Question

Fun math question

Answer 1

In an empirical trial of 100,000 triangles, about 25% of them included the center of the circle.

The probability is 25%, based on the logic that the second point together with the first define a sector of the circle in which the third point must lie if the center is to be included in the triangle. That sector cannot be more than half the circle, and is just as likely to be more than 1/4 circle as it is to be less than 1/4 circle.

Answer 2

Let A be the first point chosen,
The probability that the other two points B  and C  lie to either side of  the diameter of A  is 1/2.

Therefore the probability we are looking for amounts to 1/2 x 1/2 = 1/4 probability