[Chronicles of Darkness] Dice Probability & Formulas

Dice Probability in the Chronicles of Darkness

Originally posted on http://nwod.org/wiki/index.php/Probability_Math

Methods of Calculating, New World of Darkness

Probabilities for a single die can be calculated directly. (See below for discussion of these probabilities in the new World of Darkness.

P(at least S successes on n dice)=P(at least S successes on n-1 dice)+sum_{s=0}^{S-1} P(at least s successes on n-1 dice)P(exactly S-s successes on 1 die)

Some simple formulas:

A few things are simple enough to calculate on the fly.

If all rolls of r or better are rerolled, then the average number of successes obtained when rolling n dice is 3n/(r-1). (In particular, if 10s and only 10s are rerolled, then the average is 3n/(10-1)=n/3, and if no numbers are rerolled, then the average is 3n/(11-1)=0.3n. When using formulas, the easiest way to describe the no-rerolls situation is to call it "11-again" rather than "none-again".)

If failures are rerolled once ("rote actions"), then the average number of successes is 5.1n/(r-1) (usually 0.51n, under 10-again).

For non-rote actions, the probability of rolling no successes on n dice is 0.7^n, and so the probability of rolling at least one success is 1-0.7^n. For rote actions, failures may be rerolled once, and so the probability of rolling at least one success is 1-0.7^(2n).

With one die, the probability of rolling at least 1 success is 0.51 (rote actions) or 0.3 (anything else).

The probability of rolling at least S successes on one die under r-again, for S>1, is (11-r)/3 * 0.3 * P(at least S-1 successes) = 0.3*(1.1-r/10)^(S-1) (for non-rote actions) or 0.51*(1.1-r/10)^(S-1) (for rote actions).

10s again:

This chart shows the percentage chance of rolling at least S successes with a pool of n dice under the normal New World of Darkness rules, as described in nWoD, page 32.

Successes	1	2	3	4	5	6	7	8	9	10	11	12
1 die	30.00	3.00	0.30	0.03	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
2 dice	51.00	13.20	2.13	0.29	0.03	0.00	0.00	0.00	0.00	0.00	0.00	0.00
3 dice	65.70	26.01	6.73	1.30	0.21	0.03	0.00	0.00	0.00	0.00	0.00	0.00
4 dice	75.99	38.94	13.80	3.64	0.77	0.14	0.02	0.00	0.00	0.00	0.00	0.00
5 dice	83.19	50.77	22.53	7.56	2.02	0.45	0.09	0.01	0.00	0.00	0.00	0.00
6 dice	88.23	61.00	32.03	13.00	4.23	1.14	0.27	0.05	0.01	0.00	0.00	0.00
7 dice	91.76	69.52	41.57	19.66	7.53	2.40	0.65	0.15	0.03	0.00	0.00	0.00
8 dice	94.23	76.44	50.65	27.14	11.91	4.38	1.38	0.38	0.09	0.02	0.00	0.00
9 dice	95.96	81.95	58.94	35.02	17.27	7.17	2.56	0.79	0.22	0.05	0.01	0.00
10 dice	97.17	86.27	66.27	42.93	23.39	10.81	4.30	1.50	0.46	0.13	0.03	0.00
11 dice	98.02	89.63	72.61	50.57	30.01	15.25	6.70	2.58	0.88	0.27	0.07	0.01
12 dice	98.61	92.20	77.97	57.72	36.89	20.37	9.78	4.12	1.54	0.52	0.16	0.04
13 dice	99.03	94.17	82.44	64.24	43.79	26.01	13.52	6.19	2.53	0.92	0.31	0.09
14 dice	99.32	95.65	86.11	70.07	50.51	32.02	17.87	8.83	3.89	1.54	0.55	0.18
15 dice	99.52	96.77	89.08	75.18	56.89	38.20	22.73	12.03	5.69	2.43	0.94	0.33

9s again:

This chart shows the percentage chance of rolling at least S successes with a pool of n dice under the 9-again system, as described nWoD, page 134.

Successes	1	2	3	4	5	6	7	8	9	10	11	12
1 die	30.00	6.00	1.20	0.24	0.04	0.00	0.00	0.00	0.00	0.00	0.00	0.00
2 dice	51.00	17.40	4.92	1.27	0.31	0.07	0.01	0.00	0.00	0.00	0.00	0.00
3 dice	65.70	30.42	11.26	3.63	1.07	0.29	0.07	0.02	0.00	0.00	0.00	0.00
4 dice	75.99	43.06	19.54	7.58	2.63	0.84	0.25	0.07	0.02	0.00	0.00	0.00
5 dice	83.19	54.38	28.86	13.03	5.20	1.89	0.64	0.20	0.06	0.01	0.00	0.00
6 dice	88.23	64.03	38.44	19.69	8.88	3.62	1.36	0.47	0.15	0.05	0.01	0.00
7 dice	91.76	71.99	47.71	27.17	13.62	6.15	2.54	0.98	0.35	0.12	0.04	0.01
8 dice	94.23	78.42	56.28	35.05	19.26	9.52	4.30	1.80	0.70	0.26	0.09	0.03
9 dice	95.96	83.51	63.94	42.95	25.58	13.70	6.70	3.03	1.28	0.51	0.19	0.07
10 dice	97.17	87.49	70.61	50.58	32.32	18.61	9.78	4.75	2.15	0.91	0.37	0.14
11 dice	98.02	90.56	76.28	57.72	39.22	24.11	13.53	7.01	3.38	1.53	0.65	0.26
12 dice	98.61	92.92	81.04	64.24	46.08	30.01	17.88	9.83	5.03	2.41	1.09	0.47
13 dice	99.03	94.71	84.96	70.07	52.69	36.15	22.75	13.22	7.15	3.62	1.73	0.78
14 dice	99.32	96.06	88.16	75.17	58.93	42.36	28.01	17.13	9.75	5.20	2.61	1.24
15 dice	99.52	97.08	90.73	79.58	64.68	48.48	33.54	21.50	12.84	7.19	3.79	1.89

8s again:

This chart shows the percentage chance of rolling at least S successes with a pool of n dice under the 8-again system, as described nWoD, page 134.

Successes	1	2	3	4	5	6	7	8	9	10	11	12
1 die	30.00	9.00	2.70	0.81	0.24	0.07	0.02	0.00	0.00	0.00	0.00	0.00
2 dice	51.00	21.60	8.37	3.07	1.09	0.37	0.12	0.04	0.01	0.00	0.00	0.00
3 dice	65.70	34.83	16.30	7.04	2.87	1.12	0.42	0.15	0.05	0.02	0.00	0.00
4 dice	75.99	47.17	25.56	12.60	5.79	2.52	1.05	0.42	0.16	0.06	0.02	0.00
5 dice	83.19	57.98	35.29	19.41	9.88	4.73	2.16	0.94	0.40	0.16	0.06	0.02
6 dice	88.23	67.05	44.82	27.03	15.02	7.82	3.86	1.82	0.82	0.36	0.15	0.06
7 dice	91.76	74.47	53.71	35.03	21.03	11.78	6.23	3.14	1.52	0.71	0.32	0.14
8 dice	94.23	80.39	61.72	43.04	27.63	16.53	9.32	5.00	2.56	1.26	0.60	0.28
9 dice	95.96	85.06	68.72	50.74	34.56	21.94	13.11	7.43	4.02	2.09	1.05	0.51
10 dice	97.17	88.70	74.71	57.93	41.57	27.83	17.53	10.46	5.95	3.25	1.71	0.87
11 dice	98.02	91.49	79.75	64.48	48.45	34.02	22.47	14.06	8.39	4.79	2.63	1.40
12 dice	98.61	93.63	83.91	70.31	55.00	40.31	27.83	18.19	11.33	6.75	3.87	2.14
13 dice	99.03	95.25	87.31	75.41	61.13	46.56	33.44	22.77	14.76	9.15	5.46	3.13
14 dice	99.32	96.47	90.06	79.80	66.73	52.61	39.19	27.70	18.64	12.00	7.42	4.42
15 dice	99.52	97.38	92.26	83.54	71.77	58.36	44.94	32.87	22.91	15.27	9.78	6.03

Rote actions:

This chart shows the percentage chance of rolling at least S successes with a pool of n dice under the rote-action system, as described nWoD, page 134.

Successes	1	2	3	4	5	6	7	8	9	10	11	12
1 die	51.00	5.10	0.51	0.05	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
2 dice	75.99	31.00	5.44	0.77	0.10	0.01	0.00	0.00	0.00	0.00	0.00	0.00
3 dice	88.23	55.17	20.89	4.70	0.83	0.13	0.01	0.00	0.00	0.00	0.00	0.00
4 dice	94.23	72.63	40.12	14.88	3.82	0.79	0.14	0.02	0.00	0.00	0.00	0.00
5 dice	97.17	83.94	57.83	29.52	10.93	3.04	0.69	0.13	0.02	0.00	0.00	0.00
6 dice	98.61	90.83	71.84	45.36	21.99	8.17	2.41	0.59	0.12	0.02	0.00	0.00
7 dice	99.32	94.87	81.93	59.87	35.36	16.56	6.18	1.89	0.49	0.11	0.02	0.00
8 dice	99.66	97.17	88.76	71.80	49.06	27.52	12.57	4.72	1.49	0.41	0.10	0.02
9 dice	99.83	98.46	93.18	80.89	61.57	39.75	21.42	9.61	3.63	1.17	0.33	0.08
10 dice	99.92	99.17	95.94	87.43	72.08	51.93	31.99	16.69	7.39	2.80	0.92	0.27
11 dice	99.96	99.55	97.63	91.94	80.36	63.03	43.27	25.62	13.02	5.70	2.17	0.73
12 dice	99.98	99.76	98.63	94.94	86.57	72.49	54.29	35.72	20.46	10.18	4.42	1.69
13 dice	99.99	99.87	99.22	96.88	91.03	80.12	64.34	46.20	29.29	16.30	7.97	3.44
14 dice	99.99	99.93	99.56	98.11	94.14	85.99	72.97	56.31	38.92	23.89	12.98	6.25
15 dice	99.99	99.96	99.75	98.87	96.24	90.36	80.05	65.52	48.71	32.54	19.41	10.32