We generalize Banach's matchbox problem: demands of random size are made on one of two containers, both initially with content t, where the container is selected at random in the successive steps. Let Z
be the content of the other container at the moment when the selected container is found to be insufficient. We obtain the asymptotic distribution of Z
as t → ∞ under quite general conditions. The case of exponentially distributed demands is considered in more detail.