We introduce \emph{\(p_n\)-random \(q_n\)-proportion Bulgarian solitaire} (\(0<p_n,q_n\le 1\)), played on \(n\) cards distributed in piles. In each pile, a number of cards equal to the proportion \(q_n\) of the pile size rounded upward to the nearest integer are candidates to be picked. Each candidate card is picked with probability \(p_n\), independently of other candidate cards. This generalizes Popov's random Bulgarian solitaire, in which there is a single candidate card in each pile. Popov showed that a triangular limit shape is obtained for a fixed \(p\) as \(n\) tends to infinity. Here we let both \(p_n\) and \(q_n\) vary with \(n\). We show that under the conditions \(q_n^2 p_n n/{\log n}\rightarrow \infty\) and \(p_n q_n \rightarrow 0\) as \(n\to\infty\), the \(p_n\)-random \(q_n\)-proportion Bulgarian solitaire has an exponential limit shape.