Instructor Guide: Galton Board
Companion notes for the StatLens demo — the binomial distribution, assembled one ball at a time
Launch in Presentation Mode
Overview ~8 min
A ball meets n rows of pegs and goes right at each with probability p. The bin it lands in
is the number of rights — so the pile that builds underneath is a Binomial(n,
p) distribution being drawn in front of the class. The exact probabilities are overlaid in purple,
scaled to the number of balls dropped, and the normal approximation can be switched on in orange.
Controls worth knowing. Rows of pegs is n; the p slider tilts
the board. +1 and +10 animate, +100 and
+1000 go straight into the bins (a hundred balls falling at once is a smear), and
▶ runs a steady stream. Changing n or p clears the board, because
the earlier balls fell through a different experiment.
A run that works
- Drop one ball and narrate it: left, right, right, left … "we are flipping a coin
at every peg." Ask what the bin number is counting before you say.
- Drop ten. Ask which bin will fill fastest and why. The usual answer is "the middle";
the useful follow-up is why — there are many paths to the middle and exactly one to each
edge.
- +1000. The pile snaps onto the purple curve. That curve was there from the start; the
balls are only finding it.
- Turn on the normal approximation at n = 20, p = 0.5. It lies almost
on top of the exact values.
- Now set p = 0.15 and n = 10. The pile is visibly skewed and the normal
curve is visibly wrong — it leans past zero and sits off-centre. This is the np ≥ 10 condition
arriving as something students can see rather than a rule to remember.
Discussion question. There is exactly one path to the leftmost bin and exactly one to the
rightmost, but thousands to the middle. Where do those path counts show up in the formula? Answer:
they are the binomial coefficient — the pegs are literally building Pascal's triangle, and a bin's
probability is "how many routes lead here" times "how likely each route is".
Honesty about the physics
This board is not a physics simulation, and that is deliberate. Each ball's bounces are
decided by a seeded coin first and then drawn. Real Galton boards do not produce exact binomials
— pegs are imperfect, balls spin, neighbours interfere — so a faithful physics engine would
produce a pile that only resembles the curve beside it. If a student asks whether a real one would do this:
nearly, and the gap is a nice question. It also means p can be set to anything, which no
physical board can do.
What is honest is the motion: every hop is a projectile arc (constant horizontal speed, accelerating
downward), the ball squashes where it meets a peg, and the whole thing is paced by the clock, so it runs at the
same speed on any machine.
The Physical board ~5 min
The Board: Ideal / Physical toggle swaps the decide-then-draw board for a real little
world: a full field of pins, gravity, beads bouncing off the pins and off each other, pouring out of
a hopper at the top. Nothing decides where a bead lands — it falls, gets knocked about, and comes out of
the bottom of the field somewhere. Where it comes out is what the bars underneath count.
Its histogram is not binomial, and that is why both boards exist. The curve stays drawn
over the bars and a line underneath reports how far the observed spread has drifted from √(npq) —
saying nothing until 60 beads have fallen, because a gap at n = 10 is noise.
Watch the stream, not the bars. Before anything is counted, the plume of beads widens as
it falls and then stops widening a few rows from the bottom. That shape is the distribution,
being drawn in beads. The bars underneath are only the bookkeeping.
The demonstration worth doing. Pour 100 beads with beads hit each other turned
OFF, and note the spread the line reports. Clear, turn it ON, and pour again. Same board, same pins,
same n — and the histogram widens, because the beads are no longer falling independently.
Independence is the binomial assumption students nod past fastest, and this is the only way I know to
show it failing.
The other knobs each break a different assumption. Peg size and bead size
set the clearance between two pins relative to the bead: a bead much smaller than the gap can fall
between two pins without being deflected at all, so it stops making n decisions and the histogram
narrows toward a spike. Bounciness lets a lively bead travel sideways across several pins, or
back upward — the same violation, more obvious. Slow motion quarters the speed without
changing the physics, which is the way to actually watch one bead meet one pin.
Every seed is a different board. The pins are not on a perfect lattice — they are
set a few per cent out of true, as a hand-built board's are, from the seed. That crookedness is worth a
spread anywhere between about 1.45 and 2.05 against a √(npq) of 1.73 at n = 12, and a lean
of up to a third of a bin. A real board is not the ideal board, and two real boards are not each other.
Both boards reproduce. ?seed= gives the identical run even in Physical mode:
the world advances in fixed steps and beads are released against its own clock rather than yours, so a
slow machine gets the same histogram, just drawn less often. Verified identical at 1× and
4× CPU throttle.
Where it fits
The probability chapter, beside the binomial distribution itself. It pairs with the
Binomial Calculator (the same distribution as numbers rather than a
pile) and with the Law of Large Numbers. For the normal approximation
thread, follow with Sampling Distributions.
Links worth keeping
?rows=20&p=0.5&balls=1000 — opens with a full, symmetric pile.?rows=10&p=0.15&balls=1000&normal=true — the skewed case, with the normal
approximation failing visibly.?seed=lecture — the identical sequence of balls every time.
StatLens · Galton Board · conceptual/galton-board/