Copyright © 2009 Riad Masri and Ken Ono. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 Ramanujan- Selberg-
Gordon-Göllnitz continued fractions. One particular evaluation depends on a result
from Ramanujan's famous first letter to Hardy.