases. (30.556%)
Reach target number 7 in 1 of 36 cases. (2.778%)
Reach target number 8 in 0 of 36 cases. (0.000%)
Reach target number 9 in 0 of 36 cases. (0.000%)
Reach target number 10 in 0 of 36 cases. (0.000%)
Reach target number 11 in 0 of 36 cases. (0.000%)
Reach target number 12 in 0 of 36 cases. (0.000%)
Results for 2 dice with a promotion threshold of 5
Reach target number 6 in 12 of 36 cases. (33.333%)
Reach target number 7 in 3 of 36 cases. (8.333%)
Reach target number 8 in 0 of 36 cases. (0.000%)
Reach target number 9 in 0 of 36 cases. (0.000%)
Reach target number 10 in 0 of 36 cases. (0.000%)
Reach target number 11 in 0 of 36 cases. (0.000%)
Reach target number 12 in 0 of 36 cases. (0.000%)
Results for 2 dice with a promotion threshold of 4
Reach target number 6 in 14 of 36 cases. (38.889%)
Reach target number 7 in 5 of 36 cases. (13.889%)
Reach target number 8 in 0 of 36 cases. (0.000%)
Reach target number 9 in 0 of 36 cases. (0.000%)
Reach target number 10 in 0 of 36 cases. (0.000%)
Reach target number 11 in 0 of 36 cases. (0.000%)
Reach target number 12 in 0 of 36 cases. (0.000%)
Results for 3 dice with a promotion threshold of 6
Reach target number 6 in 91 of 216 cases. (42.130%)
Reach target number 7 in 16 of 216 cases. (7.407%)
Reach target number 8 in 1 of 216 cases. (0.463%)
Reach target number 9 in 0 of 216 cases. (0.000%)
Reach target number 10 in 0 of 216 cases. (0.000%)
Reach target number 11 in 0 of 216 cases. (0.000%)
Reach target number 12 in 0 of 216 cases. (0.000%)
Results for 3 dice with a promotion threshold of 5
Reach target number 6 in 104 of 216 cases. (48.148%)
Reach target number 7 in 44 of 216 cases. (20.370%)
Reach target number 8 in 7 of 216 cases. (3.241%)
Reach target number 9 in 0 of 216 cases. (0.000%)
Reach target number 10 in 0 of 216 cases. (0.000%)
Reach target number 11 in 0 of 216 cases. (0.000%)
Reach target number 12 in 0 of 216 cases. (0.000%)
Results for 3 dice with a promotion threshold of 4
Reach target number 6 in 126 of 216 cases. (58.333%)
Reach target number 7 in 71 of 216 cases. (32.870%)
Reach target number 8 in 19 of 216 cases. (8.796%)
Reach target number 9 in 0 of 216 cases. (0.000%)
Reach target number 10 in 0 of 216 cases. (0.000%)
Reach target number 11 in 0 of 216 cases. (0.000%)
Reach target number 12 in 0 of 216 cases. (0.000%)
Results for 4 dice with a promotion threshold of 6
Reach target number 6 in 671 of 1296 cases. (51.775%)
Reach target number 7 in 171 of 1296 cases. (13.194%)
Reach target number 8 in 21 of 1296 cases. (1.620%)
Reach target number 9 in 1 of 1296 cases. (0.077%)
Reach target number 10 in 0 of 1296 cases. (0.000%)
Reach target number 11 in 0 of 1296 cases. (0.000%)
Reach target number 12 in 0 of 1296 cases. (0.000%)
Results for 4 dice with a promotion threshold of 5
Reach target number 6 in 784 of 1296 cases. (60.494%)
Reach target number 7 in 432 of 1296 cases. (33.333%)
Reach target number 8 in 128 of 1296 cases. (9.877%)
Reach target number 9 in 15 of 1296 cases. (1.157%)
Reach target number 10 in 0 of 1296 cases. (0.000%)
Reach target number 11 in 0 of 1296 cases. (0.000%)
Reach target number 12 in 0 of 1296 cases. (0.000%)
Results for 4 dice with a promotion threshold of 4
Reach target number 6 in 945 of 1296 cases. (72.917%)
Reach target number 7 in 663 of 1296 cases. (51.157%)
Reach target number 8 in 308 of 1296 cases. (23.765%)
Reach target number 9 in 65 of 1296 cases. (5.015%)
Reach target number 10 in 0 of 1296 cases. (0.000%)
Reach target number 11 in 0 of 1296 cases. (0.000%)
Reach target number 12 in 0 of 1296 cases. (0.000%)
Results for 5 dice with a promotion threshold of 6
Reach target number 6 in 4651 of 7776 cases. (59.812%)
Reach target number 7 in 1526 of 7776 cases. (19.624%)
Reach target number 8 in 276 of 7776 cases. (3.549%)
Reach target number 9 in 26 of 7776 cases. (0.334%)
Reach target number 10 in 1 of 7776 cases. (0.013%)
Reach target number 11 in 0 of 7776 cases. (0.000%)
Reach target number 12 in 0 of 7776 cases. (0.000%)
Results for 5 dice with a promotion threshold of 5
Reach target number 6 in 5472 of 7776 cases. (70.370%)
Reach target number 7 in 3552 of 7776 cases. (45.679%)
Reach target number 8 in 1472 of 7776 cases. (18.930%)
Reach target number 9 in 332 of 7776 cases. (4.270%)
Reach target number 10 in 31 of 7776 cases. (0.399%)
Reach target number 11 in 0 of 7776 cases. (0.000%)
Reach target number 12 in 0 of 7776 cases. (0.000%)
Results for 5 dice with a promotion threshold of 4
Reach target number 6 in 6453 of 7776 cases. (82.986%)
Reach target number 7 in 5148 of 7776 cases. (66.204%)
Reach target number 8 in 3153 of 7776 cases. (40.548%)
Reach target number 9 in 1217 of 7776 cases. (15.651%)
Reach target number 10 in 211 of 7776 cases. (2.713%)
Reach target number 11 in 0 of 7776 cases. (0.000%)
Reach target number 12 in 0 of 7776 cases. (0.000%)
Results for 6 dice with a promotion threshold of 6
Reach target number 6 in 31031 of 46656 cases. (66.510%)
Reach target number 7 in 12281 of 46656 cases. (26.322%)
Reach target number 8 in 2906 of 46656 cases. (6.229%)
Reach target number 9 in 406 of 46656 cases. (0.870%)
Reach target number 10 in 31 of 46656 cases. (0.066%)
Reach target number 11 in 1 of 46656 cases. (0.002%)
Reach target number 12 in 0 of 46656 cases. (0.000%)
Results for 6 dice with a promotion threshold of 5
Reach target number 6 in 36416 of 46656 cases. (78.052%)
Reach target number 7 in 26432 of 46656 cases. (56.653%)
Reach target number 8 in 13632 of 46656 cases. (29.218%)
Reach target number 9 in 4432 of 46656 cases. (9.499%)
Reach target number 10 in 808 of 46656 cases. (1.732%)
Reach target number 11 in 63 of 46656 cases. (0.135%)
Reach target number 12 in 0 of 46656 cases. (0.000%)
Results for 6 dice with a promotion threshold of 4
Reach target number 6 in 41796 of 46656 cases. (89.583%)
Reach target number 7 in 36153 of 46656 cases. (77.488%)
Reach target number 8 in 26163 of 46656 cases. (56.076%)
Reach target number 9 in 13860 of 46656 cases. (29.707%)
Reach target number 10 in 4526 of 46656 cases. (9.701%)
Reach target number 11 in 665 of 46656 cases. (1.425%)
Reach target number 12 in 0 of 46656 cases. (0.000%)
Results for 7 dice with a promotion threshold of 6
Reach target number 6 in 201811 of 279936 cases. (72.092%)
Reach target number 7 in 92436 of 279936 cases. (33.020%)
Reach target number 8 in 26811 of 279936 cases. (9.578%)
Reach target number 9 in 4936 of 279936 cases. (1.763%)
Reach target number 10 in 561 of 279936 cases. (0.200%)
Reach target number 11 in 36 of 279936 cases. (0.013%)
Reach target number 12 in 1 of 279936 cases. (0.000%)
Results for 7 dice with a promotion threshold of 5
Reach target number 6 in 234880 of 279936 cases. (83.905%)
Reach target number 7 in 184704 of 279936 cases. (65.981%)
Reach target number 8 in 111232 of 279936 cases. (39.735%)
Reach target number 9 in 46272 of 279936 cases. (16.529%)
Reach target number 10 in 12336 of 279936 cases. (4.407%)
Reach target number 11 in 1892 of 279936 cases. (0.676%)
Reach target number 12 in 127 of 279936 cases. (0.045%)
Results for 7 dice with a promotion threshold of 4
Reach target number 6 in 262440 of 279936 cases. (93.750%)
Reach target number 7 in 239193 of 279936 cases. (85.446%)
Reach target number 8 in 192888 of 279936 cases. (68.904%)
Reach target number 9 in 124659 of 279936 cases. (44.531%)
Reach target number 10 in 57354 of 279936 cases. (20.488%)
Reach target number 11 in 16151 of 279936 cases. (5.770%)
Reach target number 12 in 2059 of 279936 cases. (0.736%)
Results for 8 dice with a promotion threshold of 6
Reach target number 6 in 1288991 of 1679616 cases. (76.743%)
Reach target number 7 in 663991 of 1679616 cases. (39.532%)
Reach target number 8 in 226491 of 1679616 cases. (13.485%)
Reach target number 9 in 51491 of 1679616 cases. (3.066%)
Reach target number 10 in 7741 of 1679616 cases. (0.461%)
Reach target number 11 in 741 of 1679616 cases. (0.044%)
Reach target number 12 in 41 of 1679616 cases. (0.002%)
Results for 8 dice with a promotion threshold of 5
Reach target number 6 in 1483008 of 1679616 cases. (88.294%)
Reach target number 7 in 1237248 of 1679616 cases. (73.663%)
Reach target number 8 in 835840 of 1679616 cases. (49.764%)
Reach target number 9 in 416512 of 1679616 cases. (24.798%)
Reach target number 10 in 144128 of 1679616 cases. (8.581%)
Reach target number 11 in 32576 of 1679616 cases. (1.939%)
Reach target number 12 in 4320 of 1679616 cases. (0.257%)
Results for 8 dice with a promotion threshold of 4
Reach target number 6 in 1617651 of 1679616 cases. (96.311%)
Reach target number 7 in 1525311 of 1679616 cases. (90.813%)
Reach target number 8 in 1322325 of 1679616 cases. (78.728%)
Reach target number 9 in 977211 of 1679616 cases. (58.181%)
Reach target number 10 in 561537 of 1679616 cases. (33.432%)
Reach target number 11 in 226605 of 1679616 cases. (13.491%)
Reach target number 12 in 55976 of 1679616 cases. (3.333%)

[Vondy]

Thanks - that is truly repworthy.

12 is the max difficulty in mind, and I was wavering between 7-8 dice for dice pool limits. The number will be modified by stats, but they will have a smaller range -3 to +3 with 0 being "joe average."

[Vondy]

Actually... I don't think I expressed myself well. I had the odds for target numbers of 1-12 for a promotion schedule of 6, but not for promotion schedules of 5 or 4. Do you have the chances of reaching target numbers 1-6 for promotion schedules of 5 or 4?

[Cancer]

A target number of 1 is always reached, right?

And you only fail to hit a target number of 2 when you roll 1's on all dice (since the promotion threshold is never as low as 1)?

Changing a couple of numbers in the script ....

Results for 2 dice with a promotion threshold of 6
Reach target number 2 in 35 of 36 cases. (97.222%)
Reach target number 3 in 32 of 36 cases. (88.889%)
Reach target number 4 in 27 of 36 cases. (75.000%)
Reach target number 5 in 20 of 36 cases. (55.556%)
Results for 2 dice with a promotion threshold of 5
Reach target number 2 in 35 of 36 cases. (97.222%)
Reach target number 3 in 32 of 36 cases. (88.889%)
Reach target number 4 in 27 of 36 cases. (75.000%)
Reach target number 5 in 20 of 36 cases. (55.556%)
Results for 2 dice with a promotion threshold of 4
Reach target number 2 in 35 of 36 cases. (97.222%)
Reach target number 3 in 32 of 36 cases. (88.889%)
Reach target number 4 in 27 of 36 cases. (75.000%)
Reach target number 5 in 21 of 36 cases. (58.333%)
Results for 3 dice with a promotion threshold of 6
Reach target number 2 in 215 of 216 cases. (99.537%)
Reach target number 3 in 208 of 216 cases. (96.296%)
Reach target number 4 in 189 of 216 cases. (87.500%)
Reach target number 5 in 152 of 216 cases. (70.370%)
Results for 3 dice with a promotion threshold of 5
Reach target number 2 in 215 of 216 cases. (99.537%)
Reach target number 3 in 208 of 216 cases. (96.296%)
Reach target number 4 in 189 of 216 cases. (87.500%)
Reach target number 5 in 152 of 216 cases. (70.370%)
Results for 3 dice with a promotion threshold of 4
Reach target number 2 in 215 of 216 cases. (99.537%)
Reach target number 3 in 208 of 216 cases. (96.296%)
Reach target number 4 in 189 of 216 cases. (87.500%)
Reach target number 5 in 162 of 216 cases. (75.000%)
Results for 4 dice with a promotion threshold of 6
Reach target number 2 in 1295 of 1296 cases. (99.923%)
Reach target number 3 in 1280 of 1296 cases. (98.765%)
Reach target number 4 in 1215 of 1296 cases. (93.750%)
Reach target number 5 in 1040 of 1296 cases. (80.247%)
Results for 4 dice with a promotion threshold of 5
Reach target number 2 in 1295 of 1296 cases. (99.923%)
Reach target number 3 in 1280 of 1296 cases. (98.765%)
Reach target number 4 in 1215 of 1296 cases. (93.750%)
Reach target number 5 in 1040 of 1296 cases. (80.247%)
Results for 4 dice with a promotion threshold of 4
Reach target number 2 in 1295 of 1296 cases. (99.923%)
Reach target number 3 in 1280 of 1296 cases. (98.765%)
Reach target number 4 in 1215 of 1296 cases. (93.750%)
Reach target number 5 in 1107 of 1296 cases. (85.417%)
Results for 5 dice with a promotion threshold of 6
Reach target number 2 in 7775 of 7776 cases. (99.987%)
Reach target number 3 in 7744 of 7776 cases. (99.588%)
Reach target number 4 in 7533 of 7776 cases. (96.875%)
Reach target number 5 in 6752 of 7776 cases. (86.831%)
Results for 5 dice with a promotion threshold of 5
Reach target number 2 in 7775 of 7776 cases. (99.987%)
Reach target number 3 in 7744 of 7776 cases. (99.588%)
Reach target number 4 in 7533 of 7776 cases. (96.875%)
Reach target number 5 in 6752 of 7776 cases. (86.831%)
Results for 5 dice with a promotion threshold of 4
Reach target number 2 in 7775 of 7776 cases. (99.987%)
Reach target number 3 in 7744 of 7776 cases. (99.588%)
Reach target number 4 in 7533 of 7776 cases. (96.875%)
Reach target number 5 in 7128 of 7776 cases. (91.667%)
Results for 6 dice with a promotion threshold of 6
Reach target number 2 in 46655 of 46656 cases. (99.998%)
Reach target number 3 in 46592 of 46656 cases. (99.863%)
Reach target number 4 in 45927 of 46656 cases. (98.438%)
Reach target number 5 in 42560 of 46656 cases. (91.221%)
Results for 6 dice with a promotion threshold of 5
Reach target number 2 in 46655 of 46656 cases. (99.998%)
Reach target number 3 in 46592 of 46656 cases. (99.863%)
Reach target number 4 in 45927 of 46656 cases. (98.438%)
Reach target number 5 in 42560 of 46656 cases. (91.221%)
Results for 6 dice with a promotion threshold of 4
Reach target number 2 in 46655 of 46656 cases. (99.998%)
Reach target number 3 in 46592 of 46656 cases. (99.863%)
Reach target number 4 in 45927 of 46656 cases. (98.438%)
Reach target number 5 in 44469 of 46656 cases. (95.313%)
Results for 7 dice with a promotion threshold of 6
Reach target number 2 in 279935 of 279936 cases. (100.000%)
Reach target number 3 in 279808 of 279936 cases. (99.954%)
Reach target number 4 in 277749 of 279936 cases. (99.219%)
Reach target number 5 in 263552 of 279936 cases. (94.147%)
Results for 7 dice with a promotion threshold of 5
Reach target number 2 in 279935 of 279936 cases. (100.000%)
Reach target number 3 in 279808 of 279936 cases. (99.954%)
Reach target number 4 in 277749 of 279936 cases. (99.219%)
Reach target number 5 in 263552 of 279936 cases. (94.147%)
Results for 7 dice with a promotion threshold of 4
Reach target number 2 in 279935 of 279936 cases. (100.000%)
Reach target number 3 in 279808 of 279936 cases. (99.954%)
Reach target number 4 in 277749 of 279936 cases. (99.219%)
Reach target number 5 in 272646 of 279936 cases. (97.396%)
Results for 8 dice with a promotion threshold of 6
Reach target number 2 in 1679615 of 1679616 cases. (100.000%)
Reach target number 3 in 1679360 of 1679616 cases. (99.985%)
Reach target number 4 in 1673055 of 1679616 cases. (99.609%)
Reach target number 5 in 1614080 of 1679616 cases. (96.098%)
Results for 8 dice with a promotion threshold of 5
Reach target number 2 in 1679615 of 1679616 cases. (100.000%)
Reach target number 3 in 1679360 of 1679616 cases. (99.985%)
Reach target number 4 in 1673055 of 1679616 cases. (99.609%)
Reach target number 5 in 1614080 of 1679616 cases. (96.098%)
Results for 8 dice with a promotion threshold of 4
Reach target number 2 in 1679615 of 1679616 cases. (100.000%)
Reach target number 3 in 1679360 of 1679616 cases. (99.985%)
Reach target number 4 in 1673055 of 1679616 cases. (99.609%)
Reach target number 5 in 1655559 of 1679616 cases. (98.568%)

[Vondy]

A target number of 1 is always reached, right?

And you only fail to hit a target number of 2 when you roll 1's on all dice (since the promotion threshold is never as low as 1)?


Correct - and thank you, again.