Posts tagged 'probability'
F.22 Functions
The standard normal cumulative distribution Phi(x) and error function erf(x) satisfy Phi(x) = 1/2 + (1/2) erf(x/sqrt(2)), where erf(x) is approximated to 1.5e-7 accuracy by Abramowitz and Stegun’s 5-term rational-exponential formula.
F.49 Vesica Piscis and Acute-Obtuse Probability
When segment AB of length c is the longest side of triangle ABC, vertex C lies inside the vesica piscis of area c^2 (4 pi - 3 sqrt(3))/6; the semicircle of diameter AB partitions each quadrant of area c^2 (4 pi - 3 sqrt(3))/24 into an obtuse region (O) = c^2 pi / 16 and an acute region (A) = c^2 (5 pi - 6 sqrt(3))/48, giving an exact obtuse-to-acute probability ratio (O)/(A) = 3 pi / (5 pi - 6 sqrt(3)) approx 1.773.