Galton Board — Help
A ball meets n pegs on the way down and goes right at each one with probability p . The bin it lands in is the number of rights — so the pile is a Binomial(n , p ) distribution being drawn by hand.
Rows of pegs is n , the number of trials. Chance of going right is p . Changing either clears the board, because balls that fell through a different board are not the same experiment.+1 and +10 animate; +100 and +1000 go straight into the bins, because a hundred balls at once is a smear rather than a demonstration. ▶ drops them in a steady stream.The purple curve is the exact Binomial(n , p ) probability for each bin, scaled to the number of balls dropped — so the pile growing into it is a fair comparison and not two y-axes that happen to agree.
The orange dashed curve is the normal approximation, mean np and SD √(npq ). Turn it on at n = 4 and at n = 20 to see when it starts being worth using.
Two boards
Ideal decides each bounce with a coin and then draws the path. The counts are exactly Binomial(n , p ), which is what makes it the board to point at when you mean “this is the binomial distribution”. The motion is still honest — projectile arcs, a squash at each peg — but nothing is being simulated.
Physical pours real beads into a real little world: a full field of pins, gravity, bounces off the pins, bounces off each other. Nothing decides where a bead should land — it falls, it gets knocked about, and it comes out of the bottom somewhere. Where it comes out is what the bars underneath count. The histogram this produces is not binomial , and that is the point of having both boards. The curve stays drawn over the bars, and a line underneath reports how far the spread has drifted from √(npq).
Watch the stream rather than the bars for a moment: the plume of beads widens as it falls, and it has stopped widening by the time it reaches the bottom. That shape is the distribution, being drawn in beads before any counting happens.
The knobs in Physical mode each break a different thing the binomial model assumes:
Bead size and peg size — what matters is the gap between two pegs. A bead too big for it can skip a row, ride a diagonal, or stop getting through at all, so it no longer makes exactly n independent decisions.Bounciness — a lively bead can travel sideways across several pegs, or back upward. Same violation, more obvious.Beads hit each other — this is the interesting one. It breaks independence between beads , the assumption students nod past fastest. Send one bead at a time and the histogram behaves; open the tap and watch it deform, on the same board with the same pins.
Slow motion quarters the speed without changing the physics, which is the way to actually see a bounce.
Both boards are reproducible: ?seed= gives the identical run every time, including in Physical mode — the world advances in fixed steps and the balls are released against its own clock rather than yours, so a slow machine gets the same pile, just drawn less often.
For the same distribution as a calculator rather than a pile, see the Binomial Distribution Calculator . For why the normal curve shows up at all, see Sampling Distributions and the Law of Large Numbers .
Instructors: there is a printable guide , and ?mode=present hides the student prompt.
Keyboard
1 / 2 / 3 / 4 — drop 1, 10, 100 or 1000 ballsp — play / stop the tap · r — clear the board · ? — this help
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