Coin toss precession: why “same side” gets extra time
This is a kinematic visualization of the Diaconis–Holmes–Montgomery idealized model:
no air resistance, no bounce, no hand motion. The center of mass is not shown; only the coin’s
orientation is animated. The fixed angular momentum vector M is tilted by ψ
from the initial heads-normal N(0)=K. The normal N(t) precesses around M.
Rendered coin orientation
The blue arrow is the heads-side normal
N(t). The dark arrow is fixed angular momentum M. The vertical arrow is “up” K.Normal vector on the unit sphere
The path of
N(t) is a circle on the sphere. Green arc = same side up. Red arc = opposite side up.Cumulative same-side time fraction
The moving curve is the observed cumulative fraction of precession-time spent with the starting side up. The horizontal line is the predicted long-run value. Use Reset to restart the observation clock.
For a heads-up start with
0° ≤ ψ ≤ 90°, the predicted same-side fraction is
p(ψ)=1 for ψ≤45°, and otherwise
p(ψ)=1/2 + asin(cot²ψ)/π.
60.0°
1.25×
Current side according to N·Kheads up
Current N·K1.000
Predicted same-side fraction0.608
Observed cumulative fraction1.000
Elapsed precession cycles0.00
Special casebiased same-side