Glossary / Poisson process
In the summer of 1944 German V-1 flying bombs were falling on London. Some districts took several hits while others stood untouched. Londoners said more and more often that the bombs fell in clusters, so they must be aimed at something. Two years later the actuary R. D. Clarke tested this with a simple table. Before finding out what he saw, take a look at what real randomness looks like.
“In plain words” is hidden on all puzzles.
A grid of 144 squares, and drops are falling on it. Each drop lands in a completely random spot, none knows about the others, and no square is any better than its neighbour. Drop some and look at the pattern.
When people are asked to draw random dots, they spread them neatly, one per cell, like the second pattern. Real chance can't do that. If the drops know nothing about each other, nothing stops two or three from landing side by side while the neighbouring square stays dry. Clumps and empty patches appear by themselves, with no aiming at all. It's the even grid that would be suspicious.
“Rigorous” is hidden on all puzzles.
A Poisson process is a stream of events that happen independently of each other, at the same average rate λ, and never two at exactly the same instant. The rate can be per hour, per page or per square kilometre. Then the number of events in a piece of size 1 has the Poisson distribution, named after the French mathematician Siméon Denis Poisson:
P(k) = λᵏ · e−λ / k!
An example. Say a book has on average 2 typos per page, λ = 2:
A page with no typos at all is almost as common as a page with four or more. Both are the ordinary play of chance, not a "careful" or "careless" day for the proofreader.
Count the squares: how many are empty, how many hold one drop, two, three… And compare with what the Poisson formula promises with λ = number of drops / 144. Now London. Clarke took 144 km² of south London, divided it into 576 squares of ¼ km² each and counted 537 hits in them, on average 537 / 576 ≈ 0.93 per square.
actual squaresby the Poisson formula
“Deeper” is hidden on all puzzles.
C(n, k) · (λ/n)ᵏ · (1 − λ/n)n−k → λᵏ · e−λ / k! as n → ∞
Split a square into n tiny cells, and let a drop land in each one independently with probability λ/n. The number of drops is then binomial. When there are very many cells, (1 − λ/n)ⁿ tends to e−λ and C(n, k) / nᵏ to 1/k!, and out comes the Poisson formula. That is why it describes anything made of many rare independent chances: thousands of callers, billions of nuclei, thousands of letters on a page. Move the slider, λ = 1.
E[K] = Σ k · λᵏe−λ/k! = λ · Σ λk−1e−λ/(k−1)! = λ
E[K(K − 1)] = λ² ⇒ Var K = λ² + λ − λ² = λ
For a Poisson process the mean and the variance are equal, and that makes a handy test. If the variance is clearly larger than the mean, events really do clump, as at rush hour or after an earthquake. If it is clearly smaller, they are spread more evenly than chance allows, as in the "chance as people imagine it" pattern. In London the mean is 0.932 hits per square and the variance 0.969, taking the largest square to have had 7 hits, which is exactly what remains of the 537 in total. The ratio is 1.04, so there are no clumps beyond chance. Clarke himself checked the fit with a chi-squared test: χ² = 1.17 with 4 degrees of freedom.
Drop some rain on the grid above, and the mean and variance of your squares will appear here.
P(K₁ + K₂ = k) = Σj e−λ₁λ₁ʲ/j! · e−λ₂λ₂k−j/(k−j)! = e−(λ₁+λ₂) (λ₁ + λ₂)ᵏ / k!
The last equality is the binomial theorem. If a call centre independently gets on average 0.5 calls a minute from Kyiv and 1 a minute from Lviv, the calls together are again a Poisson process with λ = 1.5. Check it over 10,000 minutes.
The middle value: line everyone up in order, and half are below it, half above.
A millionaire walks into a bar: the average income shoots up, the median barely moves.
The sum of all values divided by how many there are: what everyone would get if it were shared equally.
Almost everyone has more legs than average: the average is a little under two.
The average result you can expect if you repeat the same thing very many times.
A die's expected value is 3.5, yet no face shows 3.5.
A number from −1 to +1 showing how closely two quantities move together.
In summer both ice cream sales and drownings go up. Ice cream isn't to blame: the heat drives both.
A measure of spread: how far values typically stray from the mean. The standard deviation is its square root, in the same units.
Two towns with the same average temperature: one stays near +15 all year, the other swings from −20 to +50.
A bell-shaped distribution: most values sit near the mean, and the further away, the rarer.
Every ball on a Galton board falls at random, yet thousands of balls always build the same bell.
The more repetitions, the closer the share of an outcome gets to its true probability.
The share of heads creeps towards 50%, while the gap between heads and tails in coins usually grows.
The chance of an event given that something else is already known.
Almost all NBA players are tall, but almost all tall people are not NBA players.
Events are independent if knowing about one doesn't change the chances of the other.
After five heads in a row tails is no more likely: the coin remembers nothing.
How many times you multiply the base by itself to get a number: log₂ 8 = 3, because 2 · 2 · 2 = 8.
A sheet of paper folded in half 42 times would be thicker than the distance to the Moon.
A sample some people get into more easily than others, so it describes everyone badly.
In 1936 a magazine polled millions of Americans and still picked the wrong election winner.
How often chance alone would produce a gap at least as big as yours if there were really no difference.
When there is no effect at all, p below 0.05 turns up as often as p between 0.50 and 0.55.
A range around a measured number, built by a method that covers the true value in a chosen share of samples, such as 95%.
Of twenty honest 95% intervals, on average one misses the truth, and none of them knows whether it is the one.
The probability that a test notices a real effect of a given size. A power of 80% means the test misses one working effect in five.
A weak A/B test that does find an effect almost always exaggerates it: a real +10% shows up on the dashboard as +24%.
How far an average or a rate computed from a sample usually misses the truth: σ divided by the square root of the sample size.
Halving the error takes four times as much data.
Counting how many ways things can be ordered or chosen: permutations n!, combinations C(n, k).
A well-shuffled deck is almost certainly in an order nobody has ever seen.
The sum 1 + ½ + ⅓ + ¼ + …: it grows without limit, but incredibly slowly.
Identical books can be stacked so the top one overhangs the table edge as far as you like.
The rule for updating confidence when new evidence arrives: prior odds × strength of the evidence.
Two weak pieces of evidence together can be strong, while one striking piece means little if the claim was unlikely.
Growth where a quantity is multiplied by the same factor each time, so it doubles at equal intervals.
One grain on the first square of a chessboard, doubling on each next one: the last square holds as much rice as the world grows in over 300 years.
A path where each next step is taken at random: forward or back, left or right.
A drunk man will always find his way home, but a drunk bird may never return.