Statistics and research
Binomial probability calculator
Exact binomial probabilities P(X = k), P(X ≤ k), P(X ≥ k) and the full distribution table for n trials with success probability p.
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Results
P(X = 8)0.114397P(X ≤ 8)0.886669P(X ≥ 8)0.227728
- P(X = 8)
- 0.114396739705
- P(X < 8)
- 0.772271797418
- P(X ≤ 8)
- 0.886668537123
- P(X > 8)
- 0.113331462877
- P(X ≥ 8)
- 0.227728202582
- Mean np
- 6
- Variance np(1 − p)
- 4.2
- Standard deviation
- 2.049390153
| x | P(X = x) | P(X ≤ x) | P(X ≥ x) |
|---|---|---|---|
| 0 | 0.00079792266 | 0.00079792266 | 1 |
| 1 | 0.0068393371 | 0.0076372598 | 0.99920208 |
| 2 | 0.027845873 | 0.035483132 | 0.99236274 |
| 3 | 0.071603672 | 0.1070868 | 0.96451687 |
| 4 | 0.13042097 | 0.23750778 | 0.8929132 |
| 5 | 0.17886305 | 0.41637083 | 0.76249222 |
| 6 | 0.19163898 | 0.60800981 | 0.58362917 |
| 7 | 0.16426199 | 0.7722718 | 0.39199019 |
| 8 | 0.11439674 | 0.88666854 | 0.2277282 |
| 9 | 0.065369566 | 0.9520381 | 0.11333146 |
| 10 | 0.030817081 | 0.98285518 | 0.047961897 |
| 11 | 0.012006655 | 0.99486184 | 0.017144816 |
| 12 | 0.0038592819 | 0.99872112 | 0.0051381615 |
| 13 | 0.0010178326 | 0.99973895 | 0.0012788796 |
| 14 | 0.00021810699 | 0.99995706 | 0.00026104701 |
| 15 | 3.7389769e-5 | 0.99999445 | 4.2940022e-5 |
| 16 | 5.0075583e-6 | 0.99999946 | 5.5502531e-6 |
| 17 | 5.0496387e-7 | 0.99999996 | 5.4269475e-7 |
| 18 | 3.6068848e-8 | 1 | 3.7730881e-8 |
| 19 | 1.6271661e-9 | 1 | 1.6620339e-9 |
| 20 | 3.4867844e-11 | 1 | 3.4867844e-11 |
Cumulative probabilities come from the regularized incomplete beta function, so tiny tail probabilities keep their accuracy.
Method and limits
How to use it
Enter the number of independent trials n, the probability of success on each trial p, and a number of successes k. You get the probability of exactly, at most, fewer than, more than and at least k successes, and the table of the distribution.
Formulas
- P(X = k) = C(n, k) pk (1 − p)n−k, computed with Loader’s saddle-point form so it stays accurate for n in the millions.
- P(X ≤ k) = I1−p(n − k, k + 1), the regularized incomplete beta function; upper tails are computed directly, not as 1 minus a sum.
- Mean np, variance np(1 − p).
Assumptions and limits
- A fixed number of independent trials with the same probability of success.
- For sampling without replacement from a small population use the hypergeometric distribution.
- n up to 1 billion; the table shows the values with non-negligible probability (at most 401 rows).
Worked example
n = 20, p = 0.3, k = 8: P(X = 8) = 0.1144, P(X ≤ 8) = 0.8867, P(X ≥ 8) = 0.2277; mean 6, SD 2.049.
References
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