The chart will show your running heads ratio against the 95% confidence funnel. Flip enough times and you will watch certainty emerge from chaos, which is the entire job description of statistics.
why the curve always wins
Each ball is a sum of 11 coin flips. Call a right turn 1 and a left turn 0. A ball that goes right six times lands in slot six. Most balls land near the middle, because there are many paths to a middling number of rights and almost none to all-left or all-right.
That is the central limit theorem. Add up enough independent random steps and the total is normally distributed, even though each step was a coin flip. The blue line is the exact normal curve those 11 flips predict. The green bars are what actually happened. The more you drop, the closer they match.
Why it shows up in finance: asset returns, measurement errors, and a thousand other things are sums of many small random shocks. That is why the bell curve appears everywhere, whether or not it has earned the right to.
expected: switch 66.7% · stay 33.3%
BANKROLL: 1,000
WIN PROBABILITY: 60%
PAYOUT: 1:1
ROUNDS: 20
KELLY OPTIMAL BET: 20% per round
your edge is known. your sizing is not.
the tell you cannot see
what GBM gets right: the daily return distribution, the drift, and the volatility level. those three parameters are everything a standard pricing model uses.
what GBM gets wrong: real returns cluster their variance. calm days follow calm days; violent days follow violent days. real returns also have fat tails: extreme moves happen more often than a normal distribution predicts.
why you cannot see it at this scale: 120 days is half a year. vol clustering needs a long run to become visible, and fat tails need a crash to show their face. at this resolution both series are wiggly lines going somewhere. the GBM is not a bad model. it is a sufficient model. those are different things.