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How likely is it that a page has no typing errors, that a detector clicks exactly three times in a second, or that a quiet road sees no accidents this month? Each event is rare, but there are many chances for it to happen. Siméon Denis Poisson found the formula that counts such events. This post covers his life, the formula, a worked example with radioactive counts, common mistakes, and what you can learn from how he worked.
Who was Siméon Denis Poisson?
According to his MacTutor biography at the University of St Andrews, Poisson was born in Pithiviers, France, on 21 June 1781 and died near Paris on 25 April 1840. His family first apprenticed him to a surgeon uncle, but he lacked the coordination for surgery. He then came first in the entrance examination of the École Polytechnique, where Laplace and Lagrange became his teachers and lifelong supporters.
MacTutor says he published between 300 and 400 mathematical works, on topics from planetary motion to electricity, heat, sound and elasticity. His name is on Poisson's equation in potential theory and Poisson's ratio in elasticity, among others. Arago reported that he often said life is good for only two things: discovering mathematics and teaching it.
Where did the Poisson distribution come from?
MacTutor explains that the distribution first appears in Poisson's 1837 book, *Recherches sur la probabilité des jugements en matière criminelle et matière civile*, a study of the probability of court verdicts. It describes the chance that an event occurs a given number of times in an interval when the chance in any small piece is very small but the number of pieces is very large. The same book introduced the phrase "law of large numbers". MacTutor adds that the work found little favour at the time, except in Russia, where Chebyshev developed it.
For the related law, read Jacob Bernoulli and the law of large numbers.
What is the Poisson spot?
Poisson also left a mark on optics, by accident. The MacTutor biography of Augustin Fresnel describes the 1819 Paris Academy prize on diffraction. Poisson sat on the judging committee, which mostly favoured the particle model of light. Working through Fresnel's wave theory, he showed it predicted that the centre of the shadow of an opaque disc should be bright. Arago asked for the prediction to be tested, and the bright spot was seen, just as the theory said. For the wider story of light as a wave, see Thomas Young and light waves.
How do you use the Poisson distribution? A worked example
These are sample numbers for a classroom example. A detector near a weak radioactive source records on average λ = 3 clicks per second. Find the probability of 0 clicks, of exactly 3, and of at least 1.
Use e^(−3) ≈ 0.0498.
P(0) = 3⁰ × e^(−3) / 0! = 1 × 0.0498 / 1 ≈ 0.050
P(3) = 3³ × e^(−3) / 3! = 27 × 0.0498 / 6 ≈ 0.224
P(at least 1) = 1 − P(0) ≈ 1 − 0.050 = 0.950
Why does this work? Think of the second as 1,000 tiny slices, each with a chance p = 0.003 of a click, so the average is 1,000 × 0.003 = 3. The binomial formula gives P(3) = C(1000, 3) × 0.003³ × 0.997⁹⁹⁷ ≈ 0.2244, almost the same as the Poisson 0.2240. Poisson's formula is the limit of the binomial when n is large, p is small and np stays fixed. MacTutor notes that William Gosset studied this same Poisson limit to the binomial at Karl Pearson's laboratory. You can practise the binomial side in probability of repeated trials.
The Poisson distribution is not in the JEE Main syllabus (the 2024 probability unit covers probability of an event, the addition and multiplication theorems, Bayes' theorem and probability distributions of a random variate), so treat this as enrichment and check the current bulletin.
What mistakes do students make with the Poisson distribution?
Using it when events are not independent. If one event makes the next more likely, the formula does not apply.
Mismatching the interval. If λ is 3 per second, then over 2 seconds use λ = 6, not 3.
Forgetting 0! = 1. It matters every time you compute P(0).
Using it when p is not small. For 10 coin tosses, use the binomial directly.
What can a student learn from how Poisson worked?
MacTutor quotes the mathematician Libri's account that Poisson worked on one problem at a time, jotting down new ideas in a small wallet and choosing the next problem from it only when the current one was finished. That is a practical system: when a stray idea interrupts revision, write it on a list and return to your current topic.
If the step from binomial to Poisson feels like a leap, the Fermi Concept Library has a lesson for every concept in mathematics, each with its place in the map.
Recompute P(1), P(2) and P(4) for λ = 3 and check that your values from 0 to 4 add up to about 0.815.
