Binomial Distribution Table

Individual binomial probabilities P(X = k) for n = 1 to 10 trials and p = 0.10 to 0.50. Each value gives the exact probability of observing exactly k successes in n independent trials with success probability p.

What is the Binomial Distribution?

The binomial distribution models the number of successes X in n independent trials, where each trial has the same probability of success p. It requires:

Fixed number of trials n.
Two outcomes per trial: success (p) or failure (1 − p).
Independent trials: the outcome of one trial does not affect others.
Constant probability p across all trials.

The probability of exactly k successes is: P(X = k) = C(n, k) · p^k · (1 − p)^{n − k}

Symmetry: for p > 0.50, use P(X = k | n, p) = P(X = n − k | n, 1 − p). For example, P(X = 3 | n = 5, p = 0.70) = P(X = 2 | n = 5, p = 0.30) = 0.3087.

Binomial Probabilities P(X = k)

Values are rounded to 4 decimal places. Entries shown as 0.0000 are less than 0.00005.

n = 1

k	p = 0.10	p = 0.20	p = 0.25	p = 0.30	p = 0.40	p = 0.50
0	0.9000	0.8000	0.7500	0.7000	0.6000	0.5000
1	0.1000	0.2000	0.2500	0.3000	0.4000	0.5000

n = 2

k	p = 0.10	p = 0.20	p = 0.25	p = 0.30	p = 0.40	p = 0.50
0	0.8100	0.6400	0.5625	0.4900	0.3600	0.2500
1	0.1800	0.3200	0.3750	0.4200	0.4800	0.5000
2	0.0100	0.0400	0.0625	0.0900	0.1600	0.2500

n = 3

k	p = 0.10	p = 0.20	p = 0.25	p = 0.30	p = 0.40	p = 0.50
0	0.7290	0.5120	0.4219	0.3430	0.2160	0.1250
1	0.2430	0.3840	0.4219	0.4410	0.4320	0.3750
2	0.0270	0.0960	0.1406	0.1890	0.2880	0.3750
3	0.0010	0.0080	0.0156	0.0270	0.0640	0.1250

n = 4

k	p = 0.10	p = 0.20	p = 0.25	p = 0.30	p = 0.40	p = 0.50
0	0.6561	0.4096	0.3164	0.2401	0.1296	0.0625
1	0.2916	0.4096	0.4219	0.4116	0.3456	0.2500
2	0.0486	0.1536	0.2109	0.2646	0.3456	0.3750
3	0.0036	0.0256	0.0469	0.0756	0.1536	0.2500
4	0.0001	0.0016	0.0039	0.0081	0.0256	0.0625

n = 5

k	p = 0.10	p = 0.20	p = 0.25	p = 0.30	p = 0.40	p = 0.50
0	0.5905	0.3277	0.2373	0.1681	0.0778	0.0313
1	0.3281	0.4096	0.3955	0.3602	0.2592	0.1563
2	0.0729	0.2048	0.2637	0.3087	0.3456	0.3125
3	0.0081	0.0512	0.0879	0.1323	0.2304	0.3125
4	0.0005	0.0064	0.0146	0.0284	0.0768	0.1563
5	0.0000	0.0003	0.0010	0.0024	0.0102	0.0313

n = 6

k	p = 0.10	p = 0.20	p = 0.25	p = 0.30	p = 0.40	p = 0.50
0	0.5314	0.2621	0.1780	0.1176	0.0467	0.0156
1	0.3543	0.3932	0.3560	0.3025	0.1866	0.0938
2	0.0984	0.2458	0.2966	0.3241	0.3110	0.2344
3	0.0146	0.0819	0.1318	0.1852	0.2765	0.3125
4	0.0012	0.0154	0.0330	0.0595	0.1382	0.2344
5	0.0001	0.0015	0.0044	0.0102	0.0369	0.0938
6	0.0000	0.0001	0.0002	0.0007	0.0041	0.0156

n = 7

k	p = 0.10	p = 0.20	p = 0.25	p = 0.30	p = 0.40	p = 0.50
0	0.4783	0.2097	0.1335	0.0824	0.0280	0.0078
1	0.3720	0.3670	0.3115	0.2471	0.1306	0.0547
2	0.1240	0.2753	0.3115	0.3177	0.2613	0.1641
3	0.0230	0.1147	0.1730	0.2269	0.2903	0.2734
4	0.0026	0.0287	0.0577	0.0972	0.1935	0.2734
5	0.0002	0.0043	0.0115	0.0250	0.0774	0.1641
6	0.0000	0.0004	0.0013	0.0036	0.0172	0.0547
7	0.0000	0.0000	0.0001	0.0002	0.0016	0.0078

n = 8

k	p = 0.10	p = 0.20	p = 0.25	p = 0.30	p = 0.40	p = 0.50
0	0.4305	0.1678	0.1001	0.0576	0.0168	0.0039
1	0.3826	0.3355	0.2670	0.1977	0.0896	0.0313
2	0.1488	0.2936	0.3115	0.2965	0.2090	0.1094
3	0.0331	0.1468	0.2076	0.2541	0.2787	0.2188
4	0.0046	0.0459	0.0865	0.1361	0.2322	0.2734
5	0.0004	0.0092	0.0231	0.0467	0.1239	0.2188
6	0.0000	0.0011	0.0038	0.0100	0.0413	0.1094
7	0.0000	0.0001	0.0004	0.0012	0.0079	0.0313
8	0.0000	0.0000	0.0000	0.0001	0.0007	0.0039

n = 9

k	p = 0.10	p = 0.20	p = 0.25	p = 0.30	p = 0.40	p = 0.50
0	0.3874	0.1342	0.0751	0.0404	0.0101	0.0020
1	0.3874	0.3020	0.2253	0.1556	0.0605	0.0176
2	0.1722	0.3020	0.3003	0.2668	0.1612	0.0703
3	0.0446	0.1762	0.2336	0.2668	0.2508	0.1641
4	0.0074	0.0661	0.1168	0.1715	0.2508	0.2461
5	0.0008	0.0165	0.0389	0.0735	0.1672	0.2461
6	0.0001	0.0028	0.0087	0.0210	0.0743	0.1641
7	0.0000	0.0003	0.0012	0.0039	0.0212	0.0703
8	0.0000	0.0000	0.0001	0.0004	0.0035	0.0176
9	0.0000	0.0000	0.0000	0.0000	0.0003	0.0020

n = 10

k	p = 0.10	p = 0.20	p = 0.25	p = 0.30	p = 0.40	p = 0.50
0	0.3487	0.1074	0.0563	0.0282	0.0060	0.0010
1	0.3874	0.2684	0.1877	0.1211	0.0403	0.0098
2	0.1937	0.3020	0.2816	0.2335	0.1209	0.0439
3	0.0574	0.2013	0.2503	0.2668	0.2150	0.1172
4	0.0112	0.0881	0.1460	0.2001	0.2508	0.2051
5	0.0015	0.0264	0.0584	0.1029	0.2007	0.2461
6	0.0001	0.0055	0.0162	0.0368	0.1115	0.2051
7	0.0000	0.0008	0.0031	0.0090	0.0425	0.1172
8	0.0000	0.0001	0.0004	0.0014	0.0106	0.0439
9	0.0000	0.0000	0.0000	0.0001	0.0016	0.0098
10	0.0000	0.0000	0.0000	0.0000	0.0001	0.0010

How to Use This Table

Identify n and p: n is the number of trials; p is the probability of success on each trial.
Find the sub-table for your n and locate the row for k (desired number of successes).
Read the probability at the column for your p value.
For p > 0.50, use the symmetry: P(X = k | n, p) = P(X = n − k | n, 1 − p).
Cumulative probability P(X ≤ k): sum the individual probabilities from row 0 through row k.

Binomial Distribution Table

What is the Binomial Distribution?

Binomial Probabilities P(X = k)

How to Use This Table

References

See also