ფაილების კატალოგი

What is the probability that three points chosen at random on the circumference of a circle lie in one and the same semicircle? Equivalently, what is the probability that a random triangle inscribed in a circle has an obtuse angle?

PROVE: Evenly distributed in the circumference of the points. Let number points is 2n on the circumference then number of obtuse angle triangles is 2n(1+2+3+...+(n-2))=n(n-1)(n-2); number all triangls is (2n)!/(2n-3)!3!=2n(2n-1)(n-1)/3; 3n(n-1)(n-2)/2n(2n-1)(n-1)=3n(n-2)/2n(2n-1)→3/4 when n→∞ . Let number points is 2n+1 on the circumference then number of obtuse angle triangles is (2n+1)(1+2+3+...+(n-1))=n(n-1)(2n+1)/2; number all triangls is (2n+1)!/(2n-2)!3!=n(2n-1)(2n+1)/3; 3n(n-1)(2n+1)/2n(2n-1)(2n+1)=3n(n-1)/2n(2n-1)→3/4 when n→∞ .

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