Total no. of relations on a set A of cardinality n is $2^{n^{2}}$
i) No. of reflexive relations = $2^{n^{2} - n}$
Probability of reflexive relations = $\large \frac{2^{n^{2} - n}}{2^{n^{2}}}$ = $\large \frac{2^{4^{2} - 4}}{2^{4^{2}}}$
= $\large \frac{1}{16}$
= $\LARGE \frac{1}{2^{4}}$
ii) No. of symmetric relations =$\large 2^{\frac{n^{2}+n}{2}}$
Probability of symmetric relations = $\LARGE \frac{2^{\frac{n^{2}+n}{2}}}{2^{n^{2}}}$ = $\LARGE \frac{2^{\frac{4^{2}+4}{2}}}{2^{4^{2}}}$
=$\LARGE \frac{1}{64}$
= $\LARGE \frac{1}{2^{6}}$