Tables | Poisson Probabilities: r = 0.1 ~ 5.0

These tables give Poisson outcomes for selected cases where the rate r (conventionally called lambda, l) lies between 0.1 and 5.0. For other values of r, see the standard references (or our secret Forbidden Poisson table). Empty spaces in the table mean that the probability of that number of occurrences is effectively zero, and does not show up within the limits of four decimal places.

The likeliest number of occurrences (which is r, rounded down to the nearest whole number) is shown in bold. It is standard, for whole number r, that p(r) = p(r-1). Notice that in the first of the two tables, with r < 1, the likeliest occurrence per module of observation is 0. This is therefore the range that most fully meets the Poisson criterion of "rare event."

	r = 0.1	r = 0.2	r = 0.3	r = 0.4	r = 0.5	r = 0.6	r = 0.7	r = 0.8	r = 0.9
p(0)	0.9048	0.8187	0.7408	0.6703	0.6065	0.5488	0.4966	0.4493	0.4066
p(1)	0.0905	0.1637	0.2222	0.2681	0.3033	0.3293	0.3476	0.3595	0.3659
p(2)	0.0045	0.0164	0.0333	0.0536	0.0758	0.0988	0.1217	0.1438	0.1647
p(3)	0.0002	0.0011	0.0033	0.0072	0.0126	0.0198	0.0284	0.0383	0.0494
p(4)		0.0001	0.0003	0.0007	0.0016	0.0030	0.0050	0.0077	0.0111
p(5)				0.0001	0.0002	0.0004	0.0007	0.0012	0.0020
p(6)							0.0001	0.0002	0.0003

Note that values higher than r = 5 may be theoretically dubious: 5.0 is the highest value of r for which the chance of getting zero occurrences per unit of observation is even remotely likely: p ~ 0.01. If zero occurrences are unlikely to occur, then the event cannot validly be called "rare."

	r = 1.0	r = 1.5	r = 2.0	r = 2.5	r = 3.0	r = 3.5	r = 4.0	r = 4.5	r = 5.0
p(0)	0.3679	0.2231	0.1353	0.0821	0.0498	0.0302	0.0183	0.0111	0.0067
p(1)	0.3679	0.3347	0.2707	0.2052	0.1494	0.1057	0.0733	0.0500	0.0337
p(2)	0.1839	0.2510	0.2707	0.2565	0.2240	0.1850	0.1465	0.1125	0.0842
p(3)	0.0613	0.1255	0.1804	0.2138	0.2240	0.2158	0.1954	0.1687	0.1404
p(4)	0.0153	0.0471	0.0902	0.1336	0.1680	0.1888	0.1954	0.1898	0.1755
p(5)	0.0031	0.0141	0.0361	0.0668	0.1008	0.1322	0.1563	0.1708	0.1755
p(6)	0.0005	0.0035	0.0120	0.0278	0.0504	0.0771	0.1042	0.1281	0.1462
p(7)	0.0001	0.0008	0.0034	0.0099	0.0216	0.0385	0.0595	0.0824	0.1044
p(8)		0.0001	0.0009	0.0031	0.0081	0.0169	0.0298	0.0463	0.0653
p(9)			0.0002	0.0009	0.0027	0.0066	0.0132	0.0232	0.0363
p(10)				0.0002	0.0008	0.0023	0.0053	0.0104	0.0181
p(11)					0.0002	0.0007	0.0019	0.0043	0.0082
p(12)					0.0001	0.0002	0.0006	0.0016	0.0034
p(13)						0.0001	0.0002	0.0006	0.0013
p(14)							0.0001	0.0002	0.0005
p(15)								0.0001	0.0002

Notice the skew quality visible in both tables: the range of variation upward from r is greater than the variation downward from r. The possible values of r are constrained by having zero on the downward side.

Conventional Poisson tables do not go beyond r = 20, this being the point at which the Poisson Distribution becames nearly indistinguishable from the Normal Distribution. For practical purposes, then, the distinctive qualities of the Poisson are more or less lost at that point. It has moved too far out from its initial closeness to the "zero" barrier, the point below which further variation cannot occur. There is too much room on both sides of the most likely occurrence number, and the resulting curve becomes more and more symmetrical.