The probability thatn random points in a triangle are in convex position


We show thatn random points chosen independently and uniformly from a triangle are in convex position with probability

$$\frac{{2^n (3n - 3)!}}{{((n - 1)!)^3 (2n)!}}$$


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This research was supported by the Czech Republic Grant GAČR 201/94/2167 and by the Charles University grants Nos. 351 and 361.

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Valtr, P. The probability thatn random points in a triangle are in convex position. Combinatorica 16, 567–573 (1996).

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Mathematics Subject Classification (1991)

  • Primary: 60 D 05
  • Secondary: 52 A 22