A Lewis Carroll Pillow Problem: Probability of an Obtuse Triangle

Abstract

On the 100th anniversary (1993) of Lewis Carroll's Pillow Problems, Eugene Seneta presented a selection of the problems the author, Charles Dodgson, claims to have solved while in bed. The selection omits the one problem in continuous probability: "Three points are taken at random on an infinite plane. Find the chance of their being the vertices of an obtuse-angled triangle." Charles Dodgson presents a solution that involves a clear error in conditioning. An alternative solution is suggested here. This solution seems rather natural and should be especially appealing to statisticians. The nature of the solution suggests a method for using transformation groups to give meaning to the phrase "at random" in somewhat general situations.