The probability that a relation R from x ,   y to x ,   y is both symmetric and transitive, is…

Total number of elements in the Cartesian product will be $4$, so total possible relations will be $2^4=16$

The relations which will be symmetric and transitive both will be $\left\{\right\}, \left\{x,x\right\}, \left\{y,y\right\}, \left\{\left(x,x\right),\left(y,y\right)\right\}, \left\{\left(x,x\right),\left(x,y\right),\left(y,x\right)\left(y,y\right)\right\}$

 i.e. $5$ relations

Hence, required probability will be $\frac{5}{16}$