Random Thue and Fermat equations

We consider Thue equations of the form $a{x}^k+b{y}^k=1$, and assuming the truth of the $abc$-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations $a{x}^k+b{y}^k+c{z}^k=0$ of degree at least six.