Probabilities as Values of Modular Forms and Continued Fractions

Abstract. We consider certain probability problems which are naturally related to integer partitions. We show that the corresponding probabilities are values of classical modular forms. Thanks to this connection, we then show that certain ratios of probabilities are specializations of the Rogers-Ramanujan, and RamanujanSelbergGordon-Göllnitz continued fractions. One particular evaluation depends on a result from Ramanujan’s famous first letter to Hardy.