Discrete Distributions

Cumulative Binomial Distribution Table: P(X ≤ x)

The probability of at most x successes in n independent trials, for n from 1 to 20.

n = 1 to 20
15 values of p
Calculator included

Cumulative Binomial Table

Jump to a value of n below. Rows: number of successes x · Columns: probability of success p · Cells: P(X ≤ x)

n = 1

n = 1
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.9900	0.9500	0.9000	0.8000	0.7500	0.7000	0.6000	0.5000	0.4000	0.3000	0.2500	0.2000	0.1000	0.0500	0.0100
1	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 2

n = 2
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.9801	0.9025	0.8100	0.6400	0.5625	0.4900	0.3600	0.2500	0.1600	0.0900	0.0625	0.0400	0.0100	0.0025	0.0001
1	0.9999	0.9975	0.9900	0.9600	0.9375	0.9100	0.8400	0.7500	0.6400	0.5100	0.4375	0.3600	0.1900	0.0975	0.0199
2	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 3

n = 3
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.9703	0.8574	0.7290	0.5120	0.4219	0.3430	0.2160	0.1250	0.0640	0.0270	0.0156	0.0080	0.0010	0.0001	0+
1	0.9997	0.9928	0.9720	0.8960	0.8438	0.7840	0.6480	0.5000	0.3520	0.2160	0.1562	0.1040	0.0280	0.0073	0.0003
2	1−	0.9999	0.9990	0.9920	0.9844	0.9730	0.9360	0.8750	0.7840	0.6570	0.5781	0.4880	0.2710	0.1426	0.0297
3	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 4

n = 4
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.9606	0.8145	0.6561	0.4096	0.3164	0.2401	0.1296	0.0625	0.0256	0.0081	0.0039	0.0016	0.0001	0+	0+
1	0.9994	0.9860	0.9477	0.8192	0.7383	0.6517	0.4752	0.3125	0.1792	0.0837	0.0508	0.0272	0.0037	0.0005	0+
2	1−	0.9995	0.9963	0.9728	0.9492	0.9163	0.8208	0.6875	0.5248	0.3483	0.2617	0.1808	0.0523	0.0140	0.0006
3	1−	1−	0.9999	0.9984	0.9961	0.9919	0.9744	0.9375	0.8704	0.7599	0.6836	0.5904	0.3439	0.1855	0.0394
4	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 5

n = 5
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.9510	0.7738	0.5905	0.3277	0.2373	0.1681	0.0778	0.0312	0.0102	0.0024	0.0010	0.0003	0+	0+	0+
1	0.9990	0.9774	0.9185	0.7373	0.6328	0.5282	0.3370	0.1875	0.0870	0.0308	0.0156	0.0067	0.0005	0+	0+
2	1−	0.9988	0.9914	0.9421	0.8965	0.8369	0.6826	0.5000	0.3174	0.1631	0.1035	0.0579	0.0086	0.0012	0+
3	1−	1−	0.9995	0.9933	0.9844	0.9692	0.9130	0.8125	0.6630	0.4718	0.3672	0.2627	0.0815	0.0226	0.0010
4	1−	1−	1−	0.9997	0.9990	0.9976	0.9898	0.9688	0.9222	0.8319	0.7627	0.6723	0.4095	0.2262	0.0490
5	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 6

n = 6
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.9415	0.7351	0.5314	0.2621	0.1780	0.1176	0.0467	0.0156	0.0041	0.0007	0.0002	0.0001	0+	0+	0+
1	0.9985	0.9672	0.8857	0.6554	0.5339	0.4202	0.2333	0.1094	0.0410	0.0109	0.0046	0.0016	0.0001	0+	0+
2	1−	0.9978	0.9841	0.9011	0.8306	0.7443	0.5443	0.3438	0.1792	0.0705	0.0376	0.0170	0.0013	0.0001	0+
3	1−	0.9999	0.9987	0.9830	0.9624	0.9295	0.8208	0.6562	0.4557	0.2557	0.1694	0.0989	0.0158	0.0022	0+
4	1−	1−	0.9999	0.9984	0.9954	0.9891	0.9590	0.8906	0.7667	0.5798	0.4661	0.3446	0.1143	0.0328	0.0015
5	1−	1−	1−	0.9999	0.9998	0.9993	0.9959	0.9844	0.9533	0.8824	0.8220	0.7379	0.4686	0.2649	0.0585
6	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 7

n = 7
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.9321	0.6983	0.4783	0.2097	0.1335	0.0824	0.0280	0.0078	0.0016	0.0002	0.0001	0+	0+	0+	0+
1	0.9980	0.9556	0.8503	0.5767	0.4449	0.3294	0.1586	0.0625	0.0188	0.0038	0.0013	0.0004	0+	0+	0+
2	1−	0.9962	0.9743	0.8520	0.7564	0.6471	0.4199	0.2266	0.0963	0.0288	0.0129	0.0047	0.0002	0+	0+
3	1−	0.9998	0.9973	0.9667	0.9294	0.8740	0.7102	0.5000	0.2898	0.1260	0.0706	0.0333	0.0027	0.0002	0+
4	1−	1−	0.9998	0.9953	0.9871	0.9712	0.9037	0.7734	0.5801	0.3529	0.2436	0.1480	0.0257	0.0038	0+
5	1−	1−	1−	0.9996	0.9987	0.9962	0.9812	0.9375	0.8414	0.6706	0.5551	0.4233	0.1497	0.0444	0.0020
6	1−	1−	1−	1−	0.9999	0.9998	0.9984	0.9922	0.9720	0.9176	0.8665	0.7903	0.5217	0.3017	0.0679
7	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 8

n = 8
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.9227	0.6634	0.4305	0.1678	0.1001	0.0576	0.0168	0.0039	0.0007	0.0001	0+	0+	0+	0+	0+
1	0.9973	0.9428	0.8131	0.5033	0.3671	0.2553	0.1064	0.0352	0.0085	0.0013	0.0004	0.0001	0+	0+	0+
2	0.9999	0.9942	0.9619	0.7969	0.6785	0.5518	0.3154	0.1445	0.0498	0.0113	0.0042	0.0012	0+	0+	0+
3	1−	0.9996	0.9950	0.9437	0.8862	0.8059	0.5941	0.3633	0.1737	0.0580	0.0273	0.0104	0.0004	0+	0+
4	1−	1−	0.9996	0.9896	0.9727	0.9420	0.8263	0.6367	0.4059	0.1941	0.1138	0.0563	0.0050	0.0004	0+
5	1−	1−	1−	0.9988	0.9958	0.9887	0.9502	0.8555	0.6846	0.4482	0.3215	0.2031	0.0381	0.0058	0.0001
6	1−	1−	1−	0.9999	0.9996	0.9987	0.9915	0.9648	0.8936	0.7447	0.6329	0.4967	0.1869	0.0572	0.0027
7	1−	1−	1−	1−	1−	0.9999	0.9993	0.9961	0.9832	0.9424	0.8999	0.8322	0.5695	0.3366	0.0773
8	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 9

n = 9
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.9135	0.6302	0.3874	0.1342	0.0751	0.0404	0.0101	0.0020	0.0003	0+	0+	0+	0+	0+	0+
1	0.9966	0.9288	0.7748	0.4362	0.3003	0.1960	0.0705	0.0195	0.0038	0.0004	0.0001	0+	0+	0+	0+
2	0.9999	0.9916	0.9470	0.7382	0.6007	0.4628	0.2318	0.0898	0.0250	0.0043	0.0013	0.0003	0+	0+	0+
3	1−	0.9994	0.9917	0.9144	0.8343	0.7297	0.4826	0.2539	0.0994	0.0253	0.0100	0.0031	0.0001	0+	0+
4	1−	1−	0.9991	0.9804	0.9511	0.9012	0.7334	0.5000	0.2666	0.0988	0.0489	0.0196	0.0009	0+	0+
5	1−	1−	0.9999	0.9969	0.9900	0.9747	0.9006	0.7461	0.5174	0.2703	0.1657	0.0856	0.0083	0.0006	0+
6	1−	1−	1−	0.9997	0.9987	0.9957	0.9750	0.9102	0.7682	0.5372	0.3993	0.2618	0.0530	0.0084	0.0001
7	1−	1−	1−	1−	0.9999	0.9996	0.9962	0.9805	0.9295	0.8040	0.6997	0.5638	0.2252	0.0712	0.0034
8	1.0000	1−	1−	1−	1−	1−	0.9997	0.9980	0.9899	0.9596	0.9249	0.8658	0.6126	0.3698	0.0865
9	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 10

n = 10
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.9044	0.5987	0.3487	0.1074	0.0563	0.0282	0.0060	0.0010	0.0001	0+	0+	0+	0+	0+	0+
1	0.9957	0.9139	0.7361	0.3758	0.2440	0.1493	0.0464	0.0107	0.0017	0.0001	0+	0+	0+	0+	0+
2	0.9999	0.9885	0.9298	0.6778	0.5256	0.3828	0.1673	0.0547	0.0123	0.0016	0.0004	0.0001	0+	0+	0+
3	1−	0.9990	0.9872	0.8791	0.7759	0.6496	0.3823	0.1719	0.0548	0.0106	0.0035	0.0009	0+	0+	0+
4	1−	0.9999	0.9984	0.9672	0.9219	0.8497	0.6331	0.3770	0.1662	0.0473	0.0197	0.0064	0.0001	0+	0+
5	1−	1−	0.9999	0.9936	0.9803	0.9527	0.8338	0.6230	0.3669	0.1503	0.0781	0.0328	0.0016	0.0001	0+
6	1−	1−	1−	0.9991	0.9965	0.9894	0.9452	0.8281	0.6177	0.3504	0.2241	0.1209	0.0128	0.0010	0+
7	1−	1−	1−	0.9999	0.9996	0.9984	0.9877	0.9453	0.8327	0.6172	0.4744	0.3222	0.0702	0.0115	0.0001
8	1.0000	1−	1−	1−	1−	0.9999	0.9983	0.9893	0.9536	0.8507	0.7560	0.6242	0.2639	0.0861	0.0043
9	1.0000	1−	1−	1−	1−	1−	0.9999	0.9990	0.9940	0.9718	0.9437	0.8926	0.6513	0.4013	0.0956
10	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 11

n = 11
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.8953	0.5688	0.3138	0.0859	0.0422	0.0198	0.0036	0.0005	0+	0+	0+	0+	0+	0+	0+
1	0.9948	0.8981	0.6974	0.3221	0.1971	0.1130	0.0302	0.0059	0.0007	0+	0+	0+	0+	0+	0+
2	0.9998	0.9848	0.9104	0.6174	0.4552	0.3127	0.1189	0.0327	0.0059	0.0006	0.0001	0+	0+	0+	0+
3	1−	0.9984	0.9815	0.8389	0.7133	0.5696	0.2963	0.1133	0.0293	0.0043	0.0012	0.0002	0+	0+	0+
4	1−	0.9999	0.9972	0.9496	0.8854	0.7897	0.5328	0.2744	0.0994	0.0216	0.0076	0.0020	0+	0+	0+
5	1−	1−	0.9997	0.9883	0.9657	0.9218	0.7535	0.5000	0.2465	0.0782	0.0343	0.0117	0.0003	0+	0+
6	1−	1−	1−	0.9980	0.9924	0.9784	0.9006	0.7256	0.4672	0.2103	0.1146	0.0504	0.0028	0.0001	0+
7	1−	1−	1−	0.9998	0.9988	0.9957	0.9707	0.8867	0.7037	0.4304	0.2867	0.1611	0.0185	0.0016	0+
8	1.0000	1−	1−	1−	0.9999	0.9994	0.9941	0.9673	0.8811	0.6873	0.5448	0.3826	0.0896	0.0152	0.0002
9	1.0000	1−	1−	1−	1−	1−	0.9993	0.9941	0.9698	0.8870	0.8029	0.6779	0.3026	0.1019	0.0052
10	1.0000	1−	1−	1−	1−	1−	1−	0.9995	0.9964	0.9802	0.9578	0.9141	0.6862	0.4312	0.1047
11	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 12

n = 12
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.8864	0.5404	0.2824	0.0687	0.0317	0.0138	0.0022	0.0002	0+	0+	0+	0+	0+	0+	0+
1	0.9938	0.8816	0.6590	0.2749	0.1584	0.0850	0.0196	0.0032	0.0003	0+	0+	0+	0+	0+	0+
2	0.9998	0.9804	0.8891	0.5583	0.3907	0.2528	0.0834	0.0193	0.0028	0.0002	0+	0+	0+	0+	0+
3	1−	0.9978	0.9744	0.7946	0.6488	0.4925	0.2253	0.0730	0.0153	0.0017	0.0004	0.0001	0+	0+	0+
4	1−	0.9998	0.9957	0.9274	0.8424	0.7237	0.4382	0.1938	0.0573	0.0095	0.0028	0.0006	0+	0+	0+
5	1−	1−	0.9995	0.9806	0.9456	0.8822	0.6652	0.3872	0.1582	0.0386	0.0143	0.0039	0.0001	0+	0+
6	1−	1−	0.9999	0.9961	0.9857	0.9614	0.8418	0.6128	0.3348	0.1178	0.0544	0.0194	0.0005	0+	0+
7	1−	1−	1−	0.9994	0.9972	0.9905	0.9427	0.8062	0.5618	0.2763	0.1576	0.0726	0.0043	0.0002	0+
8	1−	1−	1−	0.9999	0.9996	0.9983	0.9847	0.9270	0.7747	0.5075	0.3512	0.2054	0.0256	0.0022	0+
9	1.0000	1−	1−	1−	1−	0.9998	0.9972	0.9807	0.9166	0.7472	0.6093	0.4417	0.1109	0.0196	0.0002
10	1.0000	1−	1−	1−	1−	1−	0.9997	0.9968	0.9804	0.9150	0.8416	0.7251	0.3410	0.1184	0.0062
11	1.0000	1−	1−	1−	1−	1−	1−	0.9998	0.9978	0.9862	0.9683	0.9313	0.7176	0.4596	0.1136
12	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 13

n = 13
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.8775	0.5133	0.2542	0.0550	0.0238	0.0097	0.0013	0.0001	0+	0+	0+	0+	0+	0+	0+
1	0.9928	0.8646	0.6213	0.2336	0.1267	0.0637	0.0126	0.0017	0.0001	0+	0+	0+	0+	0+	0+
2	0.9997	0.9755	0.8661	0.5017	0.3326	0.2025	0.0579	0.0112	0.0013	0.0001	0+	0+	0+	0+	0+
3	1−	0.9969	0.9658	0.7473	0.5843	0.4206	0.1686	0.0461	0.0078	0.0007	0.0001	0+	0+	0+	0+
4	1−	0.9997	0.9935	0.9009	0.7940	0.6543	0.3530	0.1334	0.0321	0.0040	0.0010	0.0002	0+	0+	0+
5	1−	1−	0.9991	0.9700	0.9198	0.8346	0.5744	0.2905	0.0977	0.0182	0.0056	0.0012	0+	0+	0+
6	1−	1−	0.9999	0.9930	0.9757	0.9376	0.7712	0.5000	0.2288	0.0624	0.0243	0.0070	0.0001	0+	0+
7	1−	1−	1−	0.9988	0.9944	0.9818	0.9023	0.7095	0.4256	0.1654	0.0802	0.0300	0.0009	0+	0+
8	1−	1−	1−	0.9998	0.9990	0.9960	0.9679	0.8666	0.6470	0.3457	0.2060	0.0991	0.0065	0.0003	0+
9	1.0000	1−	1−	1−	0.9999	0.9993	0.9922	0.9539	0.8314	0.5794	0.4157	0.2527	0.0342	0.0031	0+
10	1.0000	1−	1−	1−	1−	0.9999	0.9987	0.9888	0.9421	0.7975	0.6674	0.4983	0.1339	0.0245	0.0003
11	1.0000	1−	1−	1−	1−	1−	0.9999	0.9983	0.9874	0.9363	0.8733	0.7664	0.3787	0.1354	0.0072
12	1.0000	1.0000	1−	1−	1−	1−	1−	0.9999	0.9987	0.9903	0.9762	0.9450	0.7458	0.4867	0.1225
13	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 14

n = 14
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.8687	0.4877	0.2288	0.0440	0.0178	0.0068	0.0008	0.0001	0+	0+	0+	0+	0+	0+	0+
1	0.9916	0.8470	0.5846	0.1979	0.1010	0.0475	0.0081	0.0009	0.0001	0+	0+	0+	0+	0+	0+
2	0.9997	0.9699	0.8416	0.4481	0.2811	0.1608	0.0398	0.0065	0.0006	0+	0+	0+	0+	0+	0+
3	1−	0.9958	0.9559	0.6982	0.5213	0.3552	0.1243	0.0287	0.0039	0.0002	0+	0+	0+	0+	0+
4	1−	0.9996	0.9908	0.8702	0.7415	0.5842	0.2793	0.0898	0.0175	0.0017	0.0003	0+	0+	0+	0+
5	1−	1−	0.9985	0.9561	0.8883	0.7805	0.4859	0.2120	0.0583	0.0083	0.0022	0.0004	0+	0+	0+
6	1−	1−	0.9998	0.9884	0.9617	0.9067	0.6925	0.3953	0.1501	0.0315	0.0103	0.0024	0+	0+	0+
7	1−	1−	1−	0.9976	0.9897	0.9685	0.8499	0.6047	0.3075	0.0933	0.0383	0.0116	0.0002	0+	0+
8	1−	1−	1−	0.9996	0.9978	0.9917	0.9417	0.7880	0.5141	0.2195	0.1117	0.0439	0.0015	0+	0+
9	1.0000	1−	1−	1−	0.9997	0.9983	0.9825	0.9102	0.7207	0.4158	0.2585	0.1298	0.0092	0.0004	0+
10	1.0000	1−	1−	1−	1−	0.9998	0.9961	0.9713	0.8757	0.6448	0.4787	0.3018	0.0441	0.0042	0+
11	1.0000	1−	1−	1−	1−	1−	0.9994	0.9935	0.9602	0.8392	0.7189	0.5519	0.1584	0.0301	0.0003
12	1.0000	1−	1−	1−	1−	1−	0.9999	0.9991	0.9919	0.9525	0.8990	0.8021	0.4154	0.1530	0.0084
13	1.0000	1.0000	1−	1−	1−	1−	1−	0.9999	0.9992	0.9932	0.9822	0.9560	0.7712	0.5123	0.1313
14	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 15

n = 15
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.8601	0.4633	0.2059	0.0352	0.0134	0.0047	0.0005	0+	0+	0+	0+	0+	0+	0+	0+
1	0.9904	0.8290	0.5490	0.1671	0.0802	0.0353	0.0052	0.0005	0+	0+	0+	0+	0+	0+	0+
2	0.9996	0.9638	0.8159	0.3980	0.2361	0.1268	0.0271	0.0037	0.0003	0+	0+	0+	0+	0+	0+
3	1−	0.9945	0.9444	0.6482	0.4613	0.2969	0.0905	0.0176	0.0019	0.0001	0+	0+	0+	0+	0+
4	1−	0.9994	0.9873	0.8358	0.6865	0.5155	0.2173	0.0592	0.0093	0.0007	0.0001	0+	0+	0+	0+
5	1−	0.9999	0.9978	0.9389	0.8516	0.7216	0.4032	0.1509	0.0338	0.0037	0.0008	0.0001	0+	0+	0+
6	1−	1−	0.9997	0.9819	0.9434	0.8689	0.6098	0.3036	0.0950	0.0152	0.0042	0.0008	0+	0+	0+
7	1−	1−	1−	0.9958	0.9827	0.9500	0.7869	0.5000	0.2131	0.0500	0.0173	0.0042	0+	0+	0+
8	1−	1−	1−	0.9992	0.9958	0.9848	0.9050	0.6964	0.3902	0.1311	0.0566	0.0181	0.0003	0+	0+
9	1.0000	1−	1−	0.9999	0.9992	0.9963	0.9662	0.8491	0.5968	0.2784	0.1484	0.0611	0.0022	0.0001	0+
10	1.0000	1−	1−	1−	0.9999	0.9993	0.9907	0.9408	0.7827	0.4845	0.3135	0.1642	0.0127	0.0006	0+
11	1.0000	1−	1−	1−	1−	0.9999	0.9981	0.9824	0.9095	0.7031	0.5387	0.3518	0.0556	0.0055	0+
12	1.0000	1−	1−	1−	1−	1−	0.9997	0.9963	0.9729	0.8732	0.7639	0.6020	0.1841	0.0362	0.0004
13	1.0000	1.0000	1−	1−	1−	1−	1−	0.9995	0.9948	0.9647	0.9198	0.8329	0.4510	0.1710	0.0096
14	1.0000	1.0000	1−	1−	1−	1−	1−	1−	0.9995	0.9953	0.9866	0.9648	0.7941	0.5367	0.1399
15	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 16

n = 16
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.8515	0.4401	0.1853	0.0281	0.0100	0.0033	0.0003	0+	0+	0+	0+	0+	0+	0+	0+
1	0.9891	0.8108	0.5147	0.1407	0.0635	0.0261	0.0033	0.0003	0+	0+	0+	0+	0+	0+	0+
2	0.9995	0.9571	0.7892	0.3518	0.1971	0.0994	0.0183	0.0021	0.0001	0+	0+	0+	0+	0+	0+
3	1−	0.9930	0.9316	0.5981	0.4050	0.2459	0.0651	0.0106	0.0009	0+	0+	0+	0+	0+	0+
4	1−	0.9991	0.9830	0.7982	0.6302	0.4499	0.1666	0.0384	0.0049	0.0003	0+	0+	0+	0+	0+
5	1−	0.9999	0.9967	0.9183	0.8103	0.6598	0.3288	0.1051	0.0191	0.0016	0.0003	0+	0+	0+	0+
6	1−	1−	0.9995	0.9733	0.9204	0.8247	0.5272	0.2272	0.0583	0.0071	0.0016	0.0002	0+	0+	0+
7	1−	1−	0.9999	0.9930	0.9729	0.9256	0.7161	0.4018	0.1423	0.0257	0.0075	0.0015	0+	0+	0+
8	1−	1−	1−	0.9985	0.9925	0.9743	0.8577	0.5982	0.2839	0.0744	0.0271	0.0070	0.0001	0+	0+
9	1−	1−	1−	0.9998	0.9984	0.9929	0.9417	0.7728	0.4728	0.1753	0.0796	0.0267	0.0005	0+	0+
10	1.0000	1−	1−	1−	0.9997	0.9984	0.9809	0.8949	0.6712	0.3402	0.1897	0.0817	0.0033	0.0001	0+
11	1.0000	1−	1−	1−	1−	0.9997	0.9951	0.9616	0.8334	0.5501	0.3698	0.2018	0.0170	0.0009	0+
12	1.0000	1−	1−	1−	1−	1−	0.9991	0.9894	0.9349	0.7541	0.5950	0.4019	0.0684	0.0070	0+
13	1.0000	1−	1−	1−	1−	1−	0.9999	0.9979	0.9817	0.9006	0.8029	0.6482	0.2108	0.0429	0.0005
14	1.0000	1.0000	1−	1−	1−	1−	1−	0.9997	0.9967	0.9739	0.9365	0.8593	0.4853	0.1892	0.0109
15	1.0000	1.0000	1−	1−	1−	1−	1−	1−	0.9997	0.9967	0.9900	0.9719	0.8147	0.5599	0.1485
16	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 17

n = 17
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.8429	0.4181	0.1668	0.0225	0.0075	0.0023	0.0002	0+	0+	0+	0+	0+	0+	0+	0+
1	0.9877	0.7922	0.4818	0.1182	0.0501	0.0193	0.0021	0.0001	0+	0+	0+	0+	0+	0+	0+
2	0.9994	0.9497	0.7618	0.3096	0.1637	0.0774	0.0123	0.0012	0.0001	0+	0+	0+	0+	0+	0+
3	1−	0.9912	0.9174	0.5489	0.3530	0.2019	0.0464	0.0064	0.0005	0+	0+	0+	0+	0+	0+
4	1−	0.9988	0.9779	0.7582	0.5739	0.3887	0.1260	0.0245	0.0025	0.0001	0+	0+	0+	0+	0+
5	1−	0.9999	0.9953	0.8943	0.7653	0.5968	0.2639	0.0717	0.0106	0.0007	0.0001	0+	0+	0+	0+
6	1−	1−	0.9992	0.9623	0.8929	0.7752	0.4478	0.1662	0.0348	0.0032	0.0006	0.0001	0+	0+	0+
7	1−	1−	0.9999	0.9891	0.9598	0.8954	0.6405	0.3145	0.0919	0.0127	0.0031	0.0005	0+	0+	0+
8	1−	1−	1−	0.9974	0.9876	0.9597	0.8011	0.5000	0.1989	0.0403	0.0124	0.0026	0+	0+	0+
9	1−	1−	1−	0.9995	0.9969	0.9873	0.9081	0.6855	0.3595	0.1046	0.0402	0.0109	0.0001	0+	0+
10	1.0000	1−	1−	0.9999	0.9994	0.9968	0.9652	0.8338	0.5522	0.2248	0.1071	0.0377	0.0008	0+	0+
11	1.0000	1−	1−	1−	0.9999	0.9993	0.9894	0.9283	0.7361	0.4032	0.2347	0.1057	0.0047	0.0001	0+
12	1.0000	1−	1−	1−	1−	0.9999	0.9975	0.9755	0.8740	0.6113	0.4261	0.2418	0.0221	0.0012	0+
13	1.0000	1−	1−	1−	1−	1−	0.9995	0.9936	0.9536	0.7981	0.6470	0.4511	0.0826	0.0088	0+
14	1.0000	1.0000	1−	1−	1−	1−	0.9999	0.9988	0.9877	0.9226	0.8363	0.6904	0.2382	0.0503	0.0006
15	1.0000	1.0000	1−	1−	1−	1−	1−	0.9999	0.9979	0.9807	0.9499	0.8818	0.5182	0.2078	0.0123
16	1.0000	1.0000	1.0000	1−	1−	1−	1−	1−	0.9998	0.9977	0.9925	0.9775	0.8332	0.5819	0.1571
17	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 18

n = 18
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.8345	0.3972	0.1501	0.0180	0.0056	0.0016	0.0001	0+	0+	0+	0+	0+	0+	0+	0+
1	0.9862	0.7735	0.4503	0.0991	0.0395	0.0142	0.0013	0.0001	0+	0+	0+	0+	0+	0+	0+
2	0.9993	0.9419	0.7338	0.2713	0.1353	0.0600	0.0082	0.0007	0+	0+	0+	0+	0+	0+	0+
3	1−	0.9891	0.9018	0.5010	0.3057	0.1646	0.0328	0.0038	0.0002	0+	0+	0+	0+	0+	0+
4	1−	0.9985	0.9718	0.7164	0.5187	0.3327	0.0942	0.0154	0.0013	0+	0+	0+	0+	0+	0+
5	1−	0.9998	0.9936	0.8671	0.7175	0.5344	0.2088	0.0481	0.0058	0.0003	0+	0+	0+	0+	0+
6	1−	1−	0.9988	0.9487	0.8610	0.7217	0.3743	0.1189	0.0203	0.0014	0.0002	0+	0+	0+	0+
7	1−	1−	0.9998	0.9837	0.9431	0.8593	0.5634	0.2403	0.0576	0.0061	0.0012	0.0002	0+	0+	0+
8	1−	1−	1−	0.9957	0.9807	0.9404	0.7368	0.4073	0.1347	0.0210	0.0054	0.0009	0+	0+	0+
9	1−	1−	1−	0.9991	0.9946	0.9790	0.8653	0.5927	0.2632	0.0596	0.0193	0.0043	0+	0+	0+
10	1.0000	1−	1−	0.9998	0.9988	0.9939	0.9424	0.7597	0.4366	0.1407	0.0569	0.0163	0.0002	0+	0+
11	1.0000	1−	1−	1−	0.9998	0.9986	0.9797	0.8811	0.6257	0.2783	0.1390	0.0513	0.0012	0+	0+
12	1.0000	1−	1−	1−	1−	0.9997	0.9942	0.9519	0.7912	0.4656	0.2825	0.1329	0.0064	0.0002	0+
13	1.0000	1−	1−	1−	1−	1−	0.9987	0.9846	0.9058	0.6673	0.4813	0.2836	0.0282	0.0015	0+
14	1.0000	1.0000	1−	1−	1−	1−	0.9998	0.9962	0.9672	0.8354	0.6943	0.4990	0.0982	0.0109	0+
15	1.0000	1.0000	1−	1−	1−	1−	1−	0.9993	0.9918	0.9400	0.8647	0.7287	0.2662	0.0581	0.0007
16	1.0000	1.0000	1−	1−	1−	1−	1−	0.9999	0.9987	0.9858	0.9605	0.9009	0.5497	0.2265	0.0138
17	1.0000	1.0000	1.0000	1−	1−	1−	1−	1−	0.9999	0.9984	0.9944	0.9820	0.8499	0.6028	0.1655
18	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 19

n = 19
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.8262	0.3774	0.1351	0.0144	0.0042	0.0011	0.0001	0+	0+	0+	0+	0+	0+	0+	0+
1	0.9847	0.7547	0.4203	0.0829	0.0310	0.0104	0.0008	0+	0+	0+	0+	0+	0+	0+	0+
2	0.9991	0.9335	0.7054	0.2369	0.1113	0.0462	0.0055	0.0004	0+	0+	0+	0+	0+	0+	0+
3	1−	0.9868	0.8850	0.4551	0.2631	0.1332	0.0230	0.0022	0.0001	0+	0+	0+	0+	0+	0+
4	1−	0.9980	0.9648	0.6733	0.4654	0.2822	0.0696	0.0096	0.0006	0+	0+	0+	0+	0+	0+
5	1−	0.9998	0.9914	0.8369	0.6678	0.4739	0.1629	0.0318	0.0031	0.0001	0+	0+	0+	0+	0+
6	1−	1−	0.9983	0.9324	0.8251	0.6655	0.3081	0.0835	0.0116	0.0006	0.0001	0+	0+	0+	0+
7	1−	1−	0.9997	0.9767	0.9225	0.8180	0.4878	0.1796	0.0352	0.0028	0.0005	0+	0+	0+	0+
8	1−	1−	1−	0.9933	0.9713	0.9161	0.6675	0.3238	0.0885	0.0105	0.0023	0.0003	0+	0+	0+
9	1−	1−	1−	0.9984	0.9911	0.9674	0.8139	0.5000	0.1861	0.0326	0.0089	0.0016	0+	0+	0+
10	1.0000	1−	1−	0.9997	0.9977	0.9895	0.9115	0.6762	0.3325	0.0839	0.0287	0.0067	0+	0+	0+
11	1.0000	1−	1−	1−	0.9995	0.9972	0.9648	0.8204	0.5122	0.1820	0.0775	0.0233	0.0003	0+	0+
12	1.0000	1−	1−	1−	0.9999	0.9994	0.9884	0.9165	0.6919	0.3345	0.1749	0.0676	0.0017	0+	0+
13	1.0000	1−	1−	1−	1−	0.9999	0.9969	0.9682	0.8371	0.5261	0.3322	0.1631	0.0086	0.0002	0+
14	1.0000	1−	1−	1−	1−	1−	0.9994	0.9904	0.9304	0.7178	0.5346	0.3267	0.0352	0.0020	0+
15	1.0000	1.0000	1−	1−	1−	1−	0.9999	0.9978	0.9770	0.8668	0.7369	0.5449	0.1150	0.0132	0+
16	1.0000	1.0000	1−	1−	1−	1−	1−	0.9996	0.9945	0.9538	0.8887	0.7631	0.2946	0.0665	0.0009
17	1.0000	1.0000	1.0000	1−	1−	1−	1−	1−	0.9992	0.9896	0.9690	0.9171	0.5797	0.2453	0.0153
18	1.0000	1.0000	1.0000	1−	1−	1−	1−	1−	0.9999	0.9989	0.9958	0.9856	0.8649	0.6226	0.1738
19	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

n = 20

n = 20
Probability of success (p) → x ↓	.01	.05	.10	.20	.25	.30	.40	.50	.60	.70	.75	.80	.90	.95	.99
0	0.8179	0.3585	0.1216	0.0115	0.0032	0.0008	0+	0+	0+	0+	0+	0+	0+	0+	0+
1	0.9831	0.7358	0.3917	0.0692	0.0243	0.0076	0.0005	0+	0+	0+	0+	0+	0+	0+	0+
2	0.9990	0.9245	0.6769	0.2061	0.0913	0.0355	0.0036	0.0002	0+	0+	0+	0+	0+	0+	0+
3	1−	0.9841	0.8670	0.4114	0.2252	0.1071	0.0160	0.0013	0+	0+	0+	0+	0+	0+	0+
4	1−	0.9974	0.9568	0.6296	0.4148	0.2375	0.0510	0.0059	0.0003	0+	0+	0+	0+	0+	0+
5	1−	0.9997	0.9887	0.8042	0.6172	0.4164	0.1256	0.0207	0.0016	0+	0+	0+	0+	0+	0+
6	1−	1−	0.9976	0.9133	0.7858	0.6080	0.2500	0.0577	0.0065	0.0003	0+	0+	0+	0+	0+
7	1−	1−	0.9996	0.9679	0.8982	0.7723	0.4159	0.1316	0.0210	0.0013	0.0002	0+	0+	0+	0+
8	1−	1−	0.9999	0.9900	0.9591	0.8867	0.5956	0.2517	0.0565	0.0051	0.0009	0.0001	0+	0+	0+
9	1−	1−	1−	0.9974	0.9861	0.9520	0.7553	0.4119	0.1275	0.0171	0.0039	0.0006	0+	0+	0+
10	1.0000	1−	1−	0.9994	0.9961	0.9829	0.8725	0.5881	0.2447	0.0480	0.0139	0.0026	0+	0+	0+
11	1.0000	1−	1−	0.9999	0.9991	0.9949	0.9435	0.7483	0.4044	0.1133	0.0409	0.0100	0.0001	0+	0+
12	1.0000	1−	1−	1−	0.9998	0.9987	0.9790	0.8684	0.5841	0.2277	0.1018	0.0321	0.0004	0+	0+
13	1.0000	1−	1−	1−	1−	0.9997	0.9935	0.9423	0.7500	0.3920	0.2142	0.0867	0.0024	0+	0+
14	1.0000	1−	1−	1−	1−	1−	0.9984	0.9793	0.8744	0.5836	0.3828	0.1958	0.0113	0.0003	0+
15	1.0000	1.0000	1−	1−	1−	1−	0.9997	0.9941	0.9490	0.7625	0.5852	0.3704	0.0432	0.0026	0+
16	1.0000	1.0000	1−	1−	1−	1−	1−	0.9987	0.9840	0.8929	0.7748	0.5886	0.1330	0.0159	0+
17	1.0000	1.0000	1−	1−	1−	1−	1−	0.9998	0.9964	0.9645	0.9087	0.7939	0.3231	0.0755	0.0010
18	1.0000	1.0000	1.0000	1−	1−	1−	1−	1−	0.9995	0.9924	0.9757	0.9308	0.6083	0.2642	0.0169
19	1.0000	1.0000	1.0000	1−	1−	1−	1−	1−	1−	0.9992	0.9968	0.9885	0.8784	0.6415	0.1821
20	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Tip: click any value to highlight its row and column.

How to read this table

Find the block for your number of trials n.
Go to the row for x and the column for p.
The cell is P(X ≤ x). For P(X ≥ x) use 1 − P(X ≤ x − 1); for P(a ≤ X ≤ b) use P(X ≤ b) − P(X ≤ a − 1).
0+ means a probability that is positive but below 0.00005; 1− means one that is below 1 but above 0.99995.

Worked example

In a Phase II trial of 10 patients with a true response rate of 20%, the chance of at most 2 responses is row x = 2, column p = .20 in the n = 10 block: 0.6778. The chance of 3 or more responses is 1 − 0.6778 = 0.3222.

This is the calculation behind single-arm Phase II designs such as Simon's two-stage design, which choose sample sizes and response thresholds from binomial probabilities.

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(n, x) p^x (1 − p)^{n − x} · P(X ≤ x) = Σ_k=0^x P(X = k)

Each cell is computed exactly from the binomial formula and rounded to four decimals.

Frequently asked questions

What are the conditions for a binomial distribution?

A fixed number of trials n, two outcomes per trial, the same probability of success p on every trial, and independent trials.

What if n is larger than 20?

Use the calculator above, which handles any n. When np and n(1 − p) are both at least about 10, the normal approximation with mean np and variance np(1 − p) also works well.

Cumulative Binomial Distribution Table: P(X ≤ x)

Cumulative Binomial Table

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

n = 11

n = 12

n = 13

n = 14

n = 15

n = 16

n = 17

n = 18

n = 19

n = 20

How to read this table

Worked example

Calculator

Formula and how the values were computed

Frequently asked questions

Related calculators and tutorials