Let $S$ be the set of $n\times n$ matrices $A$, for which one of $A$'s eigenvalues has multiplicity $m$. We know the corresponding eigenspace of $A$ has dimension $1\le d\le m$. What is the likely value of $d$, as $A$ varies over $S$?
For example, out of $2\times 2$ matrices $\left[\begin{array}{cc}p&q\\r&s\end{array}\right]$ with double eigenvalue $0$, there is a single matrix (the zero matrix) with a two-dimensional eigenspace, but a two-parameter set of matrices with a one-dimensional eigenspace. You just need $p+s=0$ and $ps-qr=0$.