Paradox

LEIBNIZ’S PRINCIPLE OF SUFFICIENT REASON 245GEOMETRICAL PROBABILITYGames are usually organized into artificially discrete elements:the equal sides of a die, the uniformly shaped cards ina deck, etc. This encouraged the belief that probability alwaysreduces to questions about combinations and permutations.The theory of combinations was first presented in Leibniz’sArs Combinatoria.In 1777, the French naturalist Georges Louis Leclerc,Comte de Buffon, showed that combinations cannot be acomplete foundation for probability. His incompleteness argumentwas inspired by a popular game that involved continuousoutcomes. Gamblers throw a coin at random on a floor tiledwith congruent squares. They bet on whether the coin wouldland entirely within the boundaries of a single square tile.Buffon realized that the coin would land within the tile exactlyif the center of the coin lands within a smaller square, whoseside was equal to the side of the tile minus the diameter of thecoin. The probability of winning is simply the ratio of the areaof the small square to the area of the tile.This was the beginning of the study of “geometricprobability,” where probabilities are determined by comparingmeasurements, rather than by identifying and countingalternative, equally probable discrete events. Buffon went onto consider cases involving more complex shapes. In hisfamous Needle Problem, a needle is thrown at random on afloor marked with equidistant parallel lines. When the spacingof the lines equals the length of the needle, the probabilityof striking a line equals 2/π.This unexpected appearance of π as a measure of probabilityillustrates the interrelatedness of mathematics. Peoplewere impressed by how π, an irrational number, could