Geometric Distribution Table

Count trials including the first success

T counts trials up to and including the first success, so T starts at 1. Trials are independent and have a common probability 0 < p ≤ 1. Some software instead counts failures starting at 0; then failures = T − 1. These conventions require a shift, not identical k values.

$P(T=k)=p(1-p)^{k-1},\qquad P(T\le k)=1-(1-p)^k.$

The expected number of trials is 1/p and the variance is (1 − p)/p². The memoryless property holds under this model: failed attempts do not make success due on the next independent trial.

Exact probabilities: trials until first success

k	p = 0.05	p = 0.1	p = 0.2	p = 0.3	p = 0.5
1	0.0500	0.1000	0.2000	0.3000	0.5000
2	0.0475	0.0900	0.1600	0.2100	0.2500
3	0.0451	0.0810	0.1280	0.1470	0.1250
4	0.0429	0.0729	0.1024	0.1029	0.0625
5	0.0407	0.0656	0.0819	0.0720	0.0312
6	0.0387	0.0590	0.0655	0.0504	0.0156
7	0.0368	0.0531	0.0524	0.0353	0.0078
8	0.0349	0.0478	0.0419	0.0247	0.0039
9	0.0332	0.0430	0.0336	0.0173	0.0020
10	0.0315	0.0387	0.0268	0.0121	0.0010
11	0.0299	0.0349	0.0215	0.0085	0.0005
12	0.0284	0.0314	0.0172	0.0059	0.0002
13	0.0270	0.0282	0.0137	0.0042	0.0001
14	0.0257	0.0254	0.0110	0.0029	0.0001
15	0.0244	0.0229	0.0088	0.0020	0.0000
16	0.0232	0.0206	0.0070	0.0014	0.0000
17	0.0220	0.0185	0.0056	0.0010	0.0000
18	0.0209	0.0167	0.0045	0.0007	0.0000
19	0.0199	0.0150	0.0036	0.0005	0.0000
20	0.0189	0.0135	0.0029	0.0003	0.0000

Cumulative probabilities: trials until first success

k	p = 0.05	p = 0.1	p = 0.2	p = 0.3	p = 0.5
1	0.0500	0.1000	0.2000	0.3000	0.5000
2	0.0975	0.1900	0.3600	0.5100	0.7500
3	0.1426	0.2710	0.4880	0.6570	0.8750
4	0.1855	0.3439	0.5904	0.7599	0.9375
5	0.2262	0.4095	0.6723	0.8319	0.9688
6	0.2649	0.4686	0.7379	0.8824	0.9844
7	0.3017	0.5217	0.7903	0.9176	0.9922
8	0.3366	0.5695	0.8322	0.9424	0.9961
9	0.3698	0.6126	0.8658	0.9596	0.9980
10	0.4013	0.6513	0.8926	0.9718	0.9990
11	0.4312	0.6862	0.9141	0.9802	0.9995
12	0.4596	0.7176	0.9313	0.9862	0.9998
13	0.4867	0.7458	0.9450	0.9903	0.9999
14	0.5123	0.7712	0.9560	0.9932	0.9999
15	0.5367	0.7941	0.9648	0.9953	1.0000
16	0.5599	0.8147	0.9719	0.9967	1.0000
17	0.5819	0.8332	0.9775	0.9977	1.0000
18	0.6028	0.8499	0.9820	0.9984	1.0000
19	0.6226	0.8649	0.9856	0.9989	1.0000
20	0.6415	0.8784	0.9885	0.9992	1.0000

Worked example

With p = 0.2, exactly four trials has probability 0.8³ × 0.2 = 0.1024. Success within four trials has probability 1 − 0.8⁴ = 0.5904. More than four trials has probability 0.4096.

The expected wait is five trials, not a promise of success on trial five. Learning, fatigue or dependence between attempts can invalidate the constant-probability model.

Continue

Read discrete distributions. SciPy documents the trials-until-success convention.

Count trials including the first success

Exact probabilities: trials until first success

Cumulative probabilities: trials until first success

Worked example

Continue

Statistics formula sheet