Galton Board

The Galton board — also called a bean machine or quincunx — is a triangle of pins with a row of buckets at the bottom. Balls drop from the top and bounce randomly left or right at every pin. Although each ball takes an unpredictable path, thousands of balls together always pile up into the same, beautiful bell curve.

It is one of the clearest ways to see how randomness produces pattern, and it quietly links three big ideas: probability, Pascal’s triangle, and the normal distribution. If you enjoy this, our probability spinners demo explores chance in a different way.

How to Use

Press Run to start dropping balls, and press it again to pause. As balls land, the bars of the histogram grow to show how many balls have collected in each bucket.

The main controls are:

• Rows: sets how many rows of pins there are (and therefore how many buckets). More rows make a smoother, wider curve.
• Speed: how quickly balls are released and fall.
• Bell curve: overlays the theoretical bell curve on the histogram, so you can see the bars settling onto it.
• Path counts: labels each pin with the number of different paths a ball could take to reach it — this is Pascal’s triangle.
• Reset: clears all the balls and starts the count again.

🎲 Why a Bell Curve Forms

At every pin a ball hits, there is a 50/50 chance it goes left or right. To end up in a bucket far out to one side, a ball has to bounce the same way many times in a row — which is unlikely. To land near the middle, it just needs a roughly even mix of lefts and rights — which happens far more often.

Because there are many more ways to reach the middle buckets than the edge ones, the middle bars grow tallest and the outer bars stay short. Drop enough balls and this always settles into the symmetric bell shape.

Try setting a small number of Rows and dropping just a few balls: the histogram looks lumpy and uneven. Now increase the rows and let hundreds of balls fall — the shape becomes steadily smoother and more predictable. This is a hands-on view of the law of large numbers.

🔺 Pascal’s Triangle Hidden in the Board

Turn on Path counts to reveal the number written on every pin. Each number tells you how many different routes a ball can take from the top to that pin.

• The top pin has 1 route.
• Each pin below is the sum of the two pins above it — because a ball can only arrive from the left-above or the right-above.

That rule — add the two numbers above — builds Pascal’s triangle. The board is Pascal’s triangle made physical. The bottom row of numbers (1, n, …, n, 1) tells you exactly how likely each bucket is: buckets with bigger numbers collect more balls. This is why the histogram and the triangle share the same shape.

📈 From Binomial to Normal

The path-count numbers are the binomial coefficients, and the exact distribution of the balls is the binomial distribution for n rows with a 50/50 chance at each step.

Switch on the Bell curve to overlay the smooth curve this distribution approaches. As you add more Rows, the jagged, blocky binomial bars round off into the smooth normal distribution — the famous bell curve that appears throughout mathematics, science, and statistics, from heights and test scores to measurement errors.

Watching the bars rise and settle onto the curve is a memorable first encounter with one of the most important ideas in probability: how a pile of simple, random left-or-right choices adds up to a single, predictable shape.