Ball Triangle Picking

The determination of the probability for obtaining an Obtuse Triangle by picking 3 points at random in the unit Disk was generalized by Hall (1982) to the $n$-D Ball. Buchta (1986) subsequently gave closed form evaluations for Hall's integrals, with the first few solutions being

$$P_2 = {9\over 8}-{4\over\pi^2}\approx 0.72$$

$$P_3 = {\textstyle{37\over 70}}\approx0.53$$

$$P_4 \approx 0.39$$

$$P_5 \approx 0.29.$$

The case $P_2$ corresponds to the usual Disk case.

See also Cube Triangle Picking, Obtuse Triangle

References

Buchta, C. ``A Note on the Volume of a Random Polytope in a Tetrahedron.'' Ill. J. Math. 30, 653-659, 1986.

Hall, G. R. ``Acute Triangles in the $n$-Ball.'' J. Appl. Prob. 19, 712-715, 1982.