  We prove a Tverberg-type theorem using the probabilistic method. Given
$\varepsilon >0$, we find the smallest number of partitions of a set $X$ in
$R^d$ into $r$ parts needed in order to induce at least one Tverberg partition
on every subset of $X$ with at least $\varepsilon |X|$ elements. This
generalizes known results about Tverberg's theorem with tolerance.
