This note gives the solution of the following problem concerning geometric probabilities. What is the probability p(Bd ; 2) that the circumference determined by three points P, P 1 and P 2 chosen independently and uniformly at random in the interior of a d-dimensional unit ball Bd in Euclidean space Ed (d ≧ 2) is entirely contained in Bd ? From our result we conclude that p(Bd ; 2) →π /(3√3) as d →∞.