1,000 random bounces make a bell curve
Every ball is unpredictable. Together they always stack into the same shape.
Binomial(12, ½) → Normal1,000 balls fall through 12 rows of pegs. At every peg a ball bounces left or right with equal chance. At the bottom they land in 13 bins.
One ball, no pattern
Where a single ball lands is unpredictable. To reach bin k it must bounce right exactly k times out of 12, and there are C(12, k) different paths that do that, out of 2¹² = 4,096 possible paths.
| Bin | Paths | Chance |
|---|---|---|
| 0 or 12 (edges) | 1 | 0.02% |
| 3 or 9 | 220 | 5.4% |
| 6 (middle) | 924 | 22.6% |
The middle bin has 924 times as many paths as an edge bin, so it fills fastest.
Many balls, one shape
Stack enough balls and the bins trace the binomial distribution. With more rows it gets closer and closer to the smooth bell curve drawn on top, the normal distribution. This is the central limit theorem: add up many small random steps and the total is almost always bell-shaped, whatever the steps are.
The same bell curve shows up in heights, measurement errors and test scores for the same reason: each is the sum of many small independent effects.
