Consider a circle. Three points are chosen at random inside the circle. What is the probability that the circle which passes through these three points while lie totally inside the original circle?
In the Paul J. Nahin's book Inside Interesting Integrals(1st ed), he mentions this problem[page 25] and solves this to obtain answer of $\dfrac{2\pi}{15} (0.418879..)$. But he also mentions that this result doesn't agree with the result obtained in MATLAB i.e. $0.39992$ to $0.400972$. Hence he predicts the actual answer to be $\dfrac{2}{5}$.
$2/5$ is actually correct according to : Disturbing MATLAB Accuracy in Monte Carlo Simulation Although this question doesn't goes much into detail of how actual answer can be calculated.
I have attached the solution mentioned in Paul J. Nahin's book can be seen here(from google books sample) : https://i.sstatic.net/y0pZa.jpg
