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The science
Yerkes-Dodson inverted U
The Yerkes-Dodson law (1908) says performance rises with arousal up to an optimum and then falls. A convenient quantitative version is a Gaussian: success probability P(x) = s · exp(-(x - xopt)2 / (2w2)) with arousal x from 0 to 1, skill s (the best possible success) and tolerance width w. The optimum falls as the task gets more complex: xopt = 0.7 - 0.4 · complexity, so a simple task likes high arousal and a complex one calm focus (identical to the 2D page).
Noisy arousal
On every attempt the athlete's actual arousal is x = a + σZ with Z a standard normal. Averaging the Gaussian over this noise gives the exact success rate Pn(a) = s · w/√(w2+σ2) · exp(-(a - xopt)2 / (2(w2+σ2))): noise lowers and widens the peak.
What you see
The shooter takes free throws from 4.2 m; every attempt draws its own arousal, success is a Bernoulli trial with P(x), and the ball is thrown along a real parabola (release angle 52 degrees, speed solved for the chosen target, gravity 9.81 m/s2). Made shots aim at the rim centre; misses are aimed 0.36 m or more off centre: short when the arousal is below the optimum, long when above (a visual convention, not a biomechanical result). The thermometer follows the latest attempt, the gold ring is the optimum, the white ring the mean arousal and the green band the tolerance xopt +/- w. The ridges on the plinth are the steady (grey) and noisy (orange) curves; coloured dots are the last 60 attempts. The crowd jumps harder with higher arousal.
Independent checks
(1) A bulk run of N attempts (seeded) is compared with the exact convolution within four binomial standard errors, as in the 2D page. (2) The live free-throw tally is checked against the same exact value within four standard errors once there are 30 attempts. (3) A Simpson-rule quadrature of the Gaussian-weighted success curve is compared with the closed form to 1e-6.
Honest simplifications
The Gaussian shape and the linear optimum-complexity link are modelling conveniences; the original law is an empirical observation with large individual variation and no single universal curve. Arousal is a single number from 0 to 1 and success a single probability; real free-throw shooting also depends on fatigue, defenders and technique. The ball physics ignores air drag and spin effects, and the rim bounces are a simple restitution model. Educational model, not a coaching tool.