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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, anumber of cards equal to the proportion $q_n$ of the pile size rounded upwardto the nearest integer are candidates to be picked. Each candidate card ispicked with probability $p_n$, independently of other candidate cards. Thisgeneralizes Popov's random Bulgarian solitaire, in which there is a singlecandidate card in each pile. Popov showed that a triangular limit shape isobtained 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 $q_n^2 p_n n/{\logn}\rightarrow \infty$ and $p_n q_n \rightarrow 0$ as $n\to\infty$, the$p_n$-random $q_n$-proportion Bulgarian solitaire has an exponential limitshape.

An exponential limit shape of random $q$-proportion Bulgarian solitaire