A Galton board after 600 balls
galton is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
With nothing chosen
3 paths through a Galton board
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- a path bounces once per row ×1
- and are centred under the funnel ×1
- and ends in a bin the board has ×1
- the exact bin shares add to one ×1
- the path ends over the bin its right-turns name ×1
- the row count is a whole number between 1 and 24 ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
A walk that always comes home, until it does not
Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.
ProbabilityA bell curve assembled out of coin flips
Drop six hundred balls through a board of pegs, each bouncing left or right at random, and they pile up in a shape that can be predicted precisely. Nothing coordinated them.
ProbabilityHalf the time is the rarest answer
In a fair game of many rounds, the fraction of the time one side is ahead is not usually near a half. It is usually near nought or one, and an even split is the single least likely outcome there is.
ProbabilityHow fast the bell arrives
The limit theorem says a standardised sum approaches the bell curve and says nothing about when. The rate is one over the square root of the number of terms, the constant in front is made of the third moment, and both are visible.
ProbabilityThe average settles and the wobble does not
Two theorems are usually met a page apart and sound as though one is a sharper version of the other. They are the same sums looked at through two different magnifying glasses: divide by the number of them and everything collapses to a point, divide by its square root and a shape appears.
ProbabilityThe shape that averaging leaves alone
Adding independent quantities blurs their distributions together, and rescaling restores the width. Almost every shape is changed by that operation. Exactly one is returned unaltered, and that is why sums of unrelated things keep arriving at it.
ProbabilityTwenty-three people
A room needs 253 people before someone probably shares a birthday with you. It needs 23 before two of them probably share one with each other. The gap between those numbers is the whole problem.