Why does the normal distribution look like a curve? #maths

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[00:00:00]Every coin flip is independent. 50/50, no memory.

[00:00:04]So, why does repeating it hundreds of times produce something so structured?

[00:00:09]It's combinatorics. The number of ways to get exactly K heads from N flips is N choose K.

[00:00:16]That's it.

[00:00:17]And N choose K is not flat. 10 flips, zero heads has one path. Five heads has 252.

[00:00:24]The middle term of the binomial expansion is always the largest.

[00:00:30]So, when you histogram the outcomes, the center dominates. Not because of some law of nature, but because there are geometrically more paths to the middle.

[00:00:38]Normalize it. Take N to infinity. That's your Gaussian. Because the middle has more paths.