Poisson Distribution Table (with Examples)

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The Poisson distribution describes the probability of a given number of events occurring within a fixed interval of time or space, assuming those events happen independently and at a constant average rate. Rather than working through the formula each time, a Poisson distribution table lets you look up probabilities directly using two simple inputs.

This article walks you through how to read the table, provides three worked examples covering the most common probability questions, and points you toward calculators and software guides for when you need more flexibility.

If you want to build a stronger foundation before diving into the table, start with An Introduction to the Poisson Distribution on Statology.

The Two Inputs You Need

Every value in a Poisson distribution table is determined by exactly two parameters:

  • λ (lambda): The mean number of events expected in the interval. For example, if a call center receives an average of 3 calls per minute, then λ = 3.
  • x: The specific number of events you want to find a probability for. This is always a non-negative whole number (0, 1, 2, 3, …).

The table gives you P(X = x), the probability that exactly x events occur when the mean rate is λ.

The Poisson Distribution Table

The table below shows P(X = x) for selected values of λ (columns) and x (rows). Values are rounded to four decimal places.

x λ=0.5 λ=1.0 λ=2.0 λ=3.0 λ=4.0 λ=5.0 λ=6.0 λ=7.0 λ=8.0 λ=9.0 λ=10.0
0 0.6065 0.3679 0.1353 0.0498 0.0183 0.0067 0.0025 0.0009 0.0003 0.0001 0.0000
1 0.3033 0.3679 0.2707 0.1494 0.0733 0.0337 0.0149 0.0064 0.0027 0.0011 0.0005
2 0.0758 0.1839 0.2707 0.2240 0.1465 0.0842 0.0446 0.0223 0.0107 0.0050 0.0023
3 0.0126 0.0613 0.1804 0.2240 0.1954 0.1404 0.0892 0.0521 0.0286 0.0150 0.0076
4 0.0016 0.0153 0.0902 0.1680 0.1954 0.1755 0.1339 0.0912 0.0573 0.0337 0.0189
5 0.0002 0.0031 0.0361 0.1008 0.1563 0.1755 0.1606 0.1277 0.0916 0.0607 0.0378
6 0.0000 0.0005 0.0120 0.0504 0.1042 0.1462 0.1606 0.1490 0.1221 0.0911 0.0631
7 0.0000 0.0001 0.0034 0.0216 0.0595 0.1044 0.1377 0.1490 0.1396 0.1171 0.0901
8 0.0000 0.0000 0.0009 0.0081 0.0298 0.0653 0.1033 0.1304 0.1396 0.1318 0.1126
9 0.0000 0.0000 0.0002 0.0027 0.0132 0.0363 0.0688 0.1014 0.1241 0.1318 0.1251
10 0.0000 0.0000 0.0000 0.0008 0.0053 0.0181 0.0413 0.0710 0.0993 0.1186 0.1251
11 0.0000 0.0000 0.0000 0.0002 0.0019 0.0082 0.0225 0.0452 0.0722 0.0970 0.1137
12 0.0000 0.0000 0.0000 0.0001 0.0006 0.0034 0.0113 0.0263 0.0481 0.0728 0.0948
13 0.0000 0.0000 0.0000 0.0000 0.0002 0.0013 0.0052 0.0142 0.0296 0.0504 0.0729
14 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0022 0.0071 0.0169 0.0324 0.0521
15 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0009 0.0033 0.0090 0.0194 0.0347

How to Read the Table

Reading the table is a three-step process:

  1. Find the column that matches your value of λ.
  2. Find the row that matches your value of x.
  3. The number at their intersection is P(X = x).

That single lookup gives you the probability of observing exactly x events.

Worked Examples

Example 1: Exact Probability — P(X = x)

Scenario: A hospital emergency department receives an average of 4 patients per hour. What’s the probability that exactly 6 patients arrive in a given hour?

Here, λ = 4 and x = 6.

Locate the column for λ = 4 and the row for x = 6. The table shows:

P(X = 6) = 0.1042

There’s roughly a 10.42% chance that exactly 6 patients arrive in that hour.

Example 2: Cumulative Probability — P(X < x)

Scenario: Using the same hospital example (λ = 4), what’s the probability that fewer than 3 patients arrive in a given hour?

P(X < 3) means adding up the probabilities for x = 0, 1, and 2. You can’t just read a single cell here — you need to sum each individual value from the λ = 4 column:

  • P(X = 0) = 0.0183
  • P(X = 1) = 0.0733
  • P(X = 2) = 0.1465
 
P(X < 3) = 0.0183 + 0.0733 + 0.1465 = 0.2381

There’s roughly a 23.81% chance that fewer than 3 patients arrive.

Example 3: Cumulative Probability — P(X ≥ x)

Scenario: Again with λ = 4, what’s the probability that at least 3 patients arrive?

“At least 3” means x = 3, 4, 5, and so on indefinitely, which makes direct addition impractical. This is where the complement rule comes in handy: instead of summing an infinite series of values, you just subtract what you’ve already calculated.

 
P(X ≥ 3) = 1 − P(X < 3)

From Example 2, P(X < 3) = 0.2381, so:

 
P(X ≥ 3) = 1 − 0.2381 = 0.7619

There’s roughly a 76.19% chance that 3 or more patients arrive.

To see how these kinds of probabilities play out in everyday situations, visit 5 Real-Life Examples of the Poisson Distribution.

When the Table Isn’t Enough

The table above covers a limited range of λ and x values. For problems outside this range, or when you want to skip the arithmetic entirely, these resources will help:

Summary

The Poisson distribution table is a straightforward lookup tool for finding probabilities when you know the mean rate λ and a target number of events x. The three core question types are:

  • Exact probability: Read directly from the table at the intersection of your λ column and x row.
  • Less than: Sum the table values for all x values below your threshold.
  • At least / greater than: Use the complement rule — subtract the cumulative probability from 1.

For values outside the table or more advanced modeling work, the calculator and software guides linked above will take you further.

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