Probability Math

Methods of Calculating

Probabilities for a single die can be calculated directly. (See below for discussion of these probabilities in the new World of Darkness; in the old World of Darkness, they are obvious.)

Subsequent probabilities were calculated using a spreadsheet program, using the following formulae:

(nWoD)

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

(oWoD)

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

(Rolling a 1 is considered to be rolling -1 successes.)

New World of Darkness

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).

10 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 the World of Darkness Rulebook, 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

9 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 World of Darkness Rulebook, 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

8 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 World of Darkness Rulebook, 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 World of Darkness Rulebook, 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

Other systems

For none-again, please see User:Infornographer/Probability Math. For rolls involving a Gangrel's mental traits, please see the Old World of Darkness tables under difficulty 8.

Old World of Darkness

The probabilities in this section are based on the rules described in Mage: the Ascension, Revised Edition, page 214. As in the previous section, probabilities are the percentage chance of rolling at least S successes on n dice at difficulty d.

Probabilities for rolling successes

1 die

Difficulty	2	3	4	5	6	7	8	9	10
1 success	90	80	70	60	50	40	30	20	10

2 dice

Difficulty	2	3	4	5	6	7	8	9	10
1 success	81	80	77	72	65	56	45	32	17
2 successes	81	64	49	36	25	16	9	4	1

3 dice

Difficulty	2	3	4	5	6	7	8	9	10
1 success	97.2	92	86.8	81	74	65.2	54	39.8	22
2 successes	72.9	70.4	63.7	54	42.5	30.4	18.9	9.2	2.5
3 successes	72.9	51.2	34.3	21.6	12.5	6.4	2.7	0.8	0.1

4 dice

Difficulty	2	3	4	5	6	7	8	9	10
1 success	94.77	93.76	90.93	86.4	80.05	71.52	60.21	45.28	25.65
2 successes	94.77	85.76	76.93	66.96	55.25	41.92	27.81	14.48	4.21
3 successes	65.61	61.44	51.45	38.88	26.25	15.36	7.29	2.4	0.33
4 successes	65.61	40.96	24.01	12.96	6.25	2.56	0.81	0.16	0.01

5 dice

Difficulty	2	3	4	5	6	7	8	9	10
1 success	99.144	96.808	94.052	90.126	84.4	76.244	64.908	49.402	28.376
2 successes	91.854	89.728	84.182	75.816	64.75	51.104	35.478	19.432	5.966
3 successes	91.854	78.848	66.542	53.136	38.75	24.704	12.798	4.592	0.686
4 successes	59.049	53.248	40.817	27.216	15.625	7.424	2.673	0.592	0.041
5 successes	59.049	32.768	16.807	7.776	3.125	1.024	0.243	0.032	0.001

6 dice

Difficulty	2	3	4	5	6	7	8	9	10
1 success	98.41	97.76	95.92	92.72	87.64	79.94	68.67	52.67	30.45
2 successes	98.41	94.30	89.32	82.13	71.97	58.52	42.03	23.94	7.67
3 successes	88.57	84.99	76.31	64.15	49.43	33.53	18.58	7.16	1.14
4 successes	88.57	71.68	56.42	40.82	25.93	13.69	5.46	1.33	0.10
5 successes	53.14	45.87	31.93	18.66	9.06	3.48	0.94	0.14	0.00
6 successes	53.14	26.21	11.76	4.66	1.56	0.40	0.07	0.00	0.00

7 dice

Difficulty	2	3	4	5	6	7	8	9	10
1 success	99.72	98.71	97.22	94.59	90.12	82.91	71.79	55.39	32.08
2 successes	97.43	96.14	92.64	86.69	77.55	64.58	47.68	28.01	9.30
3 successes	97.43	91.11	83.43	72.60	58.35	41.54	24.31	9.93	1.69
4 successes	85.03	79.74	67.90	52.60	36.00	20.61	8.95	2.38	0.19
5 successes	85.03	64.55	47.05	30.55	16.75	7.26	2.21	0.36	0.01
6 successes	47.82	39.32	24.70	12.59	5.15	1.59	0.32	0.03	0.00
7 successes	47.82	20.97	8.23	2.79	0.78	0.16	0.02	0.00	0.00

8 dice

Difficulty	2	3	4	5	6	7	8	9	10
1 success	99.49	99.14	98.08	95.95	92.05	85.35	74.45	57.71	33.39
2 successes	99.49	97.69	94.92	90.02	81.92	69.61	52.58	31.68	10.82
3 successes	96.19	93.99	88.33	79.05	65.72	48.66	29.78	12.79	2.30
4 successes	96.19	87.31	76.69	62.40	45.25	27.64	12.88	3.69	0.32
5 successes	81.31	74.18	59.41	41.99	25.21	12.03	4.04	0.73	0.03
6 successes	81.31	57.67	38.70	22.39	10.51	3.71	0.86	0.09	0.00
7 successes	43.04	33.55	18.94	8.39	2.89	0.72	0.11	0.00	0.00
8 successes	43.04	16.77	5.76	1.67	0.39	0.06	0.00	0.00	0.00

9 dice

Difficulty	2	3	4	5	6	7	8	9	10
1 success	99.91	99.48	98.67	96.95	93.57	87.38	76.76	59.75	34.45
2 successes	99.16	98.48	96.47	92.48	85.37	73.81	56.86	34.99	12.22
3 successes	99.16	96.29	91.78	83.97	71.77	54.94	34.93	15.66	2.96
4 successes	94.70	91.34	83.11	70.35	53.48	34.49	17.07	5.21	0.49
5 successes	94.70	83.04	69.43	52.27	33.76	17.44	6.38	1.26	0.05
6 successes	77.48	68.47	51.22	32.75	17.10	6.74	1.74	0.21	0.00
7 successes	77.48	51.17	31.45	16.12	6.45	1.85	0.32	0.02	0.00
8 successes	38.74	28.52	14.41	5.54	1.60	0.32	0.03	0.00	0.00
9 successes	38.74	13.42	4.03	1.00	0.19	0.02	0.00	0.00	0.00

10 dice

Difficulty	2	3	4	5	6	7	8	9	10
1 success	99.83	99.65	99.07	97.69	94.78	89.08	78.77	61.56	35.34
2 successes	99.83	99.06	97.54	94.31	88.11	77.35	60.64	38.01	13.52
3 successes	98.72	97.55	94.20	87.72	76.74	60.44	39.72	18.48	3.64
4 successes	98.72	94.47	87.81	76.71	60.65	40.97	21.36	6.90	0.69
5 successes	92.98	88.23	77.18	61.17	41.95	23.19	9.12	1.94	0.09
6 successes	92.98	78.39	61.99	42.80	24.36	10.53	2.99	0.40	0.00
7 successes	73.60	62.74	43.59	25.04	11.29	3.65	0.72	0.06	0.00
8 successes	73.60	45.13	25.30	11.43	3.88	0.90	0.12	0.00	0.00
9 successes	34.86	24.15	10.89	3.62	0.87	0.14	0.01	0.00	0.00
10 successes	34.86	10.73	2.82	0.60	0.09	0.01	0.00	0.00	0.00

Botching, Second Edition

In this system, a botch is considered to be rolling more 1s than successes. It is calculated recursively as usual.

Difficulty	2	3	4	5	6	7	8	9	10
1 die	10.00	10.00	10.00	10.00	10.00	10.00	10.00	10.00	10.00
2 dice	1.00	3.00	5.00	7.00	9.00	11.00	13.00	15.00	17.00
3 dice	2.80	3.10	4.00	5.50	7.60	10.30	13.60	17.50	22.00
4 dice	0.37	1.43	2.61	4.15	6.29	9.27	13.33	18.71	25.65
5 dice	0.85	1.11	1.85	3.15	5.17	8.23	12.73	19.23	28.37
6 dice	0.12	0.61	1.27	2.39	4.25	7.26	12.02	19.37	30.45
7 dice	0.27	0.42	0.89	1.82	3.49	6.40	11.29	19.29	32.08
8 dice	0.04	0.25	0.62	1.38	2.87	5.64	10.57	19.08	33.39
9 dice	0.08	0.17	0.44	1.06	2.36	4.96	9.88	18.80	34.45
10 dice	0.01	0.10	0.31	0.81	1.95	4.37	9.22	18.46	35.34

Botching, Revised Edition

In this system, a botch is defined as rolling no successes at all and at least one 1. These probabilities are calculated according to the formula <math>((d-1)/10)^n-((d-2)/10)^n.</math>

Difficulty	2	3	4	5	6	7	8	9	10
1 die	10.00	10.00	10.00	10.00	10.00	10.00	10.00	10.00	10.00
2 dice	1.00	3.00	5.00	7.00	9.00	11.00	13.00	15.00	17.00
3 dice	0.10	0.70	1.90	3.70	6.10	9.10	12.70	16.90	21.70
4 dice	0.01	0.15	0.65	1.75	3.69	6.71	11.05	16.95	24.65
5 dice	0.00	0.03	0.21	0.78	2.10	4.65	9.03	15.96	26.28
6 dice	0.00	0.00	0.06	0.33	1.15	3.10	7.09	14.44	26.92
7 dice	0.00	0.00	0.02	0.14	0.61	2.01	5.43	12.73	26.85
8 dice	0.00	0.00	0.00	0.05	0.32	1.28	4.08	11.01	26.26
9 dice	0.00	0.00	0.00	0.02	0.16	0.81	3.02	9.38	25.32
10 dice	0.00	0.00	0.00	0.00	0.08	0.50	2.22	7.91	24.13

More thorough explanations

For a more thorough explanation of where the formulas came from, bundled in a printable PDF with more (and bigger) tables, please go here.