Order out of chaos

Each ball takes a wildly unpredictable path through the pegs. The pile they make is predictable, and it is always the same shape.

From the slides: Why should anyone care about chance?.

The Galton board

At every peg a ball bounces right with probability p and left otherwise. After R rows its bin is the number of rightward bounces.

Rows Chance of bouncing right

Balls landed

Average bin

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Most common bin

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bars: balls landed in each bin dots: the expected number, n·C(R,k) p^k(1−p)^R−k

Show the numbers

The same picture, in dice

Roll n fair dice and take their average. With one die every value from 1 to 6 is equally likely; with more dice the averages crowd around 3.5 in a bell shape that gets narrower as n grows.

Dice per roll

Rolls

Average of the averages

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exact 3.5

Spread (standard deviation)

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What to notice

One ball tells you nothing. A thousand balls always build the same hump — the binomial distribution, which we will study properly later in the course.
Move the bias away from 0.5: the hump slides but keeps its shape.
For dice, the spread of the average shrinks like 1/√n: four times as many dice, half the spread. That the hump becomes a bell whatever you average is the central limit theorem.