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If $E$ is an elliptic curve with a point of order two, then work of Klagsbrunand Lemke Oliver shows that the distribution of$\dim_{\mathbb{F}_2}\mathrm{Sel}_\phi(E^d/\mathbb{Q}) - \dim_{\mathbb{F}_2}\mathrm{Sel}_{\hat\phi}(E^{\prime d}/\mathbb{Q})$ within the quadratic twistfamily tends to the discrete normal distribution $\mathcal{N}(0,\frac{1}{2}\log \log X)$ as $X \rightarrow \infty$.  We consider the distribution of $\mathrm{dim}_{\mathbb{F}_2}\mathrm{Sel}_\phi(E^d/\mathbb{Q})$ within such a quadratic twist family when$\dim_{\mathbb{F}_2} \mathrm{Sel}_\phi(E^d/\mathbb{Q}) - \dim_{\mathbb{F}_2}\mathrm{Sel}_{\hat\phi}(E^{\prime d}/\mathbb{Q})$ has a fixed value $u$.Specifically, we show that for every $r$, the limiting probability that$\dim_{\mathbb{F}_2} \mathrm{Sel}_\phi(E^d/\mathbb{Q}) = r$ is given by anexplicit constant $\alpha_{r,u}$. The constants $\alpha_{r,u}$ are closelyrelated to the $u$-probabilities introduced in Cohen and Lenstra's work on thedistribution of class groups, and thus provide a connection between thedistribution of Selmer groups of elliptic curves and random abelian groups.  Our analysis of this problem has two steps. The first step uses algebraic andcombinatorial methods to directly relate the ranks of the Selmer groups inquestion to the dimensions of the kernels of random $\mathbb{F}_2$-matrices.This proves that the density of twists with a given $\phi$-Selmer rank $r$ isgiven by $\alpha_{r,u}$ for an unusual notion of density. The second step ofthe analysis utilizes techniques from analytic number theory to show that thisresult implies the correct asymptotics in terms of the natural notion ofdensity.

On the Joint Distribution Of $\mathrm{Sel}_\phi(E/\mathbb{Q})$ and $\mathrm{Sel}_{\hat\phi}(E^\prime/\mathbb{Q})$ in Quadratic Twist Families