Galton Board

The idea

Every peg a ball meets is one independent coin flip: left or right, and nothing about the previous bounces changes the odds. A board with n rows therefore asks each ball the same question n times, and where the ball lands is decided by nothing but how many of those flips came out right. The bin index is a count of successes, so the pile of balls in the bins is a picture of the binomial distribution B(n, p) being sampled.

The heights are not approximately binomial; they are exactly it. The chance of landing in bin k is the number of distinct left-right paths that reach it, C(n, k), times the probability of any one such path. The bell shape is a fact about counting paths before it is a fact about randomness — there is exactly one route to each outer bin and hundreds to the middle.

The smooth curve laid over the bars is the normal distribution with mean n·p and variance n·p·(1 − p). That the bars climb toward it as the board grows taller is the de Moivre–Laplace theorem, the oldest special case of the Central Limit Theorem: a sum of many small independent contributions tends to the normal regardless of what each contribution looks like. Here each contribution is as simple as a random variable gets, one step left or one step right, and the sum is still a bell.

What to look for

• Watch one ball and then watch the pile. The single ball is unpredictable at every peg; the pile arrives at the same shape every run. Nothing about the individual bounce got less random — the regularity is a property of the aggregate only.
• On the 'few' board, six rows give seven bins whose heights you can read straight off as the binomial coefficients 1, 6, 15, 20, 15, 6, 1. Count them against the bars.
• On the twelve-row 'classic' board, the middle bin has C(12, 6) = 924 paths into it while each edge bin has exactly one. That ratio is the bell.
• On the 'tall' board, the staircase of bars sits closer to the overlaid curve. The bars never stop being discrete; it is their shape that converges, not their nature.
• On the 'skew' board the pegs are biased, and the whole pile shifts off centre to sit over n·p and leans. Look at the tails: the normal approximation goes wrong there first, well before the peak looks wrong.
• The pile widens more slowly than the board grows. Doubling the rows does not double the spread.

Getting it right

• Bell-shaped does not mean normal. What is on this board is a binomial distribution, exactly, at every height. The normal is its limit and, at any finite number of rows, an approximation.
• The Central Limit Theorem is a statement about sums of independent random variables, not a claim that measurements in general are normal. Quantities built by multiplying, or with heavy tails, or with dependence between terms, do not have to land here at all.
• A ball that has bounced right five times running is no more likely to go left at the next peg. The pegs have no memory, and nothing is owed to the ball. The pile balances because the unlikely runs are rare, not because they are corrected.
• The spread grows like the square root of the number of rows: the standard deviation is √(n·p·(1 − p)). Relative to the width of the board the pile therefore gets narrower as the board grows — the Law of Large Numbers and the Central Limit Theorem showing up in the same picture.
• A physical Galton board is an idealisation. Real beads deflect off each other, pegs are not perfectly fair, and consecutive bounces can correlate. The model's claim is that each peg is one fair independent choice; the mathematics follows from that assumption, not from the hardware.

Turn the knobs

• scenario picks the board: 'few' (6 rows, coarse enough to read the coefficients), 'classic' (12, the symmetric textbook bell), 'tall' (20, visibly closer to continuous), 'skew' (biased pegs).
• rows sets the height directly, 3 to 24. Step it up and watch the width grow like its square root rather than in step with it.
• p is the chance of bouncing right, 0.1 to 0.9. Move it off 0.5 and the peak slides to n·p; the spread also narrows, because p(1 − p) is largest exactly at a half.
• scale resizes the board without changing any of the statistics.