Mathematics · Probability
Poisson Probability Calculator
Calculate the probability of exactly k events at an average rate λ.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- e^(−2)×2^3÷3!=0.1804470443154836.
- Report Exact probability=0.1804470443154836, Percent=18.04470443154836.
Understand Poisson probability
One idea, three depths
Choose how deeply to explain Poisson probability
Poisson probability: Calculate the probability of exactly k events at an average rate λ.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Poisson probability to answer this question: calculate the probability of exactly k events at an average rate λ? Enter Average rate λ and Event count k; the calculator shows Exact probability. For example: At λ=2, exactly 3 events has probability about 0.1804. The answer tells you Exact probability.
Age 15Explain it to a 15-year-oldConnect it to the formula
The Poisson distribution models independent events occurring at a constant average rate. The rule is P(X=k)=e^−λ λᵏ/k!. Its input values are Average rate λ, Event count k, and the main result is Exact probability. For example: At λ=2, exactly 3 events has probability about 0.1804.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated poisson probability relation over the valid mixed integer and real-number domain stated below. The implemented relation is P(X=k)=e^−λ λᵏ/k!, evaluated from Average rate λ, Event count k to produce Exact probability. The Poisson distribution models independent events occurring at a constant average rate. The event rate must match the interval being studied.
Inputs and valid domain
- Average rate λ must be a finite real number, at least 0.
- Event count k must be an integer, at least 0, at most 170.
Important boundary: The event rate must match the interval being studied.
The formula
P(X=k)=e^−λ λᵏ/k!
How the calculator works through it
It substitutes Average rate λ, Event count k into the formula and exposes every numerical step above. The main output is Exact probability, accompanied by Percent.
Read the result correctly
The Exact probability is the direct answer to “calculate the probability of exactly k events at an average rate λ.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
At λ=2, exactly 3 events has probability about 0.1804.
Where this model stops being reliable
The event rate must match the interval being studied.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Poisson probability works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Poisson probability uses P(X=k)=e^−λ λᵏ/k!. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Probability as a modelled proportion
Probability rules are needed to interpret what the Poisson probability result says about possible outcomes.
Review this foundation about 5 min
Optional enrichment
- Ordered arrangements
Counting ordered arrangements can extend Poisson probability to more detailed sample spaces and event models.
Review this foundation about 5 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Average rate λ, Event count k.
- Evaluate the principal relationship: P(X=k)=e^−λ λᵏ/k!.
- Return Exact probability and check the domain conditions described above.
Python
from math import *
def poisson_probability(a, n) -> float:
return ((exp((-a)) * pow(a, n)) / gamma((n + 1.0)))
assert abs(poisson_probability(2, 3) - 0.1804470443154836) < 1e-6 * max(1.0, abs(0.1804470443154836))
C
#include <assert.h>
#include <math.h>
double poisson_probability(double a, double n) {
return ((exp((-a)) * pow(a, n)) / tgamma((n + 1.0)));
}
int main(void) {
const double expected = 0.1804470443154836;
const double actual = poisson_probability(2, 3);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double poisson_probability(double a, double n) {
return ((std::exp((-a)) * std::pow(a, n)) / std::tgamma((n + 1.0)));
}
int main() {
constexpr double expected = 0.1804470443154836;
const double actual = poisson_probability(2, 3);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double poisson_probability(double a, double n)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern exp
extern pow
extern tgamma
global poisson_probability
section .text
poisson_probability:
push rbp
mov rbp, rsp
sub rsp, 80
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
pxor xmm0, xmm0
subsd xmm0, [rbp-8]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-48]
call exp wrt ..plt
movsd [rbp-40], xmm0
movsd xmm0, [rbp-8]
movsd xmm1, [rbp-16]
call pow wrt ..plt
movsd [rbp-56], xmm0
movsd xmm0, [rbp-40]
mulsd xmm0, [rbp-56]
movsd [rbp-32], xmm0
mov rax, 0x3ff0000000000000
movq xmm0, rax
movsd [rbp-80], xmm0
movsd xmm0, [rbp-16]
addsd xmm0, [rbp-80]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-72]
call tgamma wrt ..plt
movsd [rbp-64], xmm0
movsd xmm0, [rbp-32]
divsd xmm0, [rbp-64]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = poisson_probability(a, n)
result = ((exp((-a)) * (a ^ n)) / gamma((n + 1.0)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, n_] := ((Exp[(-a)] * (a ^ n)) / Gamma[(n + 1.0)]);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Introductory Statistics 2e
Read the free OpenStax statistics textbookCite this book
- APA 7
- Illowsky, B., & Dean, S. (2023). Introductory statistics 2e. OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
- MLA 9
- Illowsky, Barbara, and Susan Dean. Introductory Statistics 2e. OpenStax, 2023, https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
- Chicago author-date
- Illowsky, Barbara, and Susan Dean. 2023. Introductory Statistics 2e. Houston, TX: OpenStax. https://openstax.org/books/introductory-statistics-2e/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Poisson Probability Calculator. MW SysArc Tools. https://math.mwsysarc.com/probability/poisson-probability
MLA 9
MW SysArc. “Poisson Probability Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/probability/poisson-probability. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Poisson Probability Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/probability/poisson-probability.
Harvard
MW SysArc (2026) ‘Poisson Probability Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/probability/poisson-probability (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_poisson_probability_2026,
author = {{MW SysArc}},
title = {Poisson Probability Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/probability/poisson-probability},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Poisson Probability Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/probability/poisson-probability
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Poisson probability do?
Calculate the probability of exactly k events at an average rate λ.
How does the Poisson probability work?
The calculator applies P(X=k)=e^−λ λᵏ/k!. The Poisson distribution models independent events occurring at a constant average rate.
What can I learn from the Poisson probability?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .