Binomial Distribution Table

Binomial probability lookup

Trials (n)

Success probability (p)

Successes (k)

P(X = 3): 0.1172
P(X ≤ 3): 0.1719
P(X < 3): 0.0547
P(X ≥ 3): 0.9453
P(X > 3): 0.8281

X ~ Binomial(n = 10, p = 0.5). Exact values, not a normal approximation. Mean np = 5, SD √(np(1−p)) = 1.5811.

k	P(X = k)	P(X ≤ k)
0	0.0010	0.0010
1	0.0098	0.0107
2	0.0439	0.0547
3	0.1172	0.1719
4	0.2051	0.3770
5	0.2461	0.6230
6	0.2051	0.8281
7	0.1172	0.9453
8	0.0439	0.9893
9	0.0098	0.9990
10	0.0010	1.0000

Convention and assumptions

X counts successes in n independent trials with a common success probability p. Each trial has two outcomes, and n is fixed. The lookup above computes exact probabilities for any n up to 1,000. The printed tables below give cumulative probabilities P(X ≤ k), rounded to four decimals. A printed 0.0000 or 1.0000 can reflect rounding rather than exact impossibility or certainty.

$P(X=k)=\binom nk p^k(1-p)^{n-k}.$

Sampling without replacement from a small finite population changes the probability across draws; use the hypergeometric reference instead. Shared batch conditions or repeated users may violate independence.

Cumulative probabilities: n = 5

k	p = 0.05	p = 0.1	p = 0.2	p = 0.3	p = 0.5
0	0.7738	0.5905	0.3277	0.1681	0.0312
1	0.9774	0.9185	0.7373	0.5282	0.1875
2	0.9988	0.9914	0.9421	0.8369	0.5000
3	1.0000	0.9995	0.9933	0.9692	0.8125
4	1.0000	1.0000	0.9997	0.9976	0.9688
5	1.0000	1.0000	1.0000	1.0000	1.0000

Cumulative probabilities: n = 10

k	p = 0.05	p = 0.1	p = 0.2	p = 0.3	p = 0.5
0	0.5987	0.3487	0.1074	0.0282	0.0010
1	0.9139	0.7361	0.3758	0.1493	0.0107
2	0.9885	0.9298	0.6778	0.3828	0.0547
3	0.9990	0.9872	0.8791	0.6496	0.1719
4	0.9999	0.9984	0.9672	0.8497	0.3770
5	1.0000	0.9999	0.9936	0.9527	0.6230
6	1.0000	1.0000	0.9991	0.9894	0.8281
7	1.0000	1.0000	0.9999	0.9984	0.9453
8	1.0000	1.0000	1.0000	0.9999	0.9893
9	1.0000	1.0000	1.0000	1.0000	0.9990
10	1.0000	1.0000	1.0000	1.0000	1.0000

Cumulative probabilities: n = 15

k	p = 0.05	p = 0.1	p = 0.2	p = 0.3	p = 0.5
0	0.4633	0.2059	0.0352	0.0047	0.0000
1	0.8290	0.5490	0.1671	0.0353	0.0005
2	0.9638	0.8159	0.3980	0.1268	0.0037
3	0.9945	0.9444	0.6482	0.2969	0.0176
4	0.9994	0.9873	0.8358	0.5155	0.0592
5	0.9999	0.9978	0.9389	0.7216	0.1509
6	1.0000	0.9997	0.9819	0.8689	0.3036
7	1.0000	1.0000	0.9958	0.9500	0.5000
8	1.0000	1.0000	0.9992	0.9848	0.6964
9	1.0000	1.0000	0.9999	0.9963	0.8491
10	1.0000	1.0000	1.0000	0.9993	0.9408
11	1.0000	1.0000	1.0000	0.9999	0.9824
12	1.0000	1.0000	1.0000	1.0000	0.9963
13	1.0000	1.0000	1.0000	1.0000	0.9995
14	1.0000	1.0000	1.0000	1.0000	1.0000
15	1.0000	1.0000	1.0000	1.0000	1.0000

Cumulative probabilities: n = 20

k	p = 0.05	p = 0.1	p = 0.2	p = 0.3	p = 0.5
0	0.3585	0.1216	0.0115	0.0008	0.0000
1	0.7358	0.3917	0.0692	0.0076	0.0000
2	0.9245	0.6769	0.2061	0.0355	0.0002
3	0.9841	0.8670	0.4114	0.1071	0.0013
4	0.9974	0.9568	0.6296	0.2375	0.0059
5	0.9997	0.9887	0.8042	0.4164	0.0207
6	1.0000	0.9976	0.9133	0.6080	0.0577
7	1.0000	0.9996	0.9679	0.7723	0.1316
8	1.0000	0.9999	0.9900	0.8867	0.2517
9	1.0000	1.0000	0.9974	0.9520	0.4119
10	1.0000	1.0000	0.9994	0.9829	0.5881
11	1.0000	1.0000	0.9999	0.9949	0.7483
12	1.0000	1.0000	1.0000	0.9987	0.8684
13	1.0000	1.0000	1.0000	0.9997	0.9423
14	1.0000	1.0000	1.0000	1.0000	0.9793
15	1.0000	1.0000	1.0000	1.0000	0.9941
16	1.0000	1.0000	1.0000	1.0000	0.9987
17	1.0000	1.0000	1.0000	1.0000	0.9998
18	1.0000	1.0000	1.0000	1.0000	1.0000
19	1.0000	1.0000	1.0000	1.0000	1.0000
20	1.0000	1.0000	1.0000	1.0000	1.0000

Read the requested tail

For whole-number endpoints, P(X ≥ k) = 1 − P(X ≤ k − 1). An exact probability is P(X ≤ k) − P(X ≤ k − 1), and P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1). Treat the CDF below zero as zero. Subtracting rounded cells can lose a final decimal; use full-precision probabilities for final calculations.

Worked example

For n = 10 and p = 0.3, P(X ≤ 3) = 0.6496. Thus P(X ≥ 4) ≈ 0.3504. The exact probability P(X = 3), computed from the mass formula, is 0.2668 rounded. These are model probabilities, not evidence that real trials meet the assumptions.

The expected count is np = 3 and its standard deviation is √[np(1 − p)] ≈ 1.4491. An observed count need not equal its expectation.

Continue

Read the binomial lesson. NIST documents the model and probability functions. The general event-probability calculator does not calculate this distribution.