Poisson Distribution Explorer

Explore the Poisson distribution - see how rare events distribute and how Binomial converges to Poisson.

Worked example — Poisson Distribution Explorer

Complement	57.681%
Mean (μ)	3
Variance (λ)	3

These are the values the calculator shows when the page opens. Change any input above to recompute them.

Loading interactive simulation...

One number sets the centre and the spread at once 🖖

Poisson has a single parameter, so its mean and its variance are the same number. You cannot widen the distribution without also sliding it to the right — there is no width knob, and the tool does not offer one because the mathematics does not have one. Load the rare preset at λ = 0.5 and the standard deviation is √0.5 = 0.707 with the mode pinned at zero; load normal at λ = 10 and it is √10 = 3.162 and the shape has gone nearly symmetric, because the skewness is exactly 1/√λ. That rigidity doubles as a diagnostic: real count data whose variance runs well above its mean is overdispersed, and is telling you Poisson is the wrong model rather than a badly fitted one.

Many tiny chances become one clean curve 🖖

Picture a huge number of independent opportunities, each one very unlikely on its own - typos on a page, lightning strikes in a county, flaws on a production line. Flip a coin n times with success chance p; let n grow large while p shrinks so that n × p = λ stays fixed, and the Binomial distribution smoothly turns into Poisson. Toggle the overlay here, raise n, and watch the jagged Binomial bars settle onto the Poisson curve. Many tries, tiny odds each is exactly when Poisson is the shortcut.

Death by horse kick and the law of small numbers 🖖

In 1898 the statistician Ladislaus Bortkiewicz studied deaths from horse kicks across 10 Prussian army corps over 20 years. The yearly counts - mostly 0, occasionally 1 or 2, rarely more - matched a Poisson distribution with λ ≈ 0.61 almost perfectly. He named this pattern the law of small numbers. Poisson had derived the distribution sixty years earlier, in a book about the probability of judicial verdicts, and Bortkiewicz showed the same arithmetic governing freak accidents. Erlang would put it to work on telephone traffic eleven years later.

Problem solved in full

A process averaging 3 events per interval and getting exactly 3 5 steps
A process averages 3 events per interval. Work out how often you get at least 3, and then how often you get exactly 3. The two answers are further apart than most people expect, and the second one is why averages mislead.
1. The Poisson probability of k events is λᵏe^(−λ)/k!. Everything on this page comes out of that one expression and the single parameter λ.
  
  $P (X = k) = \frac{λ ^{k} e ^{- λ}}{k !}, λ = 3$
2. Compute the three smallest cases. e^(−3) = 0.049787 is the chance of a completely empty interval, and it is not negligible — one interval in twenty sees nothing at all, in a process averaging three.
  
  $P (0) = e^{- 3} = 0.049787, P (1) = 3 e^{- 3} = 0.149361, P (2) = 4.5 e^{- 3} = 0.224042$
3. At least 3 is the complement of at most 2, so add the three probabilities just computed and subtract from 1.
  
  $P (X \geq 3) = 1 - (0.049787 + 0.149361 + 0.224042) = 1 - 0.423190 = 0.576810$
4. Exactly 3 — the mean, the mode, the single most likely count — happens 22.4% of the time. Four times in five you get something else.
  
  $P (X = 3) = \frac{27}{6} e^{- 3} = 0.224042 = 22.4%$
5. There is no width control on this tool because the mathematics has no width parameter: the mean and the variance are the same number. Fix the rate and the spread is already decided, at σ = √3 = 1.732.
  
  $μ = σ^{2} = λ = 3 ⟹ σ = 3 = 1.732$
Answer

The tool prints 57.681% for at least 3, with mean and variance both 3. Set k = 3 and the exactly-3 case works out at 22.4%. That pairing is what to take away: "averages 3" does not mean "usually 3", it means the distribution is centred on 3 while spending more than three-quarters of its time somewhere else — and a twentieth of it empty. A ward averaging three admissions an hour sees one completely empty hour in every twenty, and a five-admission hour twice as often. Staffing for the average is staffing for an outcome that occurs one time in five.

$P (X \geq 3) = 57.681% but P (X = 3) = 22.4%; μ = σ^{2} = λ$

Learning path

From counts to curves

Leads to Normal distribution the first limit: hold np fixed, send n up and p down, and the two parameters collapse into one.

References (2)

The horse-kick data in the third block: L. von Bortkiewicz, Das Gesetz der kleinen Zahlen. B. G. Teubner, Leipzig, 1898.
The Poisson limit of the binomial, and why variance equals the mean: W. Feller, An Introduction to Probability Theory and Its Applications, Volume 1, 3rd ed., ch. VI. Wiley, 1968. ISBN 978-0-471-25708-0.

By the 42Math team · Updated 2026-06-04

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Typical λ=5 - At λ = 3 the distribution has two modes of equal height: three events and two events are both 0.2240, because an integer mean sits exactly between the values on either side of it. Nudge λ off 3 and one of them wins.
Rare events - At λ = 0.5 the most likely count is none at all, with probability 0.6065. Half an event per interval on average does not mean half an event ever happens — it means most intervals are empty.
Binomial→Poisson - Twenty binomial trials at p = 0.25 have np = 5, the same as this Poisson. Raise the trial count with that product held fixed and the jagged bars settle onto the smooth curve, which is the limit the Poisson was derived as.
Large λ→Normal - At λ = 10 the standard deviation is √10 = 3.162 and the skewness is 1/√10 = 0.316, so the shape has gone nearly symmetric. Even at its peak the single most likely count carries only 0.1251 — a wide distribution has no probable value.