Statistical Tables/Poisson Probability Table

Distributions

Poisson Probability Table

Look up Poisson probabilities for common mean rates, as exact or cumulative probabilities.

Mean rate (λ) range

Table shows

P(X = k) for each mean rate λ

k \ λ	1.5	2	2.5	3	3.5	4	4.5	5
0	0.2231	0.1353	0.0821	0.0498	0.0302	0.0183	0.0111	0.0067
1	0.3347	0.2707	0.2052	0.1494	0.1057	0.0733	0.0500	0.0337
2	0.2510	0.2707	0.2565	0.2240	0.1850	0.1465	0.1125	0.0842
3	0.1255	0.1804	0.2138	0.2240	0.2158	0.1954	0.1687	0.1404
4	0.0471	0.0902	0.1336	0.1680	0.1888	0.1954	0.1898	0.1755
5	0.0141	0.0361	0.0668	0.1008	0.1322	0.1563	0.1708	0.1755
6	0.0035	0.0120	0.0278	0.0504	0.0771	0.1042	0.1281	0.1462
7	0.0008	0.0034	0.0099	0.0216	0.0385	0.0595	0.0824	0.1044
8	0.0001	0.0009	0.0031	0.0081	0.0169	0.0298	0.0463	0.0653
9	0.0000	0.0002	0.0009	0.0027	0.0066	0.0132	0.0232	0.0363
10	0.0000	0.0000	0.0002	0.0008	0.0023	0.0053	0.0104	0.0181
11	0.0000	0.0000	0.0000	0.0002	0.0007	0.0019	0.0043	0.0082
12	0.0000	0.0000	0.0000	0.0001	0.0002	0.0006	0.0016	0.0034
13	0.0000	0.0000	0.0000	0.0000	0.0001	0.0002	0.0006	0.0013
14	0.0000	0.0000	0.0000	0.0000	0.0000	0.0001	0.0002	0.0005
15	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0001	0.0002
16	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000

Each column is a different mean number of events λ per interval. Rows stop once even the largest λ in the group has less than 0.00005 probability left in its upper tail. Values are computed exactly from the Poisson formula and rounded to four decimals.

How to use this tool

Choose the range that contains your mean rate, λ.
Switch between the exact probability P(X = k) and the cumulative probability P(X <= k).
Find your count k down the left side and your λ along the top.
The value where they meet is your probability.

Formula Reference

Poisson probability

P(X = k) = e^(−λ) λ^k / k!

Cumulative probability

P(X <= k) = sum from i=0 to k of P(X = i)

Mean and variance

E(X) = Var(X) = λ

ArticleThe Poisson Distribution CalcPoisson Calculator ToolPoisson Distribution Explorer TableBinomial Probability Table GlossaryGlossary: Poisson Distribution

Frequently Asked Questions

My λ is not in the table. What do I do?+

Use the Poisson Calculator, which computes the exact probability for any positive mean rate, not just the values listed here.

How do I find P(X >= k)?+

Switch to the cumulative view, read P(X <= k − 1), and subtract it from 1. For example, with λ = 2.5, P(X >= 4) = 1 − P(X <= 3) = 1 − 0.7576 = 0.2424.

Why do the rows stop where they do?+

Each view shows counts up to the point where even the largest λ in the selected range has less than 0.00005 probability of any higher count, so every further row would round to zero.

When should I use a Poisson table instead of a binomial one?+

Use the Poisson distribution for counts of events in a fixed interval of time or space when events happen independently at a steady average rate. It also approximates the binomial well when n is large and p is small, with λ = np.

k \ λ

1.5

2.5

3.5

4.5

0.2231

0.1353

0.0821

0.0498

0.0302

0.0183

0.0111

0.0067

0.3347

0.2707

0.2052

0.1494

0.1057

0.0733

0.0500

0.0337

0.2510

0.2707

0.2565

0.2240

0.1850

0.1465

0.1125

0.0842

0.1255

0.1804

0.2138

0.2240

0.2158

0.1954

0.1687

0.1404

0.0471

0.0902

0.1336

0.1680

0.1888

0.1954

0.1898

0.1755

0.0141

0.0361

0.0668

0.1008

0.1322

0.1563

0.1708

0.1755

0.0035

0.0120

0.0278

0.0504

0.0771

0.1042

0.1281

0.1462

0.0008

0.0034

0.0099

0.0216

0.0385

0.0595

0.0824

0.1044

0.0001

0.0009

0.0031

0.0081

0.0169

0.0298

0.0463

0.0653

0.0000

0.0002

0.0009

0.0027

0.0066

0.0132

0.0232

0.0363

0.0000

0.0002

0.0008

0.0023

0.0053

0.0104

0.0181

0.0000

0.0002

0.0007

0.0019

0.0043

0.0082

0.0000

0.0001

0.0002

0.0006

0.0016

0.0034

0.0000

0.0001

0.0002

0.0006

0.0013

0.0000

0.0001

0.0002

0.0005

0.0000

0.0001

0.0002

0.0000

Frequently Asked Questions

My λ is not in the table. What do I do?+

Use the Poisson Calculator, which computes the exact probability for any positive mean rate, not just the values listed here.

How do I find P(X >= k)?+

Switch to the cumulative view, read P(X <= k − 1), and subtract it from 1. For example, with λ = 2.5, P(X >= 4) = 1 − P(X <= 3) = 1 − 0.7576 = 0.2424.

Why do the rows stop where they do?+

Each view shows counts up to the point where even the largest λ in the selected range has less than 0.00005 probability of any higher count, so every further row would round to zero.

When should I use a Poisson table instead of a binomial one?+