Discrete Distributions

Hypergeometric Distribution Table: P(X = x)

Probabilities for the number of successes in a sample drawn without replacement from a finite population.

N = 10 and 20
Every K and n
Calculator for any N

Hypergeometric Table

Jump to a population and sample size below. Rows: successes in the sample x · Columns: successes in the population K · Cells: P(X = x)

N = 10, n = 2

N = 10, n = 2
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9
0	0.8000	0.6222	0.4667	0.3333	0.2222	0.1333	0.0667	0.0222	0
1	0.2000	0.3556	0.4667	0.5333	0.5556	0.5333	0.4667	0.3556	0.2000
2	0	0.0222	0.0667	0.1333	0.2222	0.3333	0.4667	0.6222	0.8000

N = 10, n = 3

N = 10, n = 3
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9
0	0.7000	0.4667	0.2917	0.1667	0.0833	0.0333	0.0083	0	0
1	0.3000	0.4667	0.5250	0.5000	0.4167	0.3000	0.1750	0.0667	0
2	0	0.0667	0.1750	0.3000	0.4167	0.5000	0.5250	0.4667	0.3000
3	0	0	0.0083	0.0333	0.0833	0.1667	0.2917	0.4667	0.7000

N = 10, n = 4

N = 10, n = 4
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9
0	0.6000	0.3333	0.1667	0.0714	0.0238	0.0048	0	0	0
1	0.4000	0.5333	0.5000	0.3810	0.2381	0.1143	0.0333	0	0
2	0	0.1333	0.3000	0.4286	0.4762	0.4286	0.3000	0.1333	0
3	0	0	0.0333	0.1143	0.2381	0.3810	0.5000	0.5333	0.4000
4	0	0	0	0.0048	0.0238	0.0714	0.1667	0.3333	0.6000

N = 10, n = 5

N = 10, n = 5
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9
0	0.5000	0.2222	0.0833	0.0238	0.0040	0	0	0	0
1	0.5000	0.5556	0.4167	0.2381	0.0992	0.0238	0	0	0
2	0	0.2222	0.4167	0.4762	0.3968	0.2381	0.0833	0	0
3	0	0	0.0833	0.2381	0.3968	0.4762	0.4167	0.2222	0
4	0	0	0	0.0238	0.0992	0.2381	0.4167	0.5556	0.5000
5	0	0	0	0	0.0040	0.0238	0.0833	0.2222	0.5000

N = 20, n = 2

N = 20, n = 2
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	0.9000	0.8053	0.7158	0.6316	0.5526	0.4789	0.4105	0.3474	0.2895	0.2368	0.1895	0.1474	0.1105	0.0789	0.0526	0.0316	0.0158	0.0053	0
1	0.1000	0.1895	0.2684	0.3368	0.3947	0.4421	0.4789	0.5053	0.5211	0.5263	0.5211	0.5053	0.4789	0.4421	0.3947	0.3368	0.2684	0.1895	0.1000
2	0	0.0053	0.0158	0.0316	0.0526	0.0789	0.1105	0.1474	0.1895	0.2368	0.2895	0.3474	0.4105	0.4789	0.5526	0.6316	0.7158	0.8053	0.9000

N = 20, n = 3

N = 20, n = 3
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	0.8500	0.7158	0.5965	0.4912	0.3991	0.3193	0.2509	0.1930	0.1447	0.1053	0.0737	0.0491	0.0307	0.0175	0.0088	0.0035	0.0009	0	0
1	0.1500	0.2684	0.3579	0.4211	0.4605	0.4789	0.4789	0.4632	0.4342	0.3947	0.3474	0.2947	0.2395	0.1842	0.1316	0.0842	0.0447	0.0158	0
2	0	0.0158	0.0447	0.0842	0.1316	0.1842	0.2395	0.2947	0.3474	0.3947	0.4342	0.4632	0.4789	0.4789	0.4605	0.4211	0.3579	0.2684	0.1500
3	0	0	0.0009	0.0035	0.0088	0.0175	0.0307	0.0491	0.0737	0.1053	0.1447	0.1930	0.2509	0.3193	0.3991	0.4912	0.5965	0.7158	0.8500

N = 20, n = 4

N = 20, n = 4
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	0.8000	0.6316	0.4912	0.3756	0.2817	0.2066	0.1476	0.1022	0.0681	0.0433	0.0260	0.0144	0.0072	0.0031	0.0010	0.0002	0	0	0
1	0.2000	0.3368	0.4211	0.4623	0.4696	0.4508	0.4132	0.3633	0.3065	0.2477	0.1907	0.1387	0.0939	0.0578	0.0310	0.0132	0.0035	0	0
2	0	0.0316	0.0842	0.1486	0.2167	0.2817	0.3381	0.3814	0.4087	0.4180	0.4087	0.3814	0.3381	0.2817	0.2167	0.1486	0.0842	0.0316	0
3	0	0	0.0035	0.0132	0.0310	0.0578	0.0939	0.1387	0.1907	0.2477	0.3065	0.3633	0.4132	0.4508	0.4696	0.4623	0.4211	0.3368	0.2000
4	0	0	0	0.0002	0.0010	0.0031	0.0072	0.0144	0.0260	0.0433	0.0681	0.1022	0.1476	0.2066	0.2817	0.3756	0.4912	0.6316	0.8000

N = 20, n = 5

N = 20, n = 5
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	0.7500	0.5526	0.3991	0.2817	0.1937	0.1291	0.0830	0.0511	0.0298	0.0163	0.0081	0.0036	0.0014	0.0004	0.0001	0	0	0	0
1	0.2500	0.3947	0.4605	0.4696	0.4402	0.3874	0.3228	0.2554	0.1916	0.1354	0.0894	0.0542	0.0293	0.0135	0.0048	0.0010	0	0	0
2	0	0.0526	0.1316	0.2167	0.2935	0.3522	0.3874	0.3973	0.3831	0.3483	0.2980	0.2384	0.1761	0.1174	0.0677	0.0310	0.0088	0	0
3	0	0	0.0088	0.0310	0.0677	0.1174	0.1761	0.2384	0.2980	0.3483	0.3831	0.3973	0.3874	0.3522	0.2935	0.2167	0.1316	0.0526	0
4	0	0	0	0.0010	0.0048	0.0135	0.0293	0.0542	0.0894	0.1354	0.1916	0.2554	0.3228	0.3874	0.4402	0.4696	0.4605	0.3947	0.2500
5	0	0	0	0	0.0001	0.0004	0.0014	0.0036	0.0081	0.0163	0.0298	0.0511	0.0830	0.1291	0.1937	0.2817	0.3991	0.5526	0.7500

N = 20, n = 6

N = 20, n = 6
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	0.7000	0.4789	0.3193	0.2066	0.1291	0.0775	0.0443	0.0238	0.0119	0.0054	0.0022	0.0007	0.0002	0+	0	0	0	0	0
1	0.3000	0.4421	0.4789	0.4508	0.3874	0.3099	0.2324	0.1635	0.1073	0.0650	0.0358	0.0173	0.0070	0.0022	0.0004	0	0	0	0
2	0	0.0789	0.1842	0.2817	0.3522	0.3874	0.3874	0.3576	0.3065	0.2438	0.1788	0.1192	0.0704	0.0352	0.0135	0.0031	0	0	0
3	0	0	0.0175	0.0578	0.1174	0.1878	0.2583	0.3179	0.3576	0.3715	0.3576	0.3179	0.2583	0.1878	0.1174	0.0578	0.0175	0	0
4	0	0	0	0.0031	0.0135	0.0352	0.0704	0.1192	0.1788	0.2438	0.3065	0.3576	0.3874	0.3874	0.3522	0.2817	0.1842	0.0789	0
5	0	0	0	0	0.0004	0.0022	0.0070	0.0173	0.0358	0.0650	0.1073	0.1635	0.2324	0.3099	0.3874	0.4508	0.4789	0.4421	0.3000
6	0	0	0	0	0	0+	0.0002	0.0007	0.0022	0.0054	0.0119	0.0238	0.0443	0.0775	0.1291	0.2066	0.3193	0.4789	0.7000

N = 20, n = 7

N = 20, n = 7
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	0.6500	0.4105	0.2509	0.1476	0.0830	0.0443	0.0221	0.0102	0.0043	0.0015	0.0005	0.0001	0+	0	0	0	0	0	0
1	0.3500	0.4789	0.4789	0.4132	0.3228	0.2324	0.1550	0.0954	0.0536	0.0271	0.0119	0.0043	0.0012	0.0002	0	0	0	0	0
2	0	0.1105	0.2395	0.3381	0.3874	0.3874	0.3486	0.2861	0.2146	0.1463	0.0894	0.0477	0.0211	0.0070	0.0014	0	0	0	0
3	0	0	0.0307	0.0939	0.1761	0.2583	0.3228	0.3576	0.3576	0.3251	0.2682	0.1987	0.1291	0.0704	0.0293	0.0072	0	0	0
4	0	0	0	0.0072	0.0293	0.0704	0.1291	0.1987	0.2682	0.3251	0.3576	0.3576	0.3228	0.2583	0.1761	0.0939	0.0307	0	0
5	0	0	0	0	0.0014	0.0070	0.0211	0.0477	0.0894	0.1463	0.2146	0.2861	0.3486	0.3874	0.3874	0.3381	0.2395	0.1105	0
6	0	0	0	0	0	0.0002	0.0012	0.0043	0.0119	0.0271	0.0536	0.0954	0.1550	0.2324	0.3228	0.4132	0.4789	0.4789	0.3500
7	0	0	0	0	0	0	0+	0.0001	0.0005	0.0015	0.0043	0.0102	0.0221	0.0443	0.0830	0.1476	0.2509	0.4105	0.6500

N = 20, n = 8

N = 20, n = 8
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	0.6000	0.3474	0.1930	0.1022	0.0511	0.0238	0.0102	0.0039	0.0013	0.0004	0.0001	0+	0	0	0	0	0	0	0
1	0.4000	0.5053	0.4632	0.3633	0.2554	0.1635	0.0954	0.0503	0.0236	0.0095	0.0031	0.0008	0.0001	0	0	0	0	0	0
2	0	0.1474	0.2947	0.3814	0.3973	0.3576	0.2861	0.2054	0.1320	0.0750	0.0367	0.0147	0.0043	0.0007	0	0	0	0	0
3	0	0	0.0491	0.1387	0.2384	0.3179	0.3576	0.3521	0.3081	0.2401	0.1650	0.0978	0.0477	0.0173	0.0036	0	0	0	0
4	0	0	0	0.0144	0.0542	0.1192	0.1987	0.2751	0.3301	0.3501	0.3301	0.2751	0.1987	0.1192	0.0542	0.0144	0	0	0
5	0	0	0	0	0.0036	0.0173	0.0477	0.0978	0.1650	0.2401	0.3081	0.3521	0.3576	0.3179	0.2384	0.1387	0.0491	0	0
6	0	0	0	0	0	0.0007	0.0043	0.0147	0.0367	0.0750	0.1320	0.2054	0.2861	0.3576	0.3973	0.3814	0.2947	0.1474	0
7	0	0	0	0	0	0	0.0001	0.0008	0.0031	0.0095	0.0236	0.0503	0.0954	0.1635	0.2554	0.3633	0.4632	0.5053	0.4000
8	0	0	0	0	0	0	0	0+	0.0001	0.0004	0.0013	0.0039	0.0102	0.0238	0.0511	0.1022	0.1930	0.3474	0.6000

N = 20, n = 9

N = 20, n = 9
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	0.5500	0.2895	0.1447	0.0681	0.0298	0.0119	0.0043	0.0013	0.0003	0.0001	0+	0	0	0	0	0	0	0	0
1	0.4500	0.5211	0.4342	0.3065	0.1916	0.1073	0.0536	0.0236	0.0088	0.0027	0.0006	0.0001	0	0	0	0	0	0	0
2	0	0.1895	0.3474	0.4087	0.3831	0.3065	0.2146	0.1320	0.0707	0.0322	0.0118	0.0031	0.0005	0	0	0	0	0	0
3	0	0	0.0737	0.1907	0.2980	0.3576	0.3576	0.3081	0.2311	0.1500	0.0825	0.0367	0.0119	0.0022	0	0	0	0	0
4	0	0	0	0.0260	0.0894	0.1788	0.2682	0.3301	0.3466	0.3151	0.2476	0.1650	0.0894	0.0358	0.0081	0	0	0	0
5	0	0	0	0	0.0081	0.0358	0.0894	0.1650	0.2476	0.3151	0.3466	0.3301	0.2682	0.1788	0.0894	0.0260	0	0	0
6	0	0	0	0	0	0.0022	0.0119	0.0367	0.0825	0.1500	0.2311	0.3081	0.3576	0.3576	0.2980	0.1907	0.0737	0	0
7	0	0	0	0	0	0	0.0005	0.0031	0.0118	0.0322	0.0707	0.1320	0.2146	0.3065	0.3831	0.4087	0.3474	0.1895	0
8	0	0	0	0	0	0	0	0.0001	0.0006	0.0027	0.0088	0.0236	0.0536	0.1073	0.1916	0.3065	0.4342	0.5211	0.4500
9	0	0	0	0	0	0	0	0	0+	0.0001	0.0003	0.0013	0.0043	0.0119	0.0298	0.0681	0.1447	0.2895	0.5500

N = 20, n = 10

N = 20, n = 10
Successes in the population (K) → x ↓	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
0	0.5000	0.2368	0.1053	0.0433	0.0163	0.0054	0.0015	0.0004	0.0001	0+	0	0	0	0	0	0	0	0	0
1	0.5000	0.5263	0.3947	0.2477	0.1354	0.0650	0.0271	0.0095	0.0027	0.0005	0.0001	0	0	0	0	0	0	0	0
2	0	0.2368	0.3947	0.4180	0.3483	0.2438	0.1463	0.0750	0.0322	0.0110	0.0027	0.0004	0	0	0	0	0	0	0
3	0	0	0.1053	0.2477	0.3483	0.3715	0.3251	0.2401	0.1500	0.0779	0.0322	0.0095	0.0015	0	0	0	0	0	0
4	0	0	0	0.0433	0.1354	0.2438	0.3251	0.3501	0.3151	0.2387	0.1500	0.0750	0.0271	0.0054	0	0	0	0	0
5	0	0	0	0	0.0163	0.0650	0.1463	0.2401	0.3151	0.3437	0.3151	0.2401	0.1463	0.0650	0.0163	0	0	0	0
6	0	0	0	0	0	0.0054	0.0271	0.0750	0.1500	0.2387	0.3151	0.3501	0.3251	0.2438	0.1354	0.0433	0	0	0
7	0	0	0	0	0	0	0.0015	0.0095	0.0322	0.0779	0.1500	0.2401	0.3251	0.3715	0.3483	0.2477	0.1053	0	0
8	0	0	0	0	0	0	0	0.0004	0.0027	0.0110	0.0322	0.0750	0.1463	0.2438	0.3483	0.4180	0.3947	0.2368	0
9	0	0	0	0	0	0	0	0	0.0001	0.0005	0.0027	0.0095	0.0271	0.0650	0.1354	0.2477	0.3947	0.5263	0.5000
10	0	0	0	0	0	0	0	0	0	0+	0.0001	0.0004	0.0015	0.0054	0.0163	0.0433	0.1053	0.2368	0.5000

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

How to read this table

Identify N (population size), K (successes in the population) and n (sample size).
Find the block for N and n, then the row for x and the column for K.
Add cells to get probabilities of ranges, or use the calculator for cumulative probabilities.

Worked example

A tray of 20 vials contains 4 defective ones, and an inspector samples 5 at random. The chance the sample contains no defective vials is row x = 0, column K = 4 of the N = 20, n = 5 block: 0.2817. So a 5-vial inspection misses the problem about 28% of the time.

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(K, x) C(N − K, n − x) / C(N, n)

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

Frequently asked questions

Hypergeometric or binomial?

Use the hypergeometric when sampling without replacement from a small population. When the sample is under about 10% of the population, the binomial with p = K/N is a good approximation.

How does this relate to Fisher's exact test?

Fisher's exact test uses the hypergeometric distribution of one cell of a 2 × 2 table given its margins.

Hypergeometric Distribution Table: P(X = x)

Hypergeometric Table

N = 10, n = 2

N = 10, n = 3

N = 10, n = 4

N = 10, n = 5

N = 20, n = 2

N = 20, n = 3

N = 20, n = 4

N = 20, n = 5

N = 20, n = 6

N = 20, n = 7

N = 20, n = 8

N = 20, n = 9

N = 20, n = 10

How to read this table

Worked example

Calculator

Formula and how the values were computed

Frequently asked questions

Related calculators and tutorials