Discrete Distributions
Negative Binomial Distribution Table: P(X = x)
Probabilities for the number of failures before the r-th success, including the geometric distribution (r = 1).
- r = 1 to 5
- 10 values of p
- Calculator included
Negative Binomial Table
Jump to a value of r below. Rows: failures before the r-th success x · Columns: probability of success p · Cells: P(X = x)
r = 1 success (geometric distribution)
| Probability of success (p) → x ↓ | .20 | .25 | .30 | .40 | .50 | .60 | .70 | .75 | .80 | .90 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.2000 | 0.2500 | 0.3000 | 0.4000 | 0.5000 | 0.6000 | 0.7000 | 0.7500 | 0.8000 | 0.9000 |
| 1 | 0.1600 | 0.1875 | 0.2100 | 0.2400 | 0.2500 | 0.2400 | 0.2100 | 0.1875 | 0.1600 | 0.0900 |
| 2 | 0.1280 | 0.1406 | 0.1470 | 0.1440 | 0.1250 | 0.0960 | 0.0630 | 0.0469 | 0.0320 | 0.0090 |
| 3 | 0.1024 | 0.1055 | 0.1029 | 0.0864 | 0.0625 | 0.0384 | 0.0189 | 0.0117 | 0.0064 | 0.0009 |
| 4 | 0.0819 | 0.0791 | 0.0720 | 0.0518 | 0.0313 | 0.0154 | 0.0057 | 0.0029 | 0.0013 | 0.0001 |
| 5 | 0.0655 | 0.0593 | 0.0504 | 0.0311 | 0.0156 | 0.0061 | 0.0017 | 0.0007 | 0.0003 | 0+ |
| 6 | 0.0524 | 0.0445 | 0.0353 | 0.0187 | 0.0078 | 0.0025 | 0.0005 | 0.0002 | 0.0001 | 0+ |
| 7 | 0.0419 | 0.0334 | 0.0247 | 0.0112 | 0.0039 | 0.0010 | 0.0002 | 0+ | 0+ | 0+ |
| 8 | 0.0336 | 0.0250 | 0.0173 | 0.0067 | 0.0020 | 0.0004 | 0+ | 0+ | 0+ | 0+ |
| 9 | 0.0268 | 0.0188 | 0.0121 | 0.0040 | 0.0010 | 0.0002 | 0+ | 0+ | 0+ | 0+ |
| 10 | 0.0215 | 0.0141 | 0.0085 | 0.0024 | 0.0005 | 0.0001 | 0+ | 0+ | 0+ | 0+ |
| 11 | 0.0172 | 0.0106 | 0.0059 | 0.0015 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 12 | 0.0137 | 0.0079 | 0.0042 | 0.0009 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 13 | 0.0110 | 0.0059 | 0.0029 | 0.0005 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 14 | 0.0088 | 0.0045 | 0.0020 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 15 | 0.0070 | 0.0033 | 0.0014 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 16 | 0.0056 | 0.0025 | 0.0010 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 17 | 0.0045 | 0.0019 | 0.0007 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 18 | 0.0036 | 0.0014 | 0.0005 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 19 | 0.0029 | 0.0011 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 20 | 0.0023 | 0.0008 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 21 | 0.0018 | 0.0006 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 22 | 0.0015 | 0.0004 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 23 | 0.0012 | 0.0003 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 24 | 0.0009 | 0.0003 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 25 | 0.0008 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 26 | 0.0006 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 27 | 0.0005 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 28 | 0.0004 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 29 | 0.0003 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 30 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 31 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 32 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 33 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 34 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 35 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 36 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 37 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 38 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
r = 2 successes
| Probability of success (p) → x ↓ | .20 | .25 | .30 | .40 | .50 | .60 | .70 | .75 | .80 | .90 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.0400 | 0.0625 | 0.0900 | 0.1600 | 0.2500 | 0.3600 | 0.4900 | 0.5625 | 0.6400 | 0.8100 |
| 1 | 0.0640 | 0.0938 | 0.1260 | 0.1920 | 0.2500 | 0.2880 | 0.2940 | 0.2812 | 0.2560 | 0.1620 |
| 2 | 0.0768 | 0.1055 | 0.1323 | 0.1728 | 0.1875 | 0.1728 | 0.1323 | 0.1055 | 0.0768 | 0.0243 |
| 3 | 0.0819 | 0.1055 | 0.1235 | 0.1382 | 0.1250 | 0.0922 | 0.0529 | 0.0352 | 0.0205 | 0.0032 |
| 4 | 0.0819 | 0.0989 | 0.1080 | 0.1037 | 0.0781 | 0.0461 | 0.0198 | 0.0110 | 0.0051 | 0.0004 |
| 5 | 0.0786 | 0.0890 | 0.0908 | 0.0746 | 0.0469 | 0.0221 | 0.0071 | 0.0033 | 0.0012 | 0+ |
| 6 | 0.0734 | 0.0779 | 0.0741 | 0.0523 | 0.0273 | 0.0103 | 0.0025 | 0.0010 | 0.0003 | 0+ |
| 7 | 0.0671 | 0.0667 | 0.0593 | 0.0358 | 0.0156 | 0.0047 | 0.0009 | 0.0003 | 0.0001 | 0+ |
| 8 | 0.0604 | 0.0563 | 0.0467 | 0.0242 | 0.0088 | 0.0021 | 0.0003 | 0.0001 | 0+ | 0+ |
| 9 | 0.0537 | 0.0469 | 0.0363 | 0.0161 | 0.0049 | 0.0009 | 0.0001 | 0+ | 0+ | 0+ |
| 10 | 0.0472 | 0.0387 | 0.0280 | 0.0106 | 0.0027 | 0.0004 | 0+ | 0+ | 0+ | 0+ |
| 11 | 0.0412 | 0.0317 | 0.0214 | 0.0070 | 0.0015 | 0.0002 | 0+ | 0+ | 0+ | 0+ |
| 12 | 0.0357 | 0.0257 | 0.0162 | 0.0045 | 0.0008 | 0.0001 | 0+ | 0+ | 0+ | 0+ |
| 13 | 0.0308 | 0.0208 | 0.0122 | 0.0029 | 0.0004 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 14 | 0.0264 | 0.0167 | 0.0092 | 0.0019 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 15 | 0.0225 | 0.0134 | 0.0068 | 0.0012 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 16 | 0.0191 | 0.0106 | 0.0051 | 0.0008 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 17 | 0.0162 | 0.0085 | 0.0038 | 0.0005 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 18 | 0.0137 | 0.0067 | 0.0028 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 19 | 0.0115 | 0.0053 | 0.0021 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 20 | 0.0097 | 0.0042 | 0.0015 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 21 | 0.0081 | 0.0033 | 0.0011 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 22 | 0.0068 | 0.0026 | 0.0008 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 23 | 0.0057 | 0.0020 | 0.0006 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 24 | 0.0047 | 0.0016 | 0.0004 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 25 | 0.0039 | 0.0012 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 26 | 0.0033 | 0.0010 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 27 | 0.0027 | 0.0007 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 28 | 0.0022 | 0.0006 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 29 | 0.0019 | 0.0004 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 30 | 0.0015 | 0.0003 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 31 | 0.0013 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 32 | 0.0010 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 33 | 0.0009 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 34 | 0.0007 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 35 | 0.0006 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 36 | 0.0005 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 37 | 0.0004 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 38 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 39 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 40 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
r = 3 successes
| Probability of success (p) → x ↓ | .20 | .25 | .30 | .40 | .50 | .60 | .70 | .75 | .80 | .90 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.0080 | 0.0156 | 0.0270 | 0.0640 | 0.1250 | 0.2160 | 0.3430 | 0.4219 | 0.5120 | 0.7290 |
| 1 | 0.0192 | 0.0352 | 0.0567 | 0.1152 | 0.1875 | 0.2592 | 0.3087 | 0.3164 | 0.3072 | 0.2187 |
| 2 | 0.0307 | 0.0527 | 0.0794 | 0.1382 | 0.1875 | 0.2074 | 0.1852 | 0.1582 | 0.1229 | 0.0437 |
| 3 | 0.0410 | 0.0659 | 0.0926 | 0.1382 | 0.1562 | 0.1382 | 0.0926 | 0.0659 | 0.0410 | 0.0073 |
| 4 | 0.0492 | 0.0742 | 0.0972 | 0.1244 | 0.1172 | 0.0829 | 0.0417 | 0.0247 | 0.0123 | 0.0011 |
| 5 | 0.0551 | 0.0779 | 0.0953 | 0.1045 | 0.0820 | 0.0464 | 0.0175 | 0.0087 | 0.0034 | 0.0002 |
| 6 | 0.0587 | 0.0779 | 0.0889 | 0.0836 | 0.0547 | 0.0248 | 0.0070 | 0.0029 | 0.0009 | 0+ |
| 7 | 0.0604 | 0.0751 | 0.0800 | 0.0645 | 0.0352 | 0.0127 | 0.0027 | 0.0009 | 0.0002 | 0+ |
| 8 | 0.0604 | 0.0704 | 0.0700 | 0.0484 | 0.0220 | 0.0064 | 0.0010 | 0.0003 | 0.0001 | 0+ |
| 9 | 0.0591 | 0.0645 | 0.0599 | 0.0355 | 0.0134 | 0.0031 | 0.0004 | 0.0001 | 0+ | 0+ |
| 10 | 0.0567 | 0.0581 | 0.0503 | 0.0255 | 0.0081 | 0.0015 | 0.0001 | 0+ | 0+ | 0+ |
| 11 | 0.0536 | 0.0515 | 0.0416 | 0.0181 | 0.0048 | 0.0007 | 0+ | 0+ | 0+ | 0+ |
| 12 | 0.0500 | 0.0450 | 0.0340 | 0.0127 | 0.0028 | 0.0003 | 0+ | 0+ | 0+ | 0+ |
| 13 | 0.0462 | 0.0390 | 0.0275 | 0.0088 | 0.0016 | 0.0002 | 0+ | 0+ | 0+ | 0+ |
| 14 | 0.0422 | 0.0334 | 0.0220 | 0.0060 | 0.0009 | 0.0001 | 0+ | 0+ | 0+ | 0+ |
| 15 | 0.0383 | 0.0284 | 0.0174 | 0.0041 | 0.0005 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 16 | 0.0345 | 0.0240 | 0.0137 | 0.0028 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 17 | 0.0308 | 0.0201 | 0.0107 | 0.0019 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 18 | 0.0274 | 0.0167 | 0.0084 | 0.0012 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 19 | 0.0242 | 0.0139 | 0.0065 | 0.0008 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 20 | 0.0213 | 0.0114 | 0.0050 | 0.0005 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 21 | 0.0187 | 0.0094 | 0.0038 | 0.0004 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 22 | 0.0163 | 0.0077 | 0.0029 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 23 | 0.0142 | 0.0063 | 0.0022 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 24 | 0.0123 | 0.0051 | 0.0017 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 25 | 0.0106 | 0.0041 | 0.0013 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 26 | 0.0091 | 0.0033 | 0.0010 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 27 | 0.0079 | 0.0027 | 0.0007 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 28 | 0.0067 | 0.0022 | 0.0005 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 29 | 0.0058 | 0.0017 | 0.0004 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 30 | 0.0049 | 0.0014 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 31 | 0.0042 | 0.0011 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 32 | 0.0036 | 0.0009 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 33 | 0.0030 | 0.0007 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 34 | 0.0026 | 0.0006 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 35 | 0.0022 | 0.0004 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 36 | 0.0018 | 0.0003 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 37 | 0.0015 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 38 | 0.0013 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 39 | 0.0011 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 40 | 0.0009 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
r = 4 successes
| Probability of success (p) → x ↓ | .20 | .25 | .30 | .40 | .50 | .60 | .70 | .75 | .80 | .90 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.0016 | 0.0039 | 0.0081 | 0.0256 | 0.0625 | 0.1296 | 0.2401 | 0.3164 | 0.4096 | 0.6561 |
| 1 | 0.0051 | 0.0117 | 0.0227 | 0.0614 | 0.1250 | 0.2074 | 0.2881 | 0.3164 | 0.3277 | 0.2624 |
| 2 | 0.0102 | 0.0220 | 0.0397 | 0.0922 | 0.1562 | 0.2074 | 0.2161 | 0.1978 | 0.1638 | 0.0656 |
| 3 | 0.0164 | 0.0330 | 0.0556 | 0.1106 | 0.1562 | 0.1659 | 0.1297 | 0.0989 | 0.0655 | 0.0131 |
| 4 | 0.0229 | 0.0433 | 0.0681 | 0.1161 | 0.1367 | 0.1161 | 0.0681 | 0.0433 | 0.0229 | 0.0023 |
| 5 | 0.0294 | 0.0519 | 0.0762 | 0.1115 | 0.1094 | 0.0743 | 0.0327 | 0.0173 | 0.0073 | 0.0004 |
| 6 | 0.0352 | 0.0584 | 0.0800 | 0.1003 | 0.0820 | 0.0446 | 0.0147 | 0.0065 | 0.0022 | 0.0001 |
| 7 | 0.0403 | 0.0626 | 0.0800 | 0.0860 | 0.0586 | 0.0255 | 0.0063 | 0.0023 | 0.0006 | 0+ |
| 8 | 0.0443 | 0.0645 | 0.0770 | 0.0709 | 0.0403 | 0.0140 | 0.0026 | 0.0008 | 0.0002 | 0+ |
| 9 | 0.0472 | 0.0645 | 0.0719 | 0.0568 | 0.0269 | 0.0075 | 0.0010 | 0.0003 | 0+ | 0+ |
| 10 | 0.0491 | 0.0629 | 0.0654 | 0.0443 | 0.0175 | 0.0039 | 0.0004 | 0.0001 | 0+ | 0+ |
| 11 | 0.0500 | 0.0601 | 0.0583 | 0.0338 | 0.0111 | 0.0020 | 0.0002 | 0+ | 0+ | 0+ |
| 12 | 0.0500 | 0.0563 | 0.0510 | 0.0254 | 0.0069 | 0.0010 | 0.0001 | 0+ | 0+ | 0+ |
| 13 | 0.0493 | 0.0520 | 0.0439 | 0.0187 | 0.0043 | 0.0005 | 0+ | 0+ | 0+ | 0+ |
| 14 | 0.0479 | 0.0473 | 0.0374 | 0.0136 | 0.0026 | 0.0002 | 0+ | 0+ | 0+ | 0+ |
| 15 | 0.0459 | 0.0426 | 0.0314 | 0.0098 | 0.0016 | 0.0001 | 0+ | 0+ | 0+ | 0+ |
| 16 | 0.0436 | 0.0379 | 0.0261 | 0.0070 | 0.0009 | 0.0001 | 0+ | 0+ | 0+ | 0+ |
| 17 | 0.0411 | 0.0335 | 0.0215 | 0.0049 | 0.0005 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 18 | 0.0383 | 0.0293 | 0.0175 | 0.0035 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 19 | 0.0355 | 0.0254 | 0.0142 | 0.0024 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 20 | 0.0327 | 0.0219 | 0.0114 | 0.0017 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 21 | 0.0299 | 0.0188 | 0.0092 | 0.0011 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 22 | 0.0272 | 0.0160 | 0.0073 | 0.0008 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 23 | 0.0246 | 0.0136 | 0.0058 | 0.0005 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 24 | 0.0221 | 0.0115 | 0.0045 | 0.0004 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 25 | 0.0198 | 0.0096 | 0.0036 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 26 | 0.0177 | 0.0081 | 0.0028 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 27 | 0.0157 | 0.0067 | 0.0022 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 28 | 0.0139 | 0.0056 | 0.0017 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 29 | 0.0123 | 0.0046 | 0.0013 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 30 | 0.0108 | 0.0038 | 0.0010 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 31 | 0.0095 | 0.0031 | 0.0008 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 32 | 0.0083 | 0.0026 | 0.0006 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 33 | 0.0072 | 0.0021 | 0.0004 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 34 | 0.0063 | 0.0017 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 35 | 0.0055 | 0.0014 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 36 | 0.0047 | 0.0011 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 37 | 0.0041 | 0.0009 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 38 | 0.0035 | 0.0007 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 39 | 0.0031 | 0.0006 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 40 | 0.0026 | 0.0005 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
r = 5 successes
| Probability of success (p) → x ↓ | .20 | .25 | .30 | .40 | .50 | .60 | .70 | .75 | .80 | .90 |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.0003 | 0.0010 | 0.0024 | 0.0102 | 0.0312 | 0.0778 | 0.1681 | 0.2373 | 0.3277 | 0.5905 |
| 1 | 0.0013 | 0.0037 | 0.0085 | 0.0307 | 0.0781 | 0.1555 | 0.2521 | 0.2966 | 0.3277 | 0.2952 |
| 2 | 0.0031 | 0.0082 | 0.0179 | 0.0553 | 0.1172 | 0.1866 | 0.2269 | 0.2225 | 0.1966 | 0.0886 |
| 3 | 0.0057 | 0.0144 | 0.0292 | 0.0774 | 0.1367 | 0.1742 | 0.1588 | 0.1298 | 0.0918 | 0.0207 |
| 4 | 0.0092 | 0.0216 | 0.0408 | 0.0929 | 0.1367 | 0.1393 | 0.0953 | 0.0649 | 0.0367 | 0.0041 |
| 5 | 0.0132 | 0.0292 | 0.0515 | 0.1003 | 0.1230 | 0.1003 | 0.0515 | 0.0292 | 0.0132 | 0.0007 |
| 6 | 0.0176 | 0.0365 | 0.0600 | 0.1003 | 0.1025 | 0.0669 | 0.0257 | 0.0122 | 0.0044 | 0.0001 |
| 7 | 0.0221 | 0.0430 | 0.0660 | 0.0946 | 0.0806 | 0.0420 | 0.0121 | 0.0048 | 0.0014 | 0+ |
| 8 | 0.0266 | 0.0484 | 0.0693 | 0.0851 | 0.0604 | 0.0252 | 0.0055 | 0.0018 | 0.0004 | 0+ |
| 9 | 0.0307 | 0.0524 | 0.0701 | 0.0738 | 0.0436 | 0.0146 | 0.0024 | 0.0006 | 0.0001 | 0+ |
| 10 | 0.0344 | 0.0550 | 0.0687 | 0.0620 | 0.0305 | 0.0082 | 0.0010 | 0.0002 | 0+ | 0+ |
| 11 | 0.0375 | 0.0563 | 0.0656 | 0.0507 | 0.0208 | 0.0045 | 0.0004 | 0.0001 | 0+ | 0+ |
| 12 | 0.0400 | 0.0563 | 0.0612 | 0.0406 | 0.0139 | 0.0024 | 0.0002 | 0+ | 0+ | 0+ |
| 13 | 0.0419 | 0.0552 | 0.0560 | 0.0318 | 0.0091 | 0.0012 | 0.0001 | 0+ | 0+ | 0+ |
| 14 | 0.0431 | 0.0532 | 0.0504 | 0.0246 | 0.0058 | 0.0006 | 0+ | 0+ | 0+ | 0+ |
| 15 | 0.0436 | 0.0506 | 0.0447 | 0.0187 | 0.0037 | 0.0003 | 0+ | 0+ | 0+ | 0+ |
| 16 | 0.0436 | 0.0474 | 0.0391 | 0.0140 | 0.0023 | 0.0002 | 0+ | 0+ | 0+ | 0+ |
| 17 | 0.0431 | 0.0439 | 0.0338 | 0.0104 | 0.0014 | 0.0001 | 0+ | 0+ | 0+ | 0+ |
| 18 | 0.0422 | 0.0403 | 0.0289 | 0.0076 | 0.0009 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 19 | 0.0408 | 0.0366 | 0.0245 | 0.0055 | 0.0005 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 20 | 0.0392 | 0.0329 | 0.0206 | 0.0040 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 21 | 0.0373 | 0.0294 | 0.0172 | 0.0028 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 22 | 0.0353 | 0.0260 | 0.0142 | 0.0020 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 23 | 0.0332 | 0.0229 | 0.0117 | 0.0014 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ |
| 24 | 0.0309 | 0.0201 | 0.0095 | 0.0010 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 25 | 0.0287 | 0.0175 | 0.0077 | 0.0007 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 26 | 0.0265 | 0.0151 | 0.0063 | 0.0005 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 27 | 0.0243 | 0.0130 | 0.0050 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 28 | 0.0223 | 0.0111 | 0.0040 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 29 | 0.0203 | 0.0095 | 0.0032 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 30 | 0.0184 | 0.0081 | 0.0025 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 31 | 0.0166 | 0.0068 | 0.0020 | 0.0001 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 32 | 0.0149 | 0.0058 | 0.0016 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 33 | 0.0134 | 0.0049 | 0.0012 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 34 | 0.0120 | 0.0041 | 0.0010 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 35 | 0.0107 | 0.0034 | 0.0008 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 36 | 0.0095 | 0.0028 | 0.0006 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 37 | 0.0084 | 0.0024 | 0.0005 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 38 | 0.0074 | 0.0020 | 0.0004 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 39 | 0.0066 | 0.0016 | 0.0003 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
| 40 | 0.0058 | 0.0013 | 0.0002 | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ | 0+ |
Tip: click any value to highlight its row and column.
How to read this table
- Decide on r, the number of successes you are waiting for, and p, the probability of success on each trial.
- Find the block for r, then the row for x (the number of failures) and the column for p.
- The total number of trials is x + r. Rows stop when every probability is below 0.00005 (or at x = 40).
Worked example
A site enrols eligible patients with probability 0.40 per screened patient. The chance that exactly 3 screened patients fail before the site enrols its second patient (so enrolment takes 5 screenings) is row x = 3, column p = .40 of the r = 2 block: 0.1382.
Click the underlined links to highlight the value in the table.
Calculator
For values between the rows and columns of the table. Results update as you type.
Formula and how the values were computed
P(X = x) = C(x + r − 1, x) pʳ (1 − p)ˣ · mean r(1 − p)/p
Each cell is computed exactly from the negative binomial formula and rounded to four decimals.
Frequently asked questions
How is this related to negative binomial regression?
Negative binomial regression uses a different parameterisation (a mean and a dispersion parameter) to model overdispersed counts. The distribution is the same family.
What is the geometric distribution?
The r = 1 case: the number of failures before the first success.