Long time no see. I saw a question looks something like this:
Get three points randomly on a circle and connect them to draw a triangle. What's the probability for this triangle to include the center of the circle?
The answer was said to be 0.25 but my calculation shows the answer is
Here is how I calculated. Please let me know if you observe an error. Thanks
Step1: Take a random point(A) on the circle. The probability P1=1.
Step2: Take the second point B on the circle and say the center is point O. The degree of AOB is X and 0<X<= pi. The probability of getting this point is P2= 2 * X/ (2pi) = X/pi
Step3: To get the point C making triangle ABC to include the center, we need to extend the line AO and BO respectively and intersect on the circle at A' & B'. Point C must fall on the curve A'B' and the probability of this is P3=X/(2pi).
Step4: P = P1*P2*P3 = X^2 / (2 pi^2). Do an integral