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When all ternary cubic forms over $\mathbb Z$ are ordered by the heights oftheir coefficients, we show that a positive proportion of them fail the Hasseprinciple, i.e., they have a zero over every completion of $\mathbb Q$ but nozero over $\mathbb Q$. We also show that a positive proportion of all ternarycubic forms over $\mathbb Z$ nontrivially satisfy the Hasse principle, i.e.,they possess a zero over every completion of $\mathbb Q$ and also possess azero over $\mathbb Q$. Analogous results are proven for other genus one models,namely, for equations of the form $z^2=f(x,y)$ where $f$ is a binary quarticform over $\mathbb Z$, and for intersections of pairs of quadrics in $\mathbbP^3$.

A positive proportion of plane cubics fail the Hasse principle